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More dynamics link fixes #383

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1a92ca5
Link fix
agyao2 Nov 5, 2025
e3b5931
Steering geometry list fix
agyao2 Nov 7, 2025
861bc11
Rigid body kinematics link fixe
agyao2 Nov 7, 2025
72ec6ea
Geometric properties fixes
agyao2 Nov 7, 2025
b07c7af
Contact and rolling link fixes
agyao2 Nov 7, 2025
51a0671
Work and energy fixes
agyao2 Nov 7, 2025
ef391ef
Momentum link fix
agyao2 Nov 7, 2025
f90e502
Figure particle kinematics link fix
agyao2 Nov 8, 2025
08e7d97
More rigid body kinematics link fixes
agyao2 Nov 8, 2025
ad56275
More geometric properties link fixes
agyao2 Nov 8, 2025
3a6dfaa
Warning and example links documentation
agyao2 Nov 8, 2025
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32 changes: 32 additions & 0 deletions src/pages/about/docs.astro
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Original file line number Diff line number Diff line change
Expand Up @@ -844,6 +844,38 @@ All titles should be done in sentence case.

</SubSubSection>

<SubSubSection title = "Links to Warnings" id = "links_to_warnings">
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@mrhoyle2 mrhoyle2 Nov 14, 2025

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Can you add these subsubsections to the navtree for this page?

<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>
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6 changes: 3 additions & 3 deletions src/pages/dyn/contact_and_rolling.astro
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Expand Up @@ -190,11 +190,11 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
<DisplayEquation equation="\\begin{aligned} \\vec{v}_C &amp;= r \\omega \\,\\hat{e}_t \\\\ \\vec{a}_C &amp;= r \\alpha \\,\\hat{e}_t \\end{aligned}" background="True" title="Center velocity and acceleration while rolling on a flat surface (Tangential-Normal Basis)." id="rko-ef" derivation="True">
<p>
We begin by observing that the sign conventions in
Figure <a href="#rko-ff">#rko-ff</a> mean that
Figure <a href="#rolling_flat_surface">#rko-ff</a> mean that
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Please make the text clear which figure is being referred to

<InlineEquation equation="\\vec\\omega = -\\omega\\,\\hat{e}_b" />. Now rolling without
slipping means the contact point \(A\) must
instantaneously have zero velocity, so using <a
href="rkg.html#rkg-er">#rkg-er</a> gives:
href="/dyn/rigid_body_kinematics#rkg-er">#rkg-er</a> gives:

<DisplayEquation equation="\\begin{aligned} \\vec{v}_C &amp;= \\vec{v}_A + \\vec{\\omega} \\times \\vec{r}_{AC} \\\\ &amp;= (-\\omega \\,\\hat{e}_b) \\times r \\,\\hat{e}_n \\\\ &amp;= r \\omega \\,\\hat{e}_t. \\end{aligned}" />

Expand Down Expand Up @@ -247,7 +247,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
By definition of non-slip rolling contact, the point of
contact \(P\) has zero velocity. The acceleration can be
computed from the center \(C\) with <a
href="#rkg-e2">#rkg-e2</a>:
href="/dyn/rigid_body_kinematics#rkg-e2">#rkg-e2</a>:

<DisplayEquation equation="\\begin{aligned} \\vec{a}_P &amp;= \\vec{a}_C + \\vec{\\alpha} \\times \\vec{r}_{CP} + \\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{r}_{CP}) \\\\ &amp;= \\alpha r \\,\\hat{e}_t + (-\\alpha\\,\\hat{e}_b) \\times (-r\\,\\hat{e}_n) + (-\\omega\\,\\hat{e}_b) \\times \\Big((-\\omega\\,\\hat{e}_b) \\times (-r\\,\\hat{e}_n)\\Big) \\\\ &amp;= \\alpha r \\,\\hat{e}_t - \\alpha r \\,\\hat{e}_t + \\omega^2 r \\,\\hat{e}_n \\\\ &amp;= \\omega^2 \\,\\vec{r}_{PC}. \\end{aligned}"/>
</p>
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28 changes: 14 additions & 14 deletions src/pages/dyn/geometric_properties.astro
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1103: link doesn't work

