778 lines
83 KiB
HTML
778 lines
83 KiB
HTML
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 6: Force and Motion - II</title><style>
|
||
/* cspell:disable-file */
|
||
/* webkit printing magic: print all background colors */
|
||
html {
|
||
-webkit-print-color-adjust: exact;
|
||
}
|
||
* {
|
||
box-sizing: border-box;
|
||
-webkit-print-color-adjust: exact;
|
||
}
|
||
|
||
html,
|
||
body {
|
||
margin: 0;
|
||
padding: 0;
|
||
}
|
||
@media only screen {
|
||
body {
|
||
margin: 2em auto;
|
||
max-width: 900px;
|
||
color: rgb(55, 53, 47);
|
||
}
|
||
}
|
||
|
||
body {
|
||
line-height: 1.5;
|
||
white-space: pre-wrap;
|
||
}
|
||
|
||
a,
|
||
a.visited {
|
||
color: inherit;
|
||
text-decoration: underline;
|
||
}
|
||
|
||
.pdf-relative-link-path {
|
||
font-size: 80%;
|
||
color: #444;
|
||
}
|
||
|
||
h1,
|
||
h2,
|
||
h3 {
|
||
letter-spacing: -0.01em;
|
||
line-height: 1.2;
|
||
font-weight: 600;
|
||
margin-bottom: 0;
|
||
}
|
||
|
||
.page-title {
|
||
font-size: 2.5rem;
|
||
font-weight: 700;
|
||
margin-top: 0;
|
||
margin-bottom: 0.75em;
|
||
}
|
||
|
||
h1 {
|
||
font-size: 1.875rem;
|
||
margin-top: 1.875rem;
|
||
}
|
||
|
||
h2 {
|
||
font-size: 1.5rem;
|
||
margin-top: 1.5rem;
|
||
}
|
||
|
||
h3 {
|
||
font-size: 1.25rem;
|
||
margin-top: 1.25rem;
|
||
}
|
||
|
||
.source {
|
||
border: 1px solid #ddd;
|
||
border-radius: 3px;
|
||
padding: 1.5em;
|
||
word-break: break-all;
|
||
}
|
||
|
||
.callout {
|
||
border-radius: 3px;
|
||
padding: 1rem;
|
||
}
|
||
|
||
figure {
|
||
margin: 1.25em 0;
|
||
page-break-inside: avoid;
|
||
}
|
||
|
||
figcaption {
|
||
opacity: 0.5;
|
||
font-size: 85%;
|
||
margin-top: 0.5em;
|
||
}
|
||
|
||
mark {
|
||
background-color: transparent;
|
||
}
|
||
|
||
.indented {
|
||
padding-left: 1.5em;
|
||
}
|
||
|
||
hr {
|
||
background: transparent;
|
||
display: block;
|
||
width: 100%;
|
||
height: 1px;
|
||
visibility: visible;
|
||
border: none;
|
||
border-bottom: 1px solid rgba(55, 53, 47, 0.09);
|
||
}
|
||
|
||
img {
|
||
max-width: 100%;
|
||
}
|
||
|
||
@media only print {
|
||
img {
|
||
max-height: 100vh;
|
||
object-fit: contain;
|
||
}
|
||
}
|
||
|
||
@page {
|
||
margin: 1in;
|
||
}
|
||
|
||
.collection-content {
|
||
font-size: 0.875rem;
|
||
}
|
||
|
||
.column-list {
|
||
display: flex;
|
||
justify-content: space-between;
|
||
}
|
||
|
||
.column {
|
||
padding: 0 1em;
|
||
}
|
||
|
||
.column:first-child {
|
||
padding-left: 0;
|
||
}
|
||
|
||
.column:last-child {
|
||
padding-right: 0;
|
||
}
|
||
|
||
.table_of_contents-item {
|
||
display: block;
|
||
font-size: 0.875rem;
|
||
line-height: 1.3;
|
||
padding: 0.125rem;
|
||
}
|
||
|
||
.table_of_contents-indent-1 {
|
||
margin-left: 1.5rem;
|
||
}
|
||
|
||
.table_of_contents-indent-2 {
|
||
margin-left: 3rem;
|
||
}
|
||
|
||
.table_of_contents-indent-3 {
|
||
margin-left: 4.5rem;
|
||
}
|
||
|
||
.table_of_contents-link {
|
||
text-decoration: none;
|
||
opacity: 0.7;
|
||
border-bottom: 1px solid rgba(55, 53, 47, 0.18);
|
||
}
|
||
|
||
table,
|
||
th,
|
||
td {
|
||
border: 1px solid rgba(55, 53, 47, 0.09);
|
||
border-collapse: collapse;
|
||
}
|
||
|
||
table {
|
||
border-left: none;
|
||
border-right: none;
|
||
}
|
||
|
||
th,
|
||
td {
|
||
font-weight: normal;
|
||
padding: 0.25em 0.5em;
|
||
line-height: 1.5;
|
||
min-height: 1.5em;
|
||
text-align: left;
|
||
}
|
||
|
||
th {
|
||
color: rgba(55, 53, 47, 0.6);
|
||
}
|
||
|
||
ol,
|
||
ul {
|
||
margin: 0;
|
||
margin-block-start: 0.6em;
|
||
margin-block-end: 0.6em;
|
||
}
|
||
|
||
li > ol:first-child,
|
||
li > ul:first-child {
|
||
margin-block-start: 0.6em;
|
||
}
|
||
|
||
ul > li {
|
||
list-style: disc;
|
||
}
|
||
|
||
ul.to-do-list {
|
||
text-indent: -1.7em;
|
||
}
|
||
|
||
ul.to-do-list > li {
|
||
list-style: none;
|
||
}
|
||
|
||
.to-do-children-checked {
|
||
text-decoration: line-through;
|
||
opacity: 0.375;
|
||
}
|
||
|
||
ul.toggle > li {
|
||
list-style: none;
|
||
}
|
||
|
||
ul {
|
||
padding-inline-start: 1.7em;
|
||
}
|
||
|
||
ul > li {
|
||
padding-left: 0.1em;
|
||
}
|
||
|
||
ol {
|
||
padding-inline-start: 1.6em;
|
||
}
|
||
|
||
ol > li {
|
||
padding-left: 0.2em;
|
||
}
|
||
|
||
.mono ol {
|
||
padding-inline-start: 2em;
|
||
}
|
||
|
||
.mono ol > li {
|
||
text-indent: -0.4em;
|
||
}
|
||
|
||
.toggle {
|
||
padding-inline-start: 0em;
|
||
list-style-type: none;
|
||
}
|
||
|
||
/* Indent toggle children */
|
||
.toggle > li > details {
|
||
padding-left: 1.7em;
|
||
}
|
||
|
||
.toggle > li > details > summary {
|
||
margin-left: -1.1em;
|
||
}
|
||
|
||
.selected-value {
|
||
display: inline-block;
|
||
padding: 0 0.5em;
|
||
background: rgba(206, 205, 202, 0.5);
|
||
border-radius: 3px;
|
||
margin-right: 0.5em;
|
||
margin-top: 0.3em;
|
||
margin-bottom: 0.3em;
|
||
white-space: nowrap;
|
||
}
|
||
|
||
.collection-title {
|
||
display: inline-block;
|
||
margin-right: 1em;
|
||
}
|
||
|
||
.simple-table {
|
||
margin-top: 1em;
|
||
font-size: 0.875rem;
|
||
empty-cells: show;
|
||
}
|
||
.simple-table td {
|
||
height: 29px;
|
||
min-width: 120px;
|
||
}
|
||
|
||
