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agyao2 merged 3 commits into from
Feb 18, 2026
Merged

Warning/example links documentation + #428

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4f37c60
Documentation
agyao2 Feb 18, 2026
c146655
Typo fix
agyao2 Feb 18, 2026
cb24974
Fix equation for work done by a moment
agyao2 Feb 18, 2026
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34 changes: 34 additions & 0 deletions src/pages/about/docs.astro
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Original file line number Diff line number Diff line change
Expand Up @@ -78,6 +78,8 @@ import Slideshow from "../../components/Slideshow.astro";
<ul class='list-group list-group-flush py-0'>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#same-page'> Within </a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#int-page'> MechRef </a></li>
<li class = 'list-group-item py-0'><a class = 'text-decoration-none subsubsection' href = '#links_to_warnings'>Links to warnings</a></li>
<li class = 'list-group-item py-0'><a class = 'text-decoration-none subsubsection' href = '#links_to_examples'>Links to examples</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#ext-page'> External </a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#ext-video'> Video </a></li>
</ul>
Expand Down Expand Up @@ -844,6 +846,38 @@ All titles should be done in sentence case.

</SubSubSection>

<SubSubSection title = "Links to Warnings" id = "links_to_warnings">
<p>
Tag type: Warning
</p>
Options:
<ul>
<li>href: insert the id of the warning you want to link to after '#warning_'.</li>
</ul>
Example:
<CodeBox language = "html" code = `<a href = "/xxx/CoursePage#warning_id">TEXT</a>`/>
<a href = "/dyn/vectors#warning_rvv-wl">TEXT</a>
<p>
<em>*For demonstration purposes, this links to a warning in the vectors page in dynamics.</em>
</p>
</SubSubSection>

<SubSubSection title = "Links to Examples" id = "links_to_examples">
<p>
Tag type: Example
</p>
Options:
<ul>
<li>href: insert the id of the example you want to link to after '#example_btn_'.</li>
</ul>
Example:
<CodeBox language = "html" code = `<a href = "/xxx/CoursePage#example_btn_id">TEXT</a>`/>
<a href = "/dyn/vectors#example_btn_rvv-xn">TEXT</a>
<p>
<em>*For demonstration purposes, this links to an example in the vectors page in dynamics.</em>
</p>
</SubSubSection>

<SubSubSection title="External page" id="ext-page">

<p>Tag type: Regular</p>
Expand Down
28 changes: 14 additions & 14 deletions src/pages/dyn/work_and_energy.astro
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Expand Up @@ -59,7 +59,7 @@ import Col from "../../components/Col.astro"

<DisplayEquation equation="\\begin{aligned}\\vec{F} &amp;= -\\nabla V \\\\&amp; = -\\frac{\\partial V}{\\partial x}\\hat{\\imath} -\\frac{\\partial V}{\\partial y}\\hat{\\jmath} - \\frac{\\partial V}{\\partial z}\\hat{k}\\end{aligned}" id="ren-efp" title="Force and potential energy." background="True" derivation="True">
<p>
Beginning with <a href="#ren-ec">#ren-ec</a> and generalizing it to one-dimensional displacement in the y-direction, the equation becomes:
Beginning with <a href="#ren-wc">#ren-wc</a> and generalizing it to one-dimensional displacement in the y-direction, the equation becomes:
<DisplayEquation equation="\\begin{aligned} W &amp;= \\vec{F} \\cdot \\Delta{\\vec{y}} \\end{aligned}" />

</p>
Expand Down Expand Up @@ -105,7 +105,7 @@ import Col from "../../components/Col.astro"

</p>
<p>
Since gravity only acts in one direction, we can simplify <a href="ren-efp">#ren-efp</a> to:
Since gravity only acts in one direction, we can simplify <a href="#ren-efp">#ren-efp</a> to:
<DisplayEquation equation="\\begin{aligned} F &amp;= -\\frac{dV}{dy} \\\\ -mg &amp;= -\\frac{dV}{dy} \\\\ mg \\, dy &amp;= dV \\\\ \\end{aligned}" />

</p>
Expand Down Expand Up @@ -141,7 +141,7 @@ import Col from "../../components/Col.astro"

</p>
<p>
Since the restoring force in a spring only acts in one direction, we can simplify <a href="ren-efp">#ren-efp</a> to:
Since the restoring force in a spring only acts in one direction, we can simplify <a href="#ren-efp">#ren-efp</a> to:
<DisplayEquation equation="\\begin{aligned} F &amp;= -\\frac{dV}{dx} \\\\ -kx &amp;= -\\frac{dV}{dx} \\\\ kx \\, dx &amp;= dV \\\\ \\end{aligned}" />

</p>
Expand Down Expand Up @@ -238,12 +238,12 @@ import Col from "../../components/Col.astro"

</p>
<p>
Using conservation of energy, and the fact that a vector dotted with itself equals its magnitude squared (see <a href="rvi.html#rvi-eg">#rvi-eg</a>), then:
Using conservation of energy, and the fact that a vector dotted with itself equals its magnitude squared (see <a href="/dyn/vector_calculus#rvi-eg">#rvi-eg</a>), then:
<DisplayEquation equation="\\begin{aligned} T &amp; = \\frac{1}{2}mv^2\\\\ \\end{aligned}" />

</p>
<p>
Another perhaps less intuitive way is to begin with the work definition <a href="#ren-ef">#ren-ef</a>:
Another perhaps less intuitive way is to begin with the work definition <a href="#ren-wf">#ren-wf</a>:
<DisplayEquation equation="\\begin{aligned} W &amp;= \\int_{\\vec{r}_1}^{\\vec{r}_2} \\vec{F} \\cdot d\\vec{r} \\\\ &amp; = \\int_{t_1}^{t_2} \\vec{F} \\cdot \\dot{\\vec{r}}dt \\\\ &amp; = \\int_{t_1}^{t_2} \\vec{F} \\cdot \\vec{v} \\, dt \\end{aligned}" />