Original file line number Diff line number Diff line change
Expand Up @@ -105,7 +105,7 @@ import Row from "../../components/Row.astro"
<DisplayEquation equation="\\vec{r}_C = \\frac{1}{m}\\iiint_{\\mathcal{B}} \\vec{r} dm" />
</p>
<p>
Where <InlineEquation equation="dm = \\rho \\, dV" /> from the differential of <a href="rcm.html#rcm-tm">#rcm-tm</a>. Substituting in:
Where <InlineEquation equation="dm = \\rho \\, dV" /> from the differential of <a href="/dyn/geometric_properties#rcm-tm">#rcm-tm</a>. Substituting in:
<DisplayEquation equation="\\begin{aligned} \\vec{r}_C &amp;= \\frac{1}{m}\\iiint_{\\mathcal{B}} \\vec{r} dm \\\\ &amp;= \\frac{1}{m}\\iiint_{\\mathcal{B}} \\vec{r} \\rho \\, dV \\\\ &amp;= \\frac{1}{m}\\iiint_{\\mathcal{B}} \\rho \\vec{r} \\, dV \\\\ \\end{aligned}" />
</p>
</div>
Expand Down Expand Up @@ -361,8 +361,8 @@ import Row from "../../components/Row.astro"
<DisplayEquation equation="\\begin{aligned} x_C &amp;= \\frac{\\rho}{m}\\int_{0}^{1}\\int_{0}^{1 - u} (x_P u + x_Q v) \\, J \\, dvdu \\\\ &amp;= \\frac{\\rho}{m}\\int_{0}^{1}\\int_{0}^{1 - u} (x_P u + x_Q v) \\, (x_P y_Q - x_Q y_P) \\, dvdu \\\\ &amp;= \\frac{\\rho}{m}(x_P y_Q - x_Q y_P)\\int_{0}^{1}\\int_{0}^{1 - u} (x_P u + x_Q v) \\, dvdu \\\\ &amp;= \\frac{\\rho}{m}(x_P y_Q - x_Q y_P)\\int_{0}^{1}x_P u\\left[v\\right]_{v = 0}^{v = 1-u} + \\frac{x_Q}{2}\\left[v^2\\right]_{v = 0}^{v = 1-u}du \\\\ &amp;= \\frac{\\rho}{m}(x_P y_Q - x_Q y_P)\\int_{0}^{1} x_P u(1 - u) + \\frac{x_Q}{2}(1 - u)^2 du \\\\ &amp;= \\frac{\\rho}{m}(x_P y_Q - x_Q y_P) \\left(\\frac{x_P}{2} \\left[u^2\\right]_{0}^{1} - \\frac{x_P}{3}\\left[u^3\\right]_{0}^{1} + \\frac{x_Q}{2}\\left[u\\right]_{0}^{1} - \\frac{x_Q}{2}\\left[u^2\\right]_{0}^{1} + \\frac{x_Q}{6}\\left[u^3\\right]_{0}^{1}\\right) \\\\ &amp;= \\frac{\\rho}{m}(x_P y_Q - x_Q y_P) \\left(\\frac{x_P}{2} - \\frac{x_P}{3} + \\frac{x_Q}{2} - \\frac{x_Q}{2} + \\frac{x_Q}{6}\\right) \\\\ &amp;= \\frac{\\rho}{m}(x_P y_Q - x_Q y_P) \\left(\\frac{x_P + x_Q}{6}\\right) \\end{aligned}" />
</p>
<p>
The total mass of the plate is <InlineEquation equation="m = \rho A = \frac{1}{2}\rho x_P y_Q" />, and with the chosen configuration \(y_P = 0\). Thus:
<DisplayEquation equation="x_C = \frac{x_P + x_Q}{3}" />
The total mass of the plate is <InlineEquation equation="m = \\rho A = \\frac{1}{2}\\rho x_P y_Q" />, and with the chosen configuration \(y_P = 0\). Thus:
<DisplayEquation equation="x_C = \\frac{x_P + x_Q}{3}" />
</p>
</div>
</DisplayEquationCustom>
Expand All @@ -383,7 +383,7 @@ import Row from "../../components/Row.astro"
<DisplayEquation equation="\\begin{aligned} \\vec{r}_C &amp;= \\frac{1}{m}\\iiint_\\mathcal{B} \\rho \\vec{r} \\, dV \\\\ &amp;= \\frac{1}{m}\\iint_{A} \\rho \\vec{r} \\, dA \\\\ \\end{aligned}" />
</p>
<p>
It is convenient to switch to <a href="rvy.html#rvy-ec">cylindrical coordinates</a>:
It is convenient to switch to <a href="/dyn/coordinate_systems#cylindrical_coordinates">cylindrical coordinates</a>:
<DisplayEquation equation="\\begin{aligned} x &amp;= a \\, r \\cos\\theta \\\\ y &amp;= b \\, r \\sin\\theta \\\\ \\end{aligned}" />
</p>
<p>
Expand All @@ -404,7 +404,7 @@ import Row from "../../components/Row.astro"

