The angular velocity of the coil is \omega , the radius of the inner cylinder is r, and the radius of the outer cylinders is R. What are the velocities of the coil and load axis relative to the ground?

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#1.5">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.5.2.$ The angular velocity of the coil is $\omega$, the radius of the inner cylinder is $r$, and the radius of the outer cylinders is $R$. What are the velocities of the coil and load axis relative to the ground?

    <h3>Solution</h3>
    <p>
        <p>Consider the instantaneous axis of rotation passing through the point $O_1$, then the speed of the spool axis is</p>

$$\fbox{$v_\text{reel} = \omega R$}$$

To find the speed of the load, consider the speed point $O$ as a superposition of the velocities of translational motion with the speed of the wheel center and the speed of rotation point $O$ relative to the wheel center.

$$\vec{v_O} = \vec{v}_{in} + \vec{v}_{out}$$

$$v_O = v_{out} – v_{in} = \omega R - \omega r = \omega\cