898 lines
125 KiB
HTML
898 lines
125 KiB
HTML
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<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 11: Rolling, Torque, and Angular Momentum</title><style>
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/* cspell:disable-file */
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/* webkit printing magic: print all background colors */
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html {
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-webkit-print-color-adjust: exact;
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}
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* {
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box-sizing: border-box;
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-webkit-print-color-adjust: exact;
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}
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html,
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body {
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margin: 0;
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padding: 0;
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}
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@media only screen {
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body {
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margin: 2em auto;
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max-width: 900px;
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color: rgb(55, 53, 47);
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}
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}
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body {
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line-height: 1.5;
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white-space: pre-wrap;
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}
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a,
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a.visited {
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color: inherit;
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text-decoration: underline;
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}
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.pdf-relative-link-path {
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font-size: 80%;
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color: #444;
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}
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h1,
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h2,
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h3 {
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letter-spacing: -0.01em;
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line-height: 1.2;
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font-weight: 600;
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margin-bottom: 0;
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}
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.page-title {
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font-size: 2.5rem;
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font-weight: 700;
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margin-top: 0;
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margin-bottom: 0.75em;
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}
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h1 {
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font-size: 1.875rem;
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margin-top: 1.875rem;
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}
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h2 {
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font-size: 1.5rem;
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margin-top: 1.5rem;
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}
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h3 {
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font-size: 1.25rem;
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margin-top: 1.25rem;
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}
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.source {
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border: 1px solid #ddd;
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border-radius: 3px;
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padding: 1.5em;
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word-break: break-all;
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}
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.callout {
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border-radius: 3px;
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padding: 1rem;
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}
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figure {
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margin: 1.25em 0;
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page-break-inside: avoid;
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}
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figcaption {
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opacity: 0.5;
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font-size: 85%;
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margin-top: 0.5em;
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}
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mark {
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background-color: transparent;
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}
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.indented {
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padding-left: 1.5em;
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}
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hr {
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background: transparent;
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display: block;
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width: 100%;
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height: 1px;
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visibility: visible;
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border: none;
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border-bottom: 1px solid rgba(55, 53, 47, 0.09);
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}
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img {
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max-width: 100%;
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}