.simple-table th {
|
||
height: 29px;
|
||
min-width: 120px;
|
||
}
|
||
|
||
.simple-table-header-color {
|
||
background: rgb(247, 246, 243);
|
||
color: black;
|
||
}
|
||
.simple-table-header {
|
||
font-weight: 500;
|
||
}
|
||
|
||
time {
|
||
opacity: 0.5;
|
||
}
|
||
|
||
.icon {
|
||
display: inline-block;
|
||
max-width: 1.2em;
|
||
max-height: 1.2em;
|
||
text-decoration: none;
|
||
vertical-align: text-bottom;
|
||
margin-right: 0.5em;
|
||
}
|
||
|
||
img.icon {
|
||
border-radius: 3px;
|
||
}
|
||
|
||
.user-icon {
|
||
width: 1.5em;
|
||
height: 1.5em;
|
||
border-radius: 100%;
|
||
margin-right: 0.5rem;
|
||
}
|
||
|
||
.user-icon-inner {
|
||
font-size: 0.8em;
|
||
}
|
||
|
||
.text-icon {
|
||
border: 1px solid #000;
|
||
text-align: center;
|
||
}
|
||
|
||
.page-cover-image {
|
||
display: block;
|
||
object-fit: cover;
|
||
width: 100%;
|
||
max-height: 30vh;
|
||
}
|
||
|
||
.page-header-icon {
|
||
font-size: 3rem;
|
||
margin-bottom: 1rem;
|
||
}
|
||
|
||
.page-header-icon-with-cover {
|
||
margin-top: -0.72em;
|
||
margin-left: 0.07em;
|
||
}
|
||
|
||
.page-header-icon img {
|
||
border-radius: 3px;
|
||
}
|
||
|
||
.link-to-page {
|
||
margin: 1em 0;
|
||
padding: 0;
|
||
border: none;
|
||
font-weight: 500;
|
||
}
|
||
|
||
p > .user {
|
||
opacity: 0.5;
|
||
}
|
||
|
||
td > .user,
|
||
td > time {
|
||
white-space: nowrap;
|
||
}
|
||
|
||
input[type="checkbox"] {
|
||
transform: scale(1.5);
|
||
margin-right: 0.6em;
|
||
vertical-align: middle;
|
||
}
|
||
|
||
p {
|
||
margin-top: 0.5em;
|
||
margin-bottom: 0.5em;
|
||
}
|
||
|
||
.image {
|
||
border: none;
|
||
margin: 1.5em 0;
|
||
padding: 0;
|
||
border-radius: 0;
|
||
text-align: center;
|
||
}
|
||
|
||
.code,
|
||
code {
|
||
background: rgba(135, 131, 120, 0.15);
|
||
border-radius: 3px;
|
||
padding: 0.2em 0.4em;
|
||
border-radius: 3px;
|
||
font-size: 85%;
|
||
tab-size: 2;
|
||
}
|
||
|
||
code {
|
||
color: #eb5757;
|
||
}
|
||
|
||
.code {
|
||
padding: 1.5em 1em;
|
||
}
|
||
|
||
.code-wrap {
|
||
white-space: pre-wrap;
|
||
word-break: break-all;
|
||
}
|
||
|
||
.code > code {
|
||
background: none;
|
||
padding: 0;
|
||
font-size: 100%;
|
||
color: inherit;
|
||
}
|
||
|
||
blockquote {
|
||
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="43affa15-c52d-4661-af81-ce747e327157" class="page sans"><header><h1 class="page-title">Unit 6: Force and Motion - II</h1></header><div class="page-body"><h1 id="75f53834-f123-42b3-b056-3e20b26f3755" class="">6.1 - Friction</h1><ul id="304bfb17-5dce-4328-8234-fdb6e829730f" class="bulleted-list"><li style="list-style-type:disc">When a force tends to slide a body along a surface, a frictional force from the surface acts on the body. The frictional force is parallel to the surface and directed so as to oppose the sliding. It is due to bonding between the body and the surface. If the body does not slide, the frictional force is a static frictional force <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>f</mi><mo>⃗</mo></mover><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\vec f_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1718799999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="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:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</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>. If there is sliding, the frictional force is a kinetic frictional force <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>f</mi><mo>⃗</mo></mover><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\vec f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1718799999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="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:-0.10764em;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:0.03148em;">k</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>.</li></ul><ul id="8e355376-9891-4da7-b8c5-a0811fad2d5f" class="bulleted-list"><li style="list-style-type:disc">If a body does not move, the static frictional force <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>f</mi><mo>⃗</mo></mover><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\vec f_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1718799999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="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:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</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> and the component 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>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
|
||
