A uniform heavy rope, the ends of which are fixed on the same vertical, coversa weightless hoop. How fast does the hoop fall when released?

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#2.7">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $2.7.19^*.$ A uniform heavy rope, the ends of which are fixed on the same vertical, covers

a weightless hoop. How fast does the hoop fall when released?

    </p>

    <h3>Solution</h3>
    <p>

Hoof's moment of inertia
$$I_0=mR^2$$
Parallel axis theorem for point $A$
$$I_A=2mR^2$$
We express the external gravity moment $mg$, through the moment of inertia and angular acceleration
$$I_A\cdot\varepsilon =mgR\Leftrightarrow 2mR^2\cdot\varepsilon =mgR\quad(1)$$
When there is no slippage
$$a = \varepsilon R$$
Then, the $(1)$ expression could be rewritten as
$$2a = g\Rightarrow\boxed{a=\frac{1}{2}g}$$

    <h4>Answer</h4>
    <p>
        $$a=g/2$$
    </p>
    <p style="text-align: right; font-style: italic; font-size: 14;">
      Almaskhan Arsen<br>
    </p>


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