753 lines
116 KiB
HTML
753 lines
116 KiB
HTML
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<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 15: Oscillations</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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code {
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color: #eb5757;
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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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.code-wrap {
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|||
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white-space: pre-wrap;
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word-break: break-all;
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}
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.code > code {
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background: none;
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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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blockquote {
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|||
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font-size: 1.25em;
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|||
|
|
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="98388194-e30d-4a83-bc6c-a892148a37d4" class="page sans"><header><h1 class="page-title">Unit 15: Oscillations</h1></header><div class="page-body"><h1 id="bd8c102c-b551-4d55-9e63-e3fa94e64ef3" class="">15.1 - Simple Harmonic Motion</h1><h2 id="f67c1b4a-5e03-4c11-80b5-8384d8fe00ca" class="">Important Stuff to Know</h2><ul id="4e1ae7db-4ab5-442d-9267-d9af4b23a63b" class="bulleted-list"><li style="list-style-type:disc">The frequency, <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>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span></span><span></span></span>, of periodic, or oscillatory, motion is the number of oscillations per second. In SI units, it is measured in hertz: <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><mn>1</mn><mtext> Hz</mtext><mo>=</mo><mn>1</mn><msup><mtext> s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">1 \textrm{ Hz} = 1 \textrm{ s}^{-1}</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">1</span><span class="mord text"><span class="mord textrm"> Hz</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:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord text"><span class="mord textrm"> s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;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 mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span><span></span></span>.</li></ul><ul id="0b127c55-2989-48f1-90a8-e198b57a6ae5" class="bulleted-list"><li style="list-style-type:disc">The period, <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>T</mi></mrow><annotation encoding="application/x-tex">T</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:0.13889em;">T</span></span></span></span></span><span></span></span>, is the time required for one complete oscillation, or cycle. It is related to the frequency by <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</
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h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7634050000000001em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:1.5em;"></span><span class="mord sizing reset-size6 size5"><span class="mord text"><span class="mord textrm">(angular frequency)</span></span></span></span></span></span></span></div></figure><figure id="3a2187f9-cdbe-4df1-ac65-1f7d54efd5b6" 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>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>m</mi><mi>k</mi></mfrac></msqrt><mspace width="1.5em"/><mstyle mathsize="0.9em"><mtext>(period)</mtext></mstyle></mrow><annotation encoding="application/x-tex">T = 2\pi\sqrt{\frac{m}{k}} \hspace{1.5em} \small{\textrm{(period)}}</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:0.13889em;">T</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:2.44em;vertical-align:-0.8953449999999998em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5446550000000001em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><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:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">m</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 style="top:-3.5046550000000005em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg width='400em' height='2.48em' viewBox='0 0 400000 2592' preserveAspectRatio='xMinYMin slice'><path d='M424,2478
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h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8953449999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:1.5em;"></span><span class="mord sizing reset-size6 size5"><span class="mord text"><span class="mord textrm">(period)</span></span></span></span></span></span></span></div></figure></li></ul><h1 id="20cfa54a-0cec-461a-879d-4a55ba8a6f4e" class="">15.2 - Energy in Simple Harmonic Motion</h1><p id="99df11fa-b3de-42b6-8580-0ed51286fbaf" class="">A particle in simple harmonic motion has, at any time, kinetic energy <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>K</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">K = \frac{1}{2} m v^2</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:0.07153em;">K</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:1.190108em;vertical-align:-0.345em;"></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:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></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:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;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></span></span></span></span></span></span></span><span></span></span> and potential energy <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>U</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">U = \frac{1}{2} kx^2</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;">U</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><sp
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h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7689599999999999em;"><span></span></span></span></span></span></span></span></span></span></div></figure><p id="3afb6c73-1a38-4ac4-8f99-bd5c600afced" 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>I</mi></mrow><annotation encoding="application/x-tex">I</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:0.07847em;">I</span></span></span></span></span><span></span></span> is the rotational inertia of the spinning object about the axis of rotation 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><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</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">κ</span></span></span></span></span><span></span></span> is the torsion constant of the wire. </p><h1 id="05063b2a-0c91-4721-aa18-9944ea5e3d6b" class="">15.4 - Pendulums, Circular Motion</h1><h2 id="464e68e7-bd55-42bb-9251-d0892622cd45" class="">Simple Pendulum</h2><p id="31af0233-ef29-4b7f-8341-f754130d1c5f" class="">A simple pendulum consists of a rod of negligible mass that pivots about its upper end, with a particle (the bob) attached at its lower end. If the rod swings through only small angles, its motion is approximately simple harmonic motion with a period given by:</p><figure id="cc0d55c4-4a7e-4898-aab7-7b0488d05161" 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>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>I</mi><mrow><mi>m</mi><mi>g</mi><mi>L</mi></mrow></mfrac></msqrt></mrow><annotation encoding="application/x-tex">T = 2\pi \sqrt{\frac{I}{mgL}}</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:0.13889em;">T</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:3.04em;vertical-align:-1.1661799999999998em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8738200000000003em;"><span class="svg-align" style="top:-5em;"><span class="pstrut" style="height:5em;"></span><span class="mord" style="padding-left:1em;"><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:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span cla
