807 lines
161 KiB
HTML
807 lines
161 KiB
HTML
|
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 7: Kinetic Energy and Work</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="ecc0d025-129c-4e8d-946c-b4ca882b4a1d" class="page sans"><header><h1 class="page-title">Unit 7: Kinetic Energy and Work</h1></header><div class="page-body"><h1 id="1b3b8df5-d7a8-487b-aac7-07f5658809f7" class="">7.1 - Kinetic Energy</h1><p id="194412c8-7d33-4bbe-a326-18bbca4e301b" class="">The 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></mrow><annotation encoding="application/x-tex">K</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></span></span></span><span></span></span> associated with the motion of a particle of mass <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span></span><span></span></span> and 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>, 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>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> is well below the speed of light, is:</p><figure id="ae123792-12cb-4679-b0a6-fa70c88d9fc4" 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>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}mv^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="mar
|
|||
|
|
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:0.2805559999999999em;"><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 mtight"><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> of those forces.</p><p id="07e81c13-f3c0-4371-8416-bc2f4ed29c24" class="">For a particle, a change <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 mathvariant="normal">Δ</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">\Delta K</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">Δ</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span></span><span></span></span> in the kinetic energy equals the net work <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>W</mi></mrow><annotation encoding="application/x-tex">W</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;">W</span></span></span></span></span><span></span></span> done on the particle:</p><figure id="2d80af03-7897-4683-930f-8dc36106f8c7" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>K</mi><mo>=</mo><msub><mi>K</mi><mi>f</mi></msub><mo>−</mo><msub><mi>K</mi><mi>i</mi></msub><mo>=</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">\Delta K = K_f - K_i = W</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">Δ</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:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></
|
|||
|
|
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> during displacement <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:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span><span></span></span> is:</p><figure id="151bbec5-150d-49ad-ba2a-fcebb29b0e31" 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>W</mi><mo>=</mo><mi>F</mi><mi>d</mi><mi>cos</mi><mo></mo><mi>ϕ</mi><mo>=</mo><mover accent="true"><mi>F</mi><mo>⃗</mo></mover><mo>⋅</mo><mover accent="true"><mi>d</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">W = Fd \cos \phi = \vec F \cdot \vec d</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;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ϕ</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:0.9663299999999999em;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 class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span></div></figure><p id="561f4e9f-4dd7-4bef-aa38-5ddd4e29454a" class="">in which <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>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</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">ϕ</span></span></span></span></span><span></span></span> is the constant angle between the directions 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> 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>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:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span><span></span></span>. </p><p id="c2430463-5f2f-4e42-8d7b-b68cb849f12b" class="">Only 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> that is along the displacement <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:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span><span></span></span> can do work on the object.</p><h1 id="f32ff2ac-dff7-4064-8db3-e9c81710a9da" class="">7.3 - Work Done by the Gravitational Force</h1><p id="5141de03-d1f4-4707-82f3-004bbe2551f8" class="">The work <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>W</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">W_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</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> done by 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><mi>F</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">F_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></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><span></span></span> on a particle-like object of mass <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span></span><span></span></span> as the object moves through a displacement <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-
|
|||
|
|
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
|||
|
|
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
|||
|
|
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
|||
|
|
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
|||
|
|
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
|||
|
|
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> is given by:</p><figure id="40d1e878-111b-43b2-a035-13eee166dff2" 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>W</mi><mi>g</mi></msub><mo>=</mo><mi>m</mi><mi>g</mi><mi>d</mi><mi>cos</mi><mo></mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">W_g = mgd \cos \phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ϕ</span></span></span></span></span></div></figure><p id="1b38bba5-4813-49cf-9a6b-516396e0c068" class="">in which <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>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</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">ϕ</span></span></span></span></span><span></span></span> is the angle between <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="ac
|
|||
|
|
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> 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>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:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span><span></span></span>.</p><p id="1190b6a3-92f0-4680-8df5-4955fd99a4f8" class="">The work <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>W</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">W_a</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;">W</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:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</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> done by an applied force as a particle-like object is either lifted or lowered is related to the work <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>W</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">W_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</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> done by the gravitational force and the change <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 mathvariant="normal">Δ</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">\Delta K</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">Δ</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span></span><span></span></span> in the object's kinetic energy by:</p><figure id="06db3945-33a7-47da-ad48-ddd44b30b341" 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="k
|
|||
|
|
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:0.151392em;"><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">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> from a spring is:</p><figure id="f9480ab8-411d-4e51-850e-397fe9a20ed1" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>F</mi><mo>⃗</mo></mover><mi>s</mi></msub><mo>=</mo><mo>−</mo><mi>k</mi><mover accent="true"><mi>d</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec F_s = -k\vec d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1163299999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" 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:0.151392em;"><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">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="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:1.0607699999999998em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span></div></figure><p id="7f8f363f-c6ec-4f69-82ad-c9b757204980" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>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:0.9774399999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">d</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></span></span></span></span></span></span><span></span></span> is the displacement of the spring's free end from its position when the spring is in its relaxed state (not compressed or extended), 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>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span><span></span></span> is the spring constant (a measure of the spring's stiffness). If an <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>x</mi></mrow><annotation encoding="application/x-tex">x</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">x</span></span></span></span></span><span></span></span> axis lies along the spring, with the origin at the location of the spring's free end when the spring is in its relaxed state, we can write:</p><figure id="c7e6e0e8-068d-42f3-90cf-e0a63d7bf2e8" 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>x</mi></msub><mo>=</mo><mo>−</mo><mi>k</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">F_x = -kx</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:0.151392em;"><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">x</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:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">x</span></span></span></span></span></div></figure><p id="e0faeb88-d19e-4d8b-bbba-f4ddb6308a7a" class="">A spring force is thus a variable force: It varies with the displacement of the spring's free end. (Yay calculus!)</p><p id="8162ff87-84f3-438d-8ff6-969ee7e5ba8b" class="">If an object is attached to the spring's free end, the work <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditabl
|
|||
|
|
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> on a particle-like object depends on the position of the object, the work done by <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> on the object while the object moves from an initial position <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>r</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">r_i</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:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> with coordinates (<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>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</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">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span>, <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>y</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y_i</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" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><
|
|||
|
|
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> only has an <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>x</mi></mrow><annotation encoding="application/x-tex">x</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">x</span></span></span></span></span><span></span></span> component, then this reduces to:</p><figure id="2b06078b-8365-4f64-9b12-be8e24ae3956" 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>W</mi><mo>=</mo><msubsup><mo>∫</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>x</mi><mi>f</mi></msub></msubsup><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">W = \int_{x_i}^{x_f} F(x) \space dx</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;">W</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:2.426342em;vertical-align:-1.01205em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.414292em;"><span style="top:-1.7880500000000004em;margin-left:-0.44445em;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 class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.8129000000000004em;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 class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29011428571428566em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" s
|
|||
|
|
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> at an angle <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</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">ϕ</span></span></span></span></span><span></span></span> to the direction of travel of the instantaneous 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>, the instantaneous power is:</p><figure id="46ffb129-d802-42d0-8d57-1d20d40066f6" 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>P</mi><mo>=</mo><mi>F</mi><mi>v</mi><mi>cos</mi><mo></mo><mi>ϕ</mi><mo>=</mo><mover accent="true"><mi>F</mi><mo>⃗</mo></mover><mo>⋅</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">P = Fv \cos \phi = \vec F \cdot \vec v</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;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ϕ</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:0.9663299999999999em;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 class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.714em;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></div></figure></div></article></body></html>
|