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> parallel to the surface are equal in magnitude, 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><msub><mover accent="true"><mi>f</mi><mo>⃗</mo></mover><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\vec f_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1718799999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="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:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</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 directed opposite that component. If the component increases, <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>f</mi><mo>⃗</mo></mover><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\vec f_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1718799999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="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:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</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> also increases.</li></ul><ul id="61266f09-3307-44f1-bc04-556f1ab21380" class="bulleted-list"><li style="list-style-type:disc">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><msub><mover accent="true"><mi>f</mi><mo>⃗</mo></mover><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\vec f_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1718799999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="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:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</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> has a maximum value, <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>f</mi><mo>⃗</mo></mover><mrow><mi>s</mi><mo separator="true">,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\vec f_{s,max}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.263548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;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">s</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></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><span></span></span> given by:<figure id="2a469552-6167-4596-be2e-8dbe13ebdf33" 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>f</mi><mrow><mi>s</mi><mo separator="true">,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>μ</mi><mi>s</mi></msub><msub><mi>F</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">f_{s,max} = \mu_sF_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;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">s</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></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 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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">μ</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">s</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" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><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" style="margin-right:0.10903em;">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></span></span></span></div></figure><p id="8a44955b-6004-4045-8c55-cbe3491e3722" 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>μ</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\mu_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">μ</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">s</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 coefficient of static friction 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><msub><mi>F</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">F_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><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" style="margin-right:0.10903em;">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></span></span></span><span></span></span> is the magnitude of the normal force. If the component 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>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
|
||
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> parallel to the surface exceeds <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>f</mi><mo>⃗</mo></mover><mrow><mi>s</mi><mo separator="true">,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\vec f_{s,max}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.263548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><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:0.10764em;">f</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><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="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;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">s</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></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><span></span></span>, the body slides on the surface.