</p>
Expand Down Expand Up @@ -290,14 +290,14 @@ import Col from "../../components/Col.astro"

<DisplayEquation id="ren-ea" equation="T = \\frac{1}{2} m v_Q^2 + m \\vec{v}_Q \\cdot \\left( \\vec\\omega \\times \\vec{r}_{QC} \\right) + \\frac{1}{2} I_{Q,\\hat\\omega} \\omega^2" title="Kinetic energy of a rigid body about an arbitrary body point \\(Q\\)." background="True" derivation="true">
<p>
We start with the general expression <a href="#rem-eb">#rem-eb</a>:
We start with the general expression <a href="/dyn/geometric_properties#rem-eb">#rem-eb</a>:

<DisplayEquation equation="\\begin{aligned} T &amp;= \\iiint_{\\mathcal{B}} \\frac{1}{2} \\rho v_P^2 \\, dV, \\end{aligned}" />


where we integrate over the body with a location \(P\). We
choose a point \(Q\) fixed to the body and
use <a href="rkg.html#rkg-er">#rkg-er</a> to express the
use <a href="/dyn/rigid_body_kinematics#rkg-er">#rkg-er</a> to express the
velocity of \(P\) in terms of <InlineEquation equation="\\vec{v}_Q" /> and
\(\vec\omega\), giving

Expand All @@ -317,7 +317,7 @@ import Col from "../../components/Col.astro"
orthogonal distance to point \(P\) from the line through
\(Q\) in direction <InlineEquation equation="\\vec\\omega" />
, so
from <a href="rem.html#rem-ei">#rem-ei</a> we see that
from <a href="/dyn/geometric_properties#rem-ei">#rem-ei</a> we see that
the final integral above is the moment of inertia
<InlineEquation equation="I_{Q,\\hat\\omega}" />
about the axis through
Expand All @@ -327,13 +327,13 @@ import Col from "../../components/Col.astro"

<DisplayEquation id="ren-wk" equation="W = \\Delta T" title="Work-kinetic energy theorem." background="True" derivation="true">
<p>
Skipping the derivation shown in <a href="#ren-ep">#ren-ep</a>, taking the last step with the integral:
Skipping the derivation shown in <a href="#ren-ek">#ren-ek</a>, taking the last step with the integral:
<DisplayEquation equation="\\begin{aligned} \\int dW &amp; = \\int_{\\vec{v_1}}^{\\vec{v_2}}m\\vec{v} \\, d\\vec{v} \\\\ W &amp; = \\frac{1}{2}m\\left[\\vec{v} \\cdot \\vec{v}\\right]_{\\vec{v_1}}^{\\vec{v_2}} \\\\ &amp; = \\Delta T \\end{aligned}" />
</p>
</DisplayEquation>

<DisplayEquationCustom id="ren-eco" title="Work-energy theorem." background="True">
<DisplayEquation equation="W = \Delta T + \Delta V"/>
<DisplayEquation equation="W = \\Delta T + \\Delta V"/>
<p>
where \(W\) is the work done by non-conservative forces. If non-conservative forces are not present, then
it becomes conservation of energy.
Expand Down Expand Up @@ -405,7 +405,7 @@ import Col from "../../components/Col.astro"
</div>
</DisplayEquationCustom>

<DisplayEquationCustom id="ren-wm" title="Work done by a moment \\(M\\)." background="True" derivation="true">
<DisplayEquationCustom id="ren-wm" title="Work done by a moment \\(M\\)." background="True">
<DisplayEquation equation="\\begin{aligned}W &amp;= \\int_{\\theta_1}^{\\theta_2} M \\, d\\theta \\\\&amp; = \\int_{t_1}^{t_2} M \\dot\\theta \\, dt \\\\&amp;= M \\, \\Delta \\theta \\text{ (for constant M)}\\end{aligned}" />
<p>
The rotation angle \(\theta\) is measured around the same
Expand Down Expand Up @@ -463,7 +463,7 @@ import Col from "../../components/Col.astro"

<p> Work done by friction can be positive, zero, or negative. </p>

<Example id="wdf-rws" title="Friction during rolling wihtout slip." solution=True>
<Example id="wdf-rws" title="Friction during rolling without slip." solution=True>
<div>
<Image src="/dyn/work_and_energy/WorkForce_example2.png" width="10"></Image>
<ol>
Expand Down Expand Up @@ -501,7 +501,7 @@ import Col from "../../components/Col.astro"

<SubSubSection title="Work done by a moment" id="work_moment">

<DisplayEquation equation=" W = \\int_{\\theta_0}^{\\theta_f} M_z \\,d\\theta = \\int_{t_0}^{t_f} M_z \\ \\dot{\\theta} \\,d\\theta " background="True" id="wan-wm"> </DisplayEquation>
<DisplayEquation equation=" W = \\int_{\\theta_0}^{\\theta_f} M_z \\,d\\theta = \\int_{t_0}^{t_f} M_z \\ \\dot{\\theta} \\,dt" background="True" id="wan-wm"> </DisplayEquation>

<Example id="wdm-rm" title="Rod with friction moment." solution=True>
<div>
Expand Down Expand Up @@ -608,4 +608,4 @@ import Col from "../../components/Col.astro"

</Layout>

<script src="/dyn/work_and_energy/canvases.js" is:inline></script>
<script src="/dyn/work_and_energy/canvases.js" is:inline></script>