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| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
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| <title>The log, resting its lower end in the corner between the wall and the ground, touches the bottom of the truck at a height H from the ground. Find the angular velocity of the log as a function of the angle \alpha between it and the horizontal if the truck is moving away from the wall at speed v.</title> | | <title>The log, resting its lower end in the corner between the wall and the ground, touches the bottom of the truck at a height H from the ground. Find the angular velocity of the log as a function of the angle \alpha between it and the horizontal if the truck is moving away from the wall at speed v.</title> |
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| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.17.$ The log, resting its lower end in the corner between the wall and the ground, touches the bottom of the truck at a height $H$ from the ground. Find the angular velocity of the log as a function of the angle $\alpha$ between it and the horizontal if the truck is moving away from the wall at speed $v$. | | $1.5.17.$ The log, resting its lower end in the corner between the wall and the ground, touches the bottom of the truck at a height $H$ from the ground. Find the angular velocity of the log as a function of the angle $\alpha$ between it and the horizontal if the truck is moving away from the wall at speed $v$. |
| </p> | | </p> |
| <center> | | <center> |
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| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.17$ | | For problem $1.5.17$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>NO: Before viewing the solution to this problem, I recommend that you familiarize yourself with the solution <a href="../1.5.16">1.5.16</a></p> | | <p>NO: Before viewing the solution to this problem, I recommend that you familiarize yourself with the solution <a href="../1.5.16">1.5.16</a></p> |
| <p>At time $t$ the car will travel the distance $vt$ and the bottom will have horizontal coordinate $x$</p> | | <p>At time $t$ the car will travel the distance $vt$ and the bottom will have horizontal coordinate $x$</p> |
| <p> | | <p> |
| $$ x = H \tan \alpha\;(1) $$ | | $$ x = H \tan \alpha\;(1) $$ |
| </p> | | </p> |
| <p>Let's consider the change in angle $\alpha = \arctan \frac{H}{x}$ over a small period of time $dt$</p> | | <p>Let's consider the change in angle $\alpha = \arctan \frac{H}{x}$ over a small period of time $dt$</p> |
| <p> | | <p> |
| $$ d\alpha = d\left(\arctan \frac{H}{x}\right) $$ | | $$ d\alpha = d\left(\arctan \frac{H}{x}\right) $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ d\alpha = \frac{1}{1+\frac{H^2}{x^2}} \frac{H}{x^2} $$ | | $$ d\alpha = \frac{1}{1+\frac{H^2}{x^2}} \frac{H}{x^2} $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ d\alpha = dx\frac{H}{x^2+H^2} $$ | | $$ d\alpha = dx\frac{H}{x^2+H^2} $$ |
| </p> | | </p> |
| <p>Considering that $\omega = \frac{d \alpha}{dt}$ and $v = \frac{dx}{dt}$</p> | | <p>Considering that $\omega = \frac{d \alpha}{dt}$ and $v = \frac{dx}{dt}$</p> |
| <p> | | <p> |
| $$ \omega = \frac{vH}{x^2+H^2} $$ | | $$ \omega = \frac{vH}{x^2+H^2} $$ |
| </p> | | </p> |
| <p>Substitute $(1)$</p> | | <p>Substitute $(1)$</p> |
| <p> | | <p> |
| $$ \omega = \frac{v}{H(1 + \tan^2 \alpha)} $$ | | $$ \omega = \frac{v}{H(1 + \tan^2 \alpha)} $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ \fbox{$\omega = \frac{v \sin^2 \alpha}{H}$} $$ | | $$ \fbox{$\omega = \frac{v \sin^2 \alpha}{H}$} $$ |
| </p> | | </p> |
| | | |
| | | |
| <h3> Alternative solution: </h3> | | <h3> Alternative solution: </h3> |
| <p>In time $\Delta t$ the truck will travel a distance</p> | | <p>In time $\Delta t$ the truck will travel a distance</p> |
| <p> | | <p> |
| $$ \Delta l=\upsilon \Delta t $$ | | $$ \Delta l=\upsilon \Delta t $$ |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.17/draw.png" | | <img src="https://savchenkosolutions.com/1/1.5.17/draw.png" |
| loading="lazy" width="280" /> | | loading="lazy" width="280" /> |
| <figcaption> | | <figcaption> |
| Log and truck | | Log and truck |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p>The log will drop slightly, turning at a small angle $\Delta \alpha$. If the length of the arc of rotation is $\Delta x$, then by definition of the radian measure of angle</p> | | <p>The log will drop slightly, turning at a small angle $\Delta \alpha$. If the length of the arc of rotation is $\Delta x$, then by definition of the radian measure of angle</p> |
| <p> | | <p> |
| $$ \Delta \alpha=\frac{\Delta x }{R} $$ | | $$ \Delta \alpha=\frac{\Delta x }{R} $$ |
| </p> | | </p> |
| <p>Angular velocity is defined as</p> | | <p>Angular velocity is defined as</p> |
| <p> | | <p> |
| $$ \omega=\frac{\Delta \alpha }{\Delta t }=\frac{\Delta x }{R\Delta t } $$ | | $$ \omega=\frac{\Delta \alpha }{\Delta t }=\frac{\Delta x }{R\Delta t } $$ |
| </p> | | </p> |
| <p>Но $\Delta x =\upsilon \Delta t\sin \alpha$.</p> | | <p>Но $\Delta x =\upsilon \Delta t\sin \alpha$.</p> |
| <p> | | <p> |
| $$ \omega=\frac{\upsilon \Delta t\sin \alpha }{R\Delta t }=\frac{\upsilon\sin \alpha }{R} $$ | | $$ \omega=\frac{\upsilon \Delta t\sin \alpha }{R\Delta t }=\frac{\upsilon\sin \alpha }{R} $$ |
| </p> | | </p> |
| <p>We can also write that</p> | | <p>We can also write that</p> |
| <p> | | <p> |
| $$ H=R\sin \alpha $$ | | $$ H=R\sin \alpha $$ |
| </p> | | </p> |
| <p>With this in mind</p> | | <p>With this in mind</p> |
| <p> | | <p> |
| $$ \omega=\frac{\upsilon\sin^2 \alpha}{H} $$ | | $$ \omega=\frac{\upsilon\sin^2 \alpha}{H} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\omega = \frac{v \sin^2 \alpha}{H}$$ | | $$\omega = \frac{v \sin^2 \alpha}{H}$$ |
| </p> | | </p> |
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