</p></li></ul><ul id="c1bfcfde-9a94-40a0-9c55-f3d9588f5ea8" class="bulleted-list"><li style="list-style-type:disc">If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value <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>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</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"><span class="mord mathnormal" style="margin-right:0.10764em;">f</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:-0.10764em;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:0.03148em;">k</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> given by:<figure id="ec5f3b53-0123-49b5-a4d7-303580e85857" 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>f</mi><mi>k</mi></msub><mo>=</mo><msub><mi>μ</mi><mi>k</mi></msub><msub><mi>F</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">f_k = \mu_kF_N</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"><span class="mord mathnormal" style="margin-right:0.10764em;">f</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:-0.10764em;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:0.03148em;">k</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">μ</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: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:0.03148em;">k</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" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><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" style="margin-right:0.10903em;">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></span></span></span></div></figure><p id="719990db-93ed-4c39-ae58-31f6ab26c456" 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>μ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mu_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">μ</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: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:0.03148em;">k</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 coefficient of kinetic friction.</p></li></ul><h1 id="ce4acbf8-a6f2-4e74-bacf-36eebdb79b4b" class="">6.2 - The Drag Force and Terminal Speed</h1><h2 id="c526614e-566a-4f7d-b9b5-8be91a4037b5" class="">Drag Force</h2><p id="25620780-3087-4012-b307-13c7f6c12e51" class="">When there is relative motion between air (or some other fluid) and a body, the body experiences a <strong>drag force</strong> <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>D</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec D</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:0.02778em;">D</span></span><span style="top:-3.25233em;"><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></span></span></span><span></span></span> that opposes the relative motion and points in the direction in which the fluid flows relative to the body. 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>D</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec D</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:0.02778em;">D</span></span><span style="top:-3.25233em;"><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></span></span></span><span></span></span> is related to the relative speed <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>v</mi></mrow><annotation encoding="application/x-tex">v</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;">v</span></span></span></span></span><span></span></span> by an experimentally determined drag coefficient <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>C</mi></mrow><annotation encoding="application/x-tex">C</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;">C</span></span></span></span></span><span></span></span> according to:</p><figure id="18b003fe-2e1c-4239-aac0-0fdce7d10702" 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>D</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>C</mi><mi>ρ</mi><mi>A</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">D = \frac{1}{2} C \rho A 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:0.02778em;">D</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:2.00744em;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:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</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">1</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 class="mord mathnormal">Cρ</span><span class="mord mathnormal">A</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:0.8641079999999999em;"><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">2</span></span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="0b4e48a3-1c56-4798-8160-95f0ec7b9280" 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>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">ρ</span></span></span></span></span><span></span></span> is the fluid density (mass per unit volume) 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>A</mi></mrow><annotation encoding="application/x-tex">A</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">A</span></span></span></span></span><span></span></span> is the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the relative 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>).</p><h2 id="2b439936-dff4-496f-ab04-cb304dc600b9" class="">Terminal Speed</h2><p id="bcd27380-5b2b-4e66-9054-0b9d87428c59" class="">When a blunt object has fallen far enough through the air, the magnitudes of the drag force <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>D</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec D</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:0.02778em;">D</span></span><span style="top:-3.25233em;"><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></span></span></span><span></span></span> and the gravitational force <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>F</mi><mo>⃗</mo></mover><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">\vec F_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><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" 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