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@media only print {
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img {
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max-height: 100vh;
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object-fit: contain;
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}
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}
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@page {
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margin: 1in;
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}
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.collection-content {
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font-size: 0.875rem;
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}
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.column-list {
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display: flex;
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justify-content: space-between;
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}
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.column {
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padding: 0 1em;
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}
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.column:first-child {
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padding-left: 0;
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}
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.column:last-child {
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padding-right: 0;
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}
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.table_of_contents-item {
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display: block;
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font-size: 0.875rem;
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line-height: 1.3;
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padding: 0.125rem;
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}
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.table_of_contents-indent-1 {
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margin-left: 1.5rem;
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}
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.table_of_contents-indent-2 {
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margin-left: 3rem;
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}
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.table_of_contents-indent-3 {
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margin-left: 4.5rem;
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}
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.table_of_contents-link {
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text-decoration: none;
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opacity: 0.7;
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border-bottom: 1px solid rgba(55, 53, 47, 0.18);
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}
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table,
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th,
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td {
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border: 1px solid rgba(55, 53, 47, 0.09);
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border-collapse: collapse;
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}
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table {
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border-left: none;
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border-right: none;
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}
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th,
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td {
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font-weight: normal;
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padding: 0.25em 0.5em;
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line-height: 1.5;
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min-height: 1.5em;
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text-align: left;
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}
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th {
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color: rgba(55, 53, 47, 0.6);
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}
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ol,
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ul {
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margin: 0;
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margin-block-start: 0.6em;
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margin-block-end: 0.6em;
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}
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li > ol:first-child,
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li > ul:first-child {
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margin-block-start: 0.6em;
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}
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ul > li {
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list-style: disc;
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}
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ul.to-do-list {
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text-indent: -1.7em;
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}
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ul.to-do-list > li {
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list-style: none;
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}
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.to-do-children-checked {
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text-decoration: line-through;
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opacity: 0.375;
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}
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ul.toggle > li {
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list-style: none;
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}
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ul {
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padding-inline-start: 1.7em;
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}
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ul > li {
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padding-left: 0.1em;
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}
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ol {
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padding-inline-start: 1.6em;
