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.

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    <h3 id="back-link"><a href="/#1.5">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <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 $v$.

    <h3>Solution</h3>
    <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>

At time $t$ the car will travel the distance $vt$ and the bottom will have horizontal coordinate $x$

$$x = H \tan \alpha\\;(1)$$

Let's consider the change in angle $\alpha = \arctan \frac{H}{x}$ over a small period of time $dt$

$$d\alpha = d\left(\arctan \frac{H}{x}\right)$$

$$d\alpha = \frac{1}{1+\frac{H^2}{x^2}} \frac{H}{x^2}$$

$$d\alpha = dx\frac{H}{x^2+H^2}$$

Considering that $\omega = \frac{d \alpha}{dt}$ and $v = \frac{dx}{dt}$

$$\omega = \frac{vH}{x^2+H^2}$$

Substitute $(1)$

$$\omega = \frac{v}{H(1 + \tan^2 \alpha)}$$

$$\fbox{$\omega = \frac{v \sin^2 \alpha}{H}$}$$

Alternative solution

In time $\Delta t$ the truck will travel a distance

$$\Delta l=\upsilon \Delta t$$

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

$$\Delta \alpha=\frac{\Delta x }{R}$$

Angular velocity is defined as

$$\omega=\frac{\Delta \alpha }{\Delta t }=\frac{\Delta x }{R\Delta t }$$

Но $\Delta x =\upsilon \Delta t\sin \alpha$.

$$\omega=\frac{\upsilon \Delta t\sin \alpha }{R\Delta t }=\frac{\upsilon\sin \alpha }{R}$$

We can also write that

$$H=R\sin \alpha$$

With this in mind

$$\omega=\frac{\upsilon\sin^2 \alpha}{H}$$

    <h4>Answer</h4>
    <p>
        $$\omega = \frac{v \sin^2 \alpha}{H}$$
    </p>


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