|
||
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:0.15139200000000003em;"><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" style="margin-right:0.03588em;">g</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><span></span></span> on the body become equal. The body then falls at a constant terminal speed <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><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">v_t</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:0.2805559999999999em;"><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 mathnormal mtight">t</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> given by:</p><figure id="da1d64be-c167-4ed0-a643-b87a937eff8c" 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><mi>t</mi></msub><mo>=</mo><msqrt><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mi>g</mi></msub></mrow><mrow><mi>C</mi><mi>ρ</mi><mi>A</mi></mrow></mfrac></msqrt></mrow><annotation encoding="application/x-tex">v_t = \sqrt{\frac{2F_g}{C \rho A}}</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:0.2805559999999999em;"><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 mathnormal mtight">t</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:3.04em;vertical-align:-1.1661799999999998em;"></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 class="mord mathnormal">Cρ</span><span class="mord mathnormal">A</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">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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" style="margin-right:0.03588em;">g</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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.8338200000000002em;"><span class="pstrut" style="height:5em;"></span><span class="hide-tail" style="min-width:1.02em;height:3.08em;"><svg width='400em' height='3.08em' viewBox='0 0 400000 3240' preserveAspectRatio='xMinYMin slice'><path d='M473,2793
|
||
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,
|
||
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><h1 id="25e31195-044f-4b07-a784-8beb0d2a474a" class="">6.3 - Uniform Circular Motion</h1><p id="dd9c9eb7-df34-468c-880f-8010af2eef8c" class="">If a particle moves in a circle or a circular arc 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> at constant speed <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>v</mi></mrow><annotation encoding="application/x-tex">v</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;">v</span></span></span></span></span><span></span></span>, the particle is said to be in uniform circular motion. It then has a centripetal acceleration with magnitude given by:</p><figure id="f48049be-6429-4d28-9d6d-2744a111b0b9" 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>a</mi><mo>=</mo><mfrac><msup><mi>v</mi><mn>2</mn></msup><mi>R</mi></mfrac></mrow><annotation encoding="application/x-tex">a = \frac{v^2}{R}</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">a</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:2.177108em;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:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</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"><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 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="51f2f3a2-74c0-4766-a0fe-d711baa42c9a" class="">This acceleration is due to a net centripetal force on the particle, with magnitude given by:</p><figure id="5d083d84-cf12-427c-9ef1-1498aef6b5c8" 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>F</mi><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><mi>R</mi></mfrac></mrow><annotation encoding="application/x-tex">F = \frac{mv^2}{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:0.13889em;">F</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:2.177108em;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:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</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 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 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="6ae7c463-41ea-4501-90e7-852b758d2f0f" 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:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span></span><span></span></span> is the particle's mass. The vector quantities <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>a</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec a</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">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><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> 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>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
|
||
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> are directed toward the center of the curvature of the particle's path.</p></div></article></body></html> |