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}
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ol > li {
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padding-left: 0.2em;
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}
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.mono ol {
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padding-inline-start: 2em;
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}
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.mono ol > li {
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text-indent: -0.4em;
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}
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.toggle {
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padding-inline-start: 0em;
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list-style-type: none;
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}
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/* Indent toggle children */
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.toggle > li > details {
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padding-left: 1.7em;
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}
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.toggle > li > details > summary {
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margin-left: -1.1em;
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}
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.selected-value {
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display: inline-block;
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padding: 0 0.5em;
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background: rgba(206, 205, 202, 0.5);
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border-radius: 3px;
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margin-right: 0.5em;
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margin-top: 0.3em;
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margin-bottom: 0.3em;
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white-space: nowrap;
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}
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.collection-title {
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display: inline-block;
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margin-right: 1em;
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}
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.simple-table {
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margin-top: 1em;
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font-size: 0.875rem;
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empty-cells: show;
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}
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.simple-table td {
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height: 29px;
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min-width: 120px;
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}
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.simple-table th {
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height: 29px;
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min-width: 120px;
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}
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.simple-table-header-color {
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background: rgb(247, 246, 243);
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color: black;
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}
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.simple-table-header {
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font-weight: 500;
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}
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time {
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opacity: 0.5;
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}
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.icon {
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display: inline-block;
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max-width: 1.2em;
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max-height: 1.2em;
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text-decoration: none;
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vertical-align: text-bottom;
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margin-right: 0.5em;
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}
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img.icon {
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border-radius: 3px;
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}
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.user-icon {
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width: 1.5em;
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height: 1.5em;
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border-radius: 100%;
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margin-right: 0.5rem;
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}
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.user-icon-inner {
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font-size: 0.8em;
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}
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.text-icon {
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border: 1px solid #000;
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text-align: center;
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}
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.page-cover-image {
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display: block;
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object-fit: cover;
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width: 100%;
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max-height: 30vh;
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}
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.page-header-icon {
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font-size: 3rem;
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margin-bottom: 1rem;
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}
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.page-header-icon-with-cover {
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margin-top: -0.72em;
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margin-left: 0.07em;
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}
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.page-header-icon img {
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border-radius: 3px;