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606zM1001 80h400000v40H1017.7z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1661799999999998em;"><span></span></span></span></span></span></span></span></span></span></div></figure><p id="e9ae3fd3-14a4-49d3-ba3f-7b4b1279ae10" 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>I</mi></mrow><annotation encoding="application/x-tex">I</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:0.07847em;">I</span></span></span></span></span><span></span></span> is the particle's rotational inertia about the pivot, <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> is the particle's mass, 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><mi>L</mi></mrow><annotation encoding="application/x-tex">L</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></span></span></span><span></span></span> is the rod's length.</p><h2 id="ac79ea98-0115-4b12-a261-fc5c76b4b821" class="">Physical Pendulum</h2><p id="52cc9eb8-cf4a-4c57-9392-9a70c0a97374" class="">A physical pendulum has a more complicated distribution of mass. For small angles of swinging, its motion is simple harmonic motion with a period given by:</p><figure id="aa1960f7-abe5-4788-a60f-2912b5534d3a" 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>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>I</mi><mrow><mi>m</mi><mi>g</mi><mi>h</mi></mrow></mfrac></msqrt></mrow><annotation encoding="application/x-tex">T = 2\pi \sqrt{\frac{I}{mgh}}</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:0.13889em;">T</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:3.04em;vertical-align:-1.1661799999999998em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord sqrt"><span class="vl
|
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|
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c339.3,-1799.3,509.3,-2700,510,-2702 l0 -0
|
|||
|
|
c3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7
|
|||
|
|
s-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9
|
|||
|
|
c-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200
|
|||
|
|
c0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26
|
|||
|
|
s76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,
|
|||
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|
606zM1001 80h400000v40H1017.7z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1661799999999998em;"><span></span></span></span></span></span></span></span></span></span></div></figure><p id="138d8104-c307-4179-b77f-d5608868d0ed" 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>I</mi></mrow><annotation encoding="application/x-tex">I</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:0.07847em;">I</span></span></span></span></span><span></span></span> is the pendulum's rotational inertia about the pivot, <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> is the pendulum's mass, 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><mi>h</mi></mrow><annotation encoding="application/x-tex">h</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">h</span></span></span></span></span><span></span></span> is the distance between the pivot and the pendulum's center of mass.</p><h2 id="32d1d48a-5ed1-442b-b75c-78353ccb6bff" class="">Relating SHM and UCM</h2><p id="caaad9b6-bc00-4047-a552-7c68f75353cc" class="">Simple harmonic motion corresponds to the projection of uniform circular motion onto a diameter of the circle.</p><figure id="74343ba3-4d0c-44b9-9e52-f46275c835d9" class="image"><a href="Unit%2015%20Oscillations%2098388194e30d4a83bc6ca892148a37d4/Untitled.png"><img style="width:1776px" src="Unit%2015%20Oscillations%2098388194e30d4a83bc6ca892148a37d4/Untitled.png"/></a></figure><h1 id="c1fdf250-eabc-429a-af15-752fd19fbb79" class="">15.5 - Damped Simple Harmonic Motion</h1><p id="68c6ab5c-8953-4bfc-ab24-db1b58ccea38" class="">The mechanical energy <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>E</mi></mrow><annotation encoding="application/x-tex">E</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:0.05764em;">E</span></span></span></span></span><span></span></span> in a real oscillating system decre
|
|||
|
|
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="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</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:0.79733em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathnormal">b</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>, 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>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 the velocity of the oscillator 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><mi>b</mi></mrow><annotation encoding="application/x-tex">b</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">b</span></span></span></span></span><span></span></span> is a dampening constant, then the displacement of the oscillator is given by:</p><figure id="dcc5b92e-90a7-4ab5-a350-b3c3d84e77e8" 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>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mi>m</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>b</mi><mi>t</mi><mi mathvariant="normal">/</mi><mn>2</mn><mi>m</mi></mrow></msup><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><msup><mi>ω</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>t</mi><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t) = x_m e^{-bt/2m} \cos(\omega't + \phi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</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:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</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 mathnormal mtight">m</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="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;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 mtight">−</span><span class="mord mathnormal mtight">b</span><span class="mord mathnormal mtight">t</span><span class="mord mtight">/2</span><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;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
|
|||
|
|
c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514
|
|||
|
|
c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20
|
|||
|
|
s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121
|
|||
|
|
s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081
|
|||
|
|
l0 -0c4,-6.7,10,-10,18,-10 H400000
|
|||
|
|
v40H1014.6
|
|||
|
|
s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185
|
|||
|
|
c-2,6,-10,9,-24,9
|
|||
|
|
c-8,0,-12,-0.7,-12,-2z M1001 80
|
|||
|
|
h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7405709999999996em;"><span></span></span></span></span></span></span></span></span></span></div></figure><p id="ef1cbc29-47e9-47b0-b422-c52f06687192" class="">If the dampening constant is small (<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>b</mi><mo>≪</mo><msqrt><mrow><mi>k</mi><mi>m</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">b \ll \sqrt{km}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</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:1.04em;vertical-align:-0.10777999999999999em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.93222em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">km</span></span></span><span style="top:-2.89222em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
|
|||
|
|
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
|
|||
|
|
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
|
|||
|
|
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
|
|||
|
|
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
|
|||
|
|
c69,-144,104.5,-217.7,106.5,-221
|
|||
|
|
l0 -0
|
|||
|
|
c5.3,-9.3,12,-14,20,-14
|
|||
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H400000v40H845.2724
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s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
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M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.10777999999999999em;"><span></span></span></span></span></span></span></span></span></span><span></span></span>), then <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><msup><mi>ω</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>≈</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega' \approx \omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;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 mtight">′</span></span></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:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span></span><span></span></span>, 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>ω</mi></mrow><annotation encoding="application/x-tex">\omega</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:0.03588em;">ω</span></span></span></span></span><span></span></span> is the angular frequency of the undampened oscillator. For a small <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>b</mi></mrow><annotation encoding="application/x-tex">b</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">b</span></span></span></span></span><span></span></span>, the mechanical energy <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>E</mi></mrow><annotation encoding="application/x-tex">E</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:0.05764em;">E<
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