<SubSubSection title="Simplified shapes" id="simple_shapes_com">
<p>
The centers of mass listed below are all special cases of the basic shapes given in Section <a href="#rcm-bs">#rcm-bs</a>. Other
The centers of mass listed below are all special cases of the basic shapes given in Section <a href="/dyn/geometric_properties#basic_shapes_com">#rcm-bs</a>. Other
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Can you change the "#id" to the name of the section?

special cases can be easily obtained by similar methods, or directly computing the integral.
</p>

Expand All @@ -421,7 +421,7 @@ import Row from "../../components/Row.astro"

<div slot="derivation">
<p>
See <a href="#rcm-xc">example problem</a> on how to derive it by directly computing the integrals.
See <a href="/dyn/geometric_properties#example_btn_rcm-xc">example problem</a> on how to derive it by directly computing the integrals.
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Can you make the text clearer for which example problem?

</p>
<p>
The other perhaps simpler approach is to let \(x_Q = 0\) in <a href="#rcm-et">#rcm-et</a>, which forms a right triangle if the configuration
Expand All @@ -444,7 +444,7 @@ import Row from "../../components/Row.astro"

<div slot="derivation">
<p>
See <a href="#rcm-xc">example problem</a> on how to derive it by directly computing the integrals.
See <a href="/dyn/geometric_properties#example_btn_rcm-xc">example problem</a> on how to derive it by directly computing the integrals.
</p>
<p>
The other perhaps simpler approach is to let \(x_Q = 0\) in <a href="#rcm-et">#rcm-et</a>, which forms a right triangle if the configuration
Expand Down Expand Up @@ -473,7 +473,7 @@ import Row from "../../components/Row.astro"
<DisplayEquation equation="\\begin{aligned} \\vec{r}_C &amp;= \\frac{1}{m}\\iiint_\\mathcal{B} \\rho \\vec{r} \\, dV \\\\ &amp;= \\frac{1}{m}\\iint_{A} \\rho \\vec{r} \\, dA \\\\ \\end{aligned}" />
</p>
<p>
It is convenient to switch to <a href="rvy.html#rvy-ec">cylindrical coordinates</a>:
It is convenient to switch to <a href="/dyn/coordinate_systems#cylindrical_coordinates">cylindrical coordinates</a>:
<DisplayEquation equation="\\begin{aligned} x &amp;= r \\cos\\theta \\\\ y &amp;= r \\sin\\theta \\\\ \\end{aligned}"/>
</p>
<p>
Expand Down Expand Up @@ -823,7 +823,7 @@ import Row from "../../components/Row.astro"

<p>
Use the answer to Example Problem <a
href="#rem-xs">#rem-xs</a> and the parallel axis
href="#example_btn_rem-xs">#rem-xs</a> and the parallel axis
theorem <a href="#rem-el">#rem-el</a>.
</p>
</div>
Expand All @@ -834,7 +834,7 @@ import Row from "../../components/Row.astro"

<div slot="solution">
<p>
In Example Problem <a href="#rem-xs">#rem-xs</a> we
In Example Problem <a href="#example_btn_rem-xs">#rem-xs</a> we
computed the moment of inertia of a square place
about the center to be <InlineEquation equation="I_{C,z} = \\frac{1}{6} m \\ell^2" />. The parallel axis theorem <a
href="#rem-el">#rem-el</a> now gives:
Expand Down Expand Up @@ -963,7 +963,7 @@ import Row from "../../components/Row.astro"
<div class="w-50">
<p>
Recall that in Example Problem <a
href="#rem-xs">#rem-xs</a> we computed the
href="#example_btn_rem-xs">#rem-xs</a> we computed the
moment of inertia of a square place about the center
to be <InlineEquation equation="I_{C,z} = \\frac{1}{6} m \\ell^2" />.
</p>
Expand Down Expand Up @@ -1077,7 +1077,7 @@ import Row from "../../components/Row.astro"
<p>
To compute the integrals in <a
href="#rem-ec">#rem-ec</a> it is convenient to switch to
<a href="rvy.html">cylindrical coordinates</a>:
<a href="/dyn/coordinate_systems#cylindrical_coordinates">cylindrical coordinates</a>:
</p>
<DisplayEquation equation="\\begin{aligned}x &= r \\cos\\theta \\\\y &= r \\sin\\theta \\\\z &= z.\\end{aligned}\\" />