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}
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.link-to-page {
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margin: 1em 0;
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padding: 0;
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border: none;
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font-weight: 500;
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}
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p > .user {
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opacity: 0.5;
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}
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td > .user,
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td > time {
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white-space: nowrap;
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}
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|||
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input[type="checkbox"] {
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transform: scale(1.5);
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margin-right: 0.6em;
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vertical-align: middle;
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}
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p {
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margin-top: 0.5em;
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margin-bottom: 0.5em;
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}
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.image {
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border: none;
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margin: 1.5em 0;
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padding: 0;
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border-radius: 0;
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text-align: center;
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}
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.code,
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code {
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background: rgba(135, 131, 120, 0.15);
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border-radius: 3px;
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padding: 0.2em 0.4em;
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border-radius: 3px;
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font-size: 85%;
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tab-size: 2;
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}
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|||
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code {
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|||
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color: #eb5757;
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|||
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}
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|||
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.code {
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padding: 1.5em 1em;
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|||
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}
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|||
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.code-wrap {
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|||
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white-space: pre-wrap;
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|||
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word-break: break-all;
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|||
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}
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.code > code {
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|||
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background: none;
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|||
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padding: 0;
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|||
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font-size: 100%;
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|||
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color: inherit;
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|||
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}
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|||
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|||
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blockquote {
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|||
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font-size: 1.25em;
|
|||
|
|
margin: 1em 0;
|
|||
|
|
padding-left: 1em;
|
|||
|
|
border-left: 3px solid rgb(55, 53, 47);
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark {
|
|||
|
|
text-decoration: none;
|
|||
|
|
max-height: 8em;
|
|||
|
|
padding: 0;
|
|||
|
|
display: flex;
|
|||
|
|
width: 100%;
|
|||
|
|
align-items: stretch;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark-title {
|
|||
|
|
font-size: 0.85em;
|
|||
|
|
overflow: hidden;
|
|||
|
|
text-overflow: ellipsis;
|
|||
|
|
height: 1.75em;
|
|||
|
|
white-space: nowrap;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark-text {
|
|||
|
|
display: flex;
|
|||
|
|
flex-direction: column;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark-info {
|
|||
|
|
flex: 4 1 180px;
|
|||
|
|
padding: 12px 14px 14px;
|
|||
|
|
display: flex;
|
|||
|
|
flex-direction: column;
|
|||
|
|
justify-content: space-between;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark-image {
|
|||
|
|
width: 33%;
|
|||
|
|
flex: 1 1 180px;
|
|||
|
|
display: block;
|
|||
|
|
position: relative;
|
|||
|
|
object-fit: cover;
|
|||
|
|
border-radius: 1px;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark-description {
|
|||
|
|
color: rgba(55, 53, 47, 0.6);
|
|||
|
|
font-size: 0.75em;
|
|||
|
|
overflow: hidden;
|
|||
|
|
max-height: 4.5em;
|
|||
|
|
word-break: break-word;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.bookmark-href {
|
|||
|
|
font-size: 0.75em;
|
|||
|
|
margin-top: 0.25em;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.sans { font-family: ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol"; }
|
|||
|
|
.code { font-family: "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace; }
|
|||
|
|
.serif { font-family: Lyon-Text, Georgia, ui-serif, serif; }
|
|||
|
|
.mono { font-family: iawriter-mono, Nitti, Menlo, Courier, monospace; }
|
|||
|
|
.pdf .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK JP'; }
|
|||
|
|
.pdf:lang(zh-CN) .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK SC'; }
|
|||
|
|
.pdf:lang(zh-TW) .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK TC'; }
|
|||
|
|
.pdf:lang(ko-KR) .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK KR'; }
|
|||
|
|
.pdf .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK JP'; }
|
|||
|
|
.pdf:lang(zh-CN) .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK SC'; }
|
|||
|
|
.pdf:lang(zh-TW) .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK TC'; }
|
|||
|
|
.pdf:lang(ko-KR) .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK KR'; }
|
|||
|
|
.pdf .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK JP'; }
|
|||
|
|
.pdf:lang(zh-CN) .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK SC'; }
|
|||
|
|
.pdf:lang(zh-TW) .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK TC'; }
|
|||
|
|
.pdf:lang(ko-KR) .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK KR'; }
|
|||
|
|
.pdf .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK JP'; }
|
|||
|
|
.pdf:lang(zh-CN) .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK SC'; }
|
|||
|
|
.pdf:lang(zh-TW) .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK TC'; }
|
|||
|
|
.pdf:lang(ko-KR) .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK KR'; }
|
|||
|
|
.highlight-default {
|
|||
|
|
color: rgba(55, 53, 47, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-gray {
|
|||
|
|
color: rgba(120, 119, 116, 1);
|
|||
|
|
fill: rgba(120, 119, 116, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-brown {
|
|||
|
|
color: rgba(159, 107, 83, 1);
|
|||
|
|
fill: rgba(159, 107, 83, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-orange {
|
|||
|
|
color: rgba(217, 115, 13, 1);
|
|||
|
|
fill: rgba(217, 115, 13, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-yellow {
|
|||
|
|
color: rgba(203, 145, 47, 1);
|
|||
|
|
fill: rgba(203, 145, 47, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-teal {
|
|||
|
|
color: rgba(68, 131, 97, 1);
|
|||
|
|
fill: rgba(68, 131, 97, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-blue {
|
|||
|
|
color: rgba(51, 126, 169, 1);
|
|||
|
|
fill: rgba(51, 126, 169, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-purple {
|
|||
|
|
color: rgba(144, 101, 176, 1);
|
|||
|
|
fill: rgba(144, 101, 176, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-pink {
|
|||
|
|
color: rgba(193, 76, 138, 1);
|
|||
|
|
fill: rgba(193, 76, 138, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-red {
|
|||
|
|
color: rgba(212, 76, 71, 1);
|
|||
|
|
fill: rgba(212, 76, 71, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-gray_background {
|
|||
|
|
background: rgba(241, 241, 239, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-brown_background {
|
|||
|
|
background: rgba(244, 238, 238, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-orange_background {
|
|||
|
|
background: rgba(251, 236, 221, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-yellow_background {
|
|||
|
|
background: rgba(251, 243, 219, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-teal_background {
|
|||
|
|
background: rgba(237, 243, 236, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-blue_background {
|
|||
|
|
background: rgba(231, 243, 248, 1);
|
|||
|
|
}
|
|||
|
|
.highlight-purple_background {
|
|||
|
|
background: rgba(244, 240, 247, 0.8);
|
|||
|
|
}
|
|||
|
|
.highlight-pink_background {
|
|||
|
|
background: rgba(249, 238, 243, 0.8);
|
|||
|
|
}
|
|||
|
|
.highlight-red_background {
|
|||
|
|
background: rgba(253, 235, 236, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-default {
|
|||
|
|
color: inherit;
|
|||
|
|
fill: inherit;
|
|||
|
|
}
|
|||
|
|
.block-color-gray {
|
|||
|
|
color: rgba(120, 119, 116, 1);
|
|||
|
|
fill: rgba(120, 119, 116, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-brown {
|
|||
|
|
color: rgba(159, 107, 83, 1);
|
|||
|
|
fill: rgba(159, 107, 83, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-orange {
|
|||
|
|
color: rgba(217, 115, 13, 1);
|
|||
|
|
fill: rgba(217, 115, 13, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-yellow {
|
|||
|
|
color: rgba(203, 145, 47, 1);
|
|||
|
|
fill: rgba(203, 145, 47, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-teal {
|
|||
|
|
color: rgba(68, 131, 97, 1);
|
|||
|
|
fill: rgba(68, 131, 97, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-blue {
|
|||
|
|
color: rgba(51, 126, 169, 1);
|
|||
|
|
fill: rgba(51, 126, 169, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-purple {
|
|||
|
|
color: rgba(144, 101, 176, 1);
|
|||
|
|
fill: rgba(144, 101, 176, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-pink {
|
|||
|
|
color: rgba(193, 76, 138, 1);
|
|||
|
|
fill: rgba(193, 76, 138, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-red {
|
|||
|
|
color: rgba(212, 76, 71, 1);
|
|||
|
|
fill: rgba(212, 76, 71, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-gray_background {
|
|||
|
|
background: rgba(241, 241, 239, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-brown_background {
|
|||
|
|
background: rgba(244, 238, 238, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-orange_background {
|
|||
|
|
background: rgba(251, 236, 221, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-yellow_background {
|
|||
|
|
background: rgba(251, 243, 219, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-teal_background {
|
|||
|
|
background: rgba(237, 243, 236, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-blue_background {
|
|||
|
|
background: rgba(231, 243, 248, 1);
|
|||
|
|
}
|
|||
|
|
.block-color-purple_background {
|
|||
|
|
background: rgba(244, 240, 247, 0.8);
|
|||
|
|
}
|
|||
|
|
.block-color-pink_background {
|
|||
|
|
background: rgba(249, 238, 243, 0.8);
|
|||
|
|
}
|
|||
|
|
.block-color-red_background {
|
|||
|
|
background: rgba(253, 235, 236, 1);
|
|||
|
|
}
|
|||
|
|
.select-value-color-pink { background-color: rgba(245, 224, 233, 1); }
|
|||
|
|
.select-value-color-purple { background-color: rgba(232, 222, 238, 1); }
|
|||
|
|
.select-value-color-green { background-color: rgba(219, 237, 219, 1); }
|
|||
|
|
.select-value-color-gray { background-color: rgba(227, 226, 224, 1); }
|
|||
|
|
.select-value-color-opaquegray { background-color: rgba(255, 255, 255, 0.0375); }
|
|||
|
|
.select-value-color-orange { background-color: rgba(250, 222, 201, 1); }
|
|||
|
|
.select-value-color-brown { background-color: rgba(238, 224, 218, 1); }
|
|||
|
|
.select-value-color-red { background-color: rgba(255, 226, 221, 1); }
|
|||
|
|
.select-value-color-yellow { background-color: rgba(253, 236, 200, 1); }
|
|||
|
|