Expand Down Expand Up @@ -1131,7 +1131,7 @@ import Row from "../../components/Row.astro"
<p>
To compute the integrals in <a
href="#rem-ec">#rem-ec</a> it is convenient to switch to
<a href="rvs.html">spherical coordinates</a>:
<a href="/dyn/coordinate_systems#spherical_coordinates">spherical coordinates</a>:
</p>
<DisplayEquation equation="\\begin{aligned}x &= r \\cos\\theta \\sin\\phi \\\\y &= r \\sin\\theta \\sin\\phi \\\\z &= r \\cos\\phi.\\end{aligned}\\" />

Expand Down Expand Up @@ -1160,7 +1160,7 @@ import Row from "../../components/Row.astro"
<p>
The moments of inertia listed below are all special cases of
the basic shapes given in Section <a
href="#rem-sb">#rem-sb</a>. Other special cases can be
href="#basic_shapes_moi">#rem-sb</a>. Other special cases can be
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Please make the text clearer which section is being referred to here.

easily obtained by similar methods.
</p>

Expand Down
4 changes: 2 additions & 2 deletions src/pages/dyn/momentum.astro
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128: wrong link
652: move container to top of section
657: 404 link error
661: Can we embed this video?

Original file line number Diff line number Diff line change
Expand Up @@ -255,7 +255,7 @@ import Col from "../../components/Col.astro"
</p>
<p>
The cross product of a vector with itself is zero
(<a href="rvi.html#rvi-ez">cross product self-annihilation</a>). So the
(<a href="/dyn/vector_calculus#rvi-ez">cross product self-annihilation</a>). So the
integral of <InlineEquation equation="\\int m\\vec{v}_P \\times \\vec{v}_P \\, dt" /> is a constant, which is
the initial angular momentum. Assuming this is zero, we arrive at the
desired expression.
Expand Down Expand Up @@ -299,7 +299,7 @@ import Col from "../../components/Col.astro"

<Example title="The Rocket Equation" id="rec-xr" solution="false">
<p>
As stated in the warning at <a href="#rec-wm">#rec-wm</a>, the
As stated in the warning at <a href="#warning_rec-wm">#rec-wm</a>, the
relationship between Newton's equations and linear momentum are
not applicable for a system whose mass is changing with time. To use
that relationship, a few things must be changed. Consider the figure
Expand Down
4 changes: 2 additions & 2 deletions src/pages/dyn/particle_kinematics.astro
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There are a couple of places where it says "figure #id" and it is unclear to the reader what figure it is referring to. Please change the displayed text to indicate more clearly the figure being referenced.

line 495 and 500: links to wrong equation
line 510: link doesn't work
lines 656 and 666: #rvi-es reword so its clear you're leaving the pages, or link to an existing equation

Original file line number Diff line number Diff line change
Expand Up @@ -198,7 +198,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
</table>
<p class="figureCaption mt-2">
Velocity and acceleration of various movements. Compare to
Figure <a href="rvc.html#rvc-fp">#rvc-fp</a>.
Figure <a href="/dyn/vector_calculus#derivatives_and_vector_positions">#rvc-fp</a>.
</p>
</PrairieDrawCanvas>

Expand Down Expand Up @@ -551,7 +551,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
Take <InlineEquation equation="\\hat{a}" /> to be a unit vector rotating in the 2D
<InlineEquation equation="\\hat\\imath–\\hat\\jmath" /> plane, making an angle of
\(\theta\) with the \(x\)-axis, as in Figure <a
href="#rkr-f2">#rkr-f2</a>. Then:
href="#rkr-fe-c">#rkr-fe-c</a>. Then:

<DisplayEquation equation="\\hat{a} = \\cos\\theta \\,\\hat\\imath + \\sin\\theta \\,\\hat\\jmath." />

Expand Down
14 changes: 7 additions & 7 deletions src/pages/dyn/rigid_body_kinematics.astro
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Line 471: link does not work
line 855: please change the link to something more specific on that page.
line 926: link does not work

Original file line number Diff line number Diff line change
Expand Up @@ -229,7 +229,7 @@ import DisplayTable from "../../components/DisplayTable.astro"
<DisplayEquation equation="\\begin{aligned} \\vec{r}_Q &amp;= \\vec{r}_P + \\vec{r}_{PQ} \\\\ \\dot{\\vec{r}}_Q &amp;= \\dot{\\vec{r}}_P + \\dot{\\vec{r}}_{PQ} \\\\ \\vec{v}_Q &amp;= \\vec{v}_P + \\vec{\\omega} \\times \\vec{r}_{PQ}, \\end{aligned}" />