.select-value-color-blue { background-color: rgba(211, 229, 239, 1); }
|
|||
|
|
|
|||
|
|
.checkbox {
|
|||
|
|
display: inline-flex;
|
|||
|
|
vertical-align: text-bottom;
|
|||
|
|
width: 16;
|
|||
|
|
height: 16;
|
|||
|
|
background-size: 16px;
|
|||
|
|
margin-left: 2px;
|
|||
|
|
margin-right: 5px;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.checkbox-on {
|
|||
|
|
background-image: url("data:image/svg+xml;charset=UTF-8,%3Csvg%20width%3D%2216%22%20height%3D%2216%22%20viewBox%3D%220%200%2016%2016%22%20fill%3D%22none%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%3E%0A%3Crect%20width%3D%2216%22%20height%3D%2216%22%20fill%3D%22%2358A9D7%22%2F%3E%0A%3Cpath%20d%3D%22M6.71429%2012.2852L14%204.9995L12.7143%203.71436L6.71429%209.71378L3.28571%206.2831L2%207.57092L6.71429%2012.2852Z%22%20fill%3D%22white%22%2F%3E%0A%3C%2Fsvg%3E");
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
.checkbox-off {
|
|||
|
|
background-image: url("data:image/svg+xml;charset=UTF-8,%3Csvg%20width%3D%2216%22%20height%3D%2216%22%20viewBox%3D%220%200%2016%2016%22%20fill%3D%22none%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%3E%0A%3Crect%20x%3D%220.75%22%20y%3D%220.75%22%20width%3D%2214.5%22%20height%3D%2214.5%22%20fill%3D%22white%22%20stroke%3D%22%2336352F%22%20stroke-width%3D%221.5%22%2F%3E%0A%3C%2Fsvg%3E");
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
</style></head><body><article id="fd5a4dd4-aad8-4649-b2f3-144d69299a33" class="page sans"><header><h1 class="page-title">Unit 11: Rolling, Torque, and Angular Momentum</h1></header><div class="page-body"><h1 id="3bac228b-0d57-4c4d-946c-9af471b84756" class="">11.1 - Rolling as Translation and Rotation Combined</h1><p id="e63c9f8c-7575-4302-a8be-d805f3262683" class="">For a wheel of radius <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span><span></span></span> rolling smoothly,</p><figure id="e7ea4cf3-a6e3-4075-be72-5030b7865c13" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>v</mi><mrow><mi>c</mi><mi>o</mi><mi>m</mi></mrow></msub><mo>=</mo><mi>ω</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">v_{com} = \omega R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">co</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span></div></figure><p id="692016cf-1511-4295-8788-9cb81cdc1c52" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mrow><mi>c</mi><mi>o</mi><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">v_{com}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-siz
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">co</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> is related to the angular acceleration <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span></span><span></span></span> about the center with:</p><figure id="9b50c247-2b2c-45e8-9a71-5be4df9d463a" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mrow><mi>c</mi><mi>o</mi><mi>m</mi></mrow></msub><mo>=</mo><mi>α</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">a_{com} = \alpha R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">co</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span></div></figure><h2 id="c10f42bf-4b5c-4ddc-8b6f-36ba1d02a275" class="">Rolling Down a Ramp</h2><p id="a1838bf6-646c-4a73-8e61-85d52502823c" class="">If a wheel is rolling smoothly down a ramp of angle <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is a vector quantity defined relative to a fixed point (usually an origin); it is:</p><figure id="fd7fe4aa-9e9d-4cd2-8387-8f67dfe03b33" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>τ</mi><mo>⃗</mo></mover><mo>=</mo><mover accent="true"><mi>r</mi><mo>⃗</mo></mover><mo>×</mo><mover accent="true"><mi>F</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec \tau = \vec r \times \vec F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.714em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79733em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="a60dc0f0-a159-422f-a839-60bbcfd465c9" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>F</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is a force applied to a particle and <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>r</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.714em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is a position vector locating the particle relative to the fixed point.</p><h1 id="c5e78a87-657b-43f3-be5c-d902ea404893" class="">11.5 - Angular Momentum</h1><p id="2f4fa542-9609-4196-8294-be6e7f3a2947" class="">The angular momentum <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> of a particle with linear momentum <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>p</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084399999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span></span></span></span><span></span></span>, mass <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span></span><span></span></span>, and linear velocity <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.714em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is a vector quantity defined relative to a fixed point (usually an origin) as:</p><figure id="93966b15-f9e0-4d67-a71e-ddb512b40cd1" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>l</mi><mo>⃗</mo></mover><mo>=</mo><mover accent="true"><mi>r</mi><mo>⃗</mo></mover><mo>×</mo><mover accent="true"><mi>p</mi><mo>⃗</mo></mover><mo>=</mo><mi>m</mi><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>⃗</mo></mover><mo>×</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vec l = \vec r \times \vec p = m(\vec r \times \vec v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79733em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9084399999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></div></figure><p id="b763901c-6c25-4f45-aeaa-57a4d2b8a53e" class="">The magnitude of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is given by:</p><figure id="0df9a329-917a-4d99-9d3b-5952a2dbaa96" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>l</mi><mo>=</mo><mi>r</mi><mi>m</mi><mi>v</mi><mi>sin</mi><mo></mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">l = rmv \sin \phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ϕ</span></span></span></span></span></div></figure><p id="28bccf52-1e2e-4c14-8f58-370c5002891e" class="">The direction of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is given by the right-hand rule: Position your right hand so that the fingers are in the direction of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>r</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.714em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> .Then rotate them around the palm to be in the direction of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>p</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084399999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span></span></span></span><span></span></span>. Your outstretched thumb gives the direction of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span>.