where the derivative of <InlineEquation equation="\\vec{r}_{PQ}" /> comes from the <a
href="rkr.html#rkr-ew">rotation formula</a>, given that
href="/dyn/particle_kinematics/#rkr-ew">rotation formula</a>, given that
this offset vector is simply rotating with the rigid
body.
</p>
Expand All @@ -246,7 +246,7 @@ import DisplayTable from "../../components/DisplayTable.astro"
<Warning id="rkg-wc" title="Cross product order">
<p>
Because cross products are <a
href="/dyn/vector_calculus#rvv-wc">not associative</a>, it is very
href="/dyn/vector_calculus#warning_rvv-wc">not associative</a>, it is very
important to compute the centripetal acceleration term
with the parentheses as shown. That is, we must not
compute <InlineEquation equation="(\\vec{\\omega} \\times \\vec{\\omega}) \\times \\vec{r}_{PQ}" />, as this is always zero.
Expand Down Expand Up @@ -415,7 +415,7 @@ import DisplayTable from "../../components/DisplayTable.astro"

Evaluating <a href="#rkg-er">#rkg-er</a> and using the
cross-product expression <a
href="rvv.html#rvv-e9">#rvv-e9</a> now gives the
href="/dyn/vector_calculus#rvv-e9">#rvv-e9</a> now gives the
velocity expression:

<DisplayEquation equation="\\begin{aligned} \\vec{v}_Q &amp;= \\vec{v}_P + \\vec{\\omega} \\times \\vec{r}_{PQ} \\\\ &amp;= \\vec{v}_P + \\omega\\,\\hat{k} \\times \\vec{r}_{PQ} \\\\ &amp;= \\vec{v}_P + \\omega \\,\\vec{r}_{PQ}^\\perp, \\end{aligned}" />
Expand Down Expand Up @@ -937,13 +937,13 @@ import DisplayTable from "../../components/DisplayTable.astro"
We can use conservation of energy to obtain an expression for <InlineEquation equation="\\dot\\theta" />.
</p>
<p>
Using <a href="#">the work-energy theorem</a>:
Using <a href="/dyn/work_and_energy#ren-eco">the work-energy theorem</a>:
<DisplayEquation equation=" W = \\Delta T + \\Delta V " />

</p>
<p>
The rod is sliding along frictionless surfaces, so there are no non-conservative forces present,
and constraint forces do no work (See <a href="#">work done by a constraint force</a>)
and constraint forces do no work (See <a href="/dyn/work_and_energy#ren-wt">work done by a constraint force</a>)
</p>
<p>
Therefore:
Expand Down Expand Up @@ -1014,7 +1014,7 @@ import DisplayTable from "../../components/DisplayTable.astro"

<div slot="solution">
<p>
We added one more assumption to the ones made in <a href="#rkc-xcp">#rkc-xcp</a>. Namely, the rope can only slip.
We added one more assumption to the ones made in <a href="#example_btn_rkc-xcp">#rkc-xcp</a>. Namely, the rope can only slip.
This simply means that the tension is the same at any point in the rope.
</p>
<p>
Expand Down Expand Up @@ -1052,7 +1052,7 @@ import DisplayTable from "../../components/DisplayTable.astro"
<DisplayEquation equation=" T - m_2 g = m_2 a_2 " />
</p>
<p>
We have 3 unknowns, 2 equations. We apply our constraint equation from <a href="#rkc-xcp">#rkc-xcp</a> to relate
We have 3 unknowns, 2 equations. We apply our constraint equation from <a href="#example_btn_rkc-xcp">#rkc-xcp</a> to relate
the accelerations of <InlineEquation equation="m_1" /> and <InlineEquation equation="m_2" />:
<DisplayEquation equation=" 2a_1 + a_2 = 0 " />

Expand Down
6 changes: 4 additions & 2 deletions src/pages/dyn/steering_geometry.astro
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222: container moved to top of section
249: no link
250: video unavailable. If you find a new link to put here, can we embed the video?
269: can we embed the link?

Original file line number Diff line number Diff line change
Expand Up @@ -68,8 +68,10 @@ import DisplayTable from "../../components/DisplayTable.astro"

<CalloutContainer slot="cards">
<CalloutCard title="Reference material">
<li><a href = "/dyn/rigid_body_kinematics#rigid_bodies">Rigid bodies</a></li>
<li><a href = "/dyn/rigid_body_kinematics#constrained_motion">Constrained motion</a></li>
<ul>
<li><a href = "/dyn/rigid_body_kinematics#rigid_bodies">Rigid bodies</a></li>
<li><a href = "/dyn/rigid_body_kinematics#constrained_motion">Constrained motion</a></li>
</ul>
</CalloutCard>
</CalloutContainer>
</Section>
Expand Down
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