</p><h1 id="9a89ea61-b4aa-4c42-8fdf-d26124d2c985" class="">11.6 - Newton's Law in Angular Form</h1><p id="ec49eb55-45be-4390-94b4-aff19b712824" class="">Newton's Second Law can be written in angular form as:</p><figure id="44924bfe-f02e-4c35-a388-89297ff16f23" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>τ</mi><mo>⃗</mo></mover><mrow><mi>n</mi><mi>e</mi><mi>t</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>d</mi><mover accent="true"><mi>l</mi><mo>⃗</mo></mover></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\vec \tau_{net} = \frac{d \vec l}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.864em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.34044em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.65444em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><p id="594b4fe2-d451-4893-afd6-6232db9ce7dc" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>τ</mi><mo>⃗</mo></mover><mrow><mi>n</mi><mi>e</mi><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\vec \tau_{net}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.864em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> is the net torque acting on the particle and <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is the angular momentum of the particle.</p><h1 id="4c34a281-c311-41bb-8149-1723c6bf4c30" class="">11.7 - Angular Momentum of a Rigid Body</h1><p id="971291e8-7d15-4e4d-b9de-b2b7b498e0be" class="">The angular momentum, <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>L</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> of a system of particles is the vector sum of the angular momenta of the individual particles:</p><figure id="35ea4732-7b5a-4548-b924-419d08835ae1" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>L</mi><mo>⃗</mo></mover><mo>=</mo><msub><mover accent="true"><mi>l</mi><mo>⃗</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mover accent="true"><mi>l</mi><mo>⃗</mo></mover><mn>2</mn></msub><mo>+</mo><msub><mover accent="true"><mi>l</mi><mo>⃗</mo></mover><mn>3</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mover accent="true"><mi>l</mi><mo>⃗</mo></mover><mi>n</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mover accent="true"><mi>l</mi><mo>⃗</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\vec L = \vec l_1 + \vec l_2 + \vec l_3 + \dots + \vec l_n = \sum_{i = 1}^n \vec l_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1274399999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1274399999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1274399999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></div></figure><h2 id="a4f6106d-2c05-42a2-b013-dc18e9b41fae" class="">Effect of Torque on a System of Particles</h2><p id="acc6a6a5-ebbe-4786-89fa-725e374cf898" class="">The time rate of change of this angular momentum is equal to the net external torque on the system (the vector sum of the torques due to interactions of the particles of the system with particles external to the system):</p><figure id="869eed19-31fd-4e90-8342-b34468b9bab9" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>τ</mi><mo>⃗</mo></mover><mrow><mi>n</mi><mi>e</mi><mi>t</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>d</mi><mover accent="true"><mi>L</mi><mo>⃗</mo></mover></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\vec \tau_{net} = \frac{d\vec L}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.864em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><h2 id="3d1d924c-f93f-416a-a961-962a1d2fdc90" class="">Angular Momentum of a Rigid Body Rotating About a Fixed Axis</h2><p id="87a354ec-535d-4517-85fa-b7924b0b9ea0" class="">For a rigid body rotating about a fixed axis, the component of its angular momentum parallel to the rotation axis is:</p><figure id="6f18e8cc-ebd0-4df6-a329-fc7992f4d8cd" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo>=</mo><mi>I</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">L = I\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span></span></div></figure><h1 id="933358b1-66de-4be7-b449-641717791e8a" class="">11.8 - Conservation of Angular Momentum</h1><p id="6ecc0686-4d93-4c33-b10b-b3e05b1cbc9e" class="">The angular momentum <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>L</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> of a system remains constant if the net external torque acting on the system is zero. Therefore,</p><figure id="e6fa7ded-eed5-46a9-a25a-c8c12ce9de76" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>L</mi><mo>⃗</mo></mover><mi>i</mi></msub><mo>=</mo><msub><mover accent="true"><mi>L</mi><mo>⃗</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\vec L_i = \vec L_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1163299999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.252438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="042ea0e1-2892-40e7-8421-46d7b3e574db" class="">in an isolated system. This is the law of conservation of angular momentum.</p><h1 id="611a8a8e-f5d0-4a44-a2ce-0f0429426a6b" class="">11.9 - Precession of a Gyroscope</h1><p id="d8290918-8684-476f-86ad-570779e9b3a0" class="">A spinning gyroscope can precess about a vertical axis through its support at the rate:</p><figure id="5e882a98-8769-487f-89a2-005718436b26" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mfrac><mrow><mi>M</mi><mi>g</mi><mi>r</mi></mrow><mrow><mi>I</mi><mi>ω</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\Omega = \frac{Mgr}{I\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.0463299999999998em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><p id="18d8cc36-3248-43e6-8748-97ba22568c01" class="">Where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span></span><span></span></span> is the gyroscope's mass, <style>@import
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> of the gyroscope, and is caused by the downward force of gravity <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mover accent="true"><mi>g</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">M\vec g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084399999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span></span></span></span><span></span></span>. </p><p id="3cebccc0-c717-404a-9513-667a9cf47ec6" class="">
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</p></div></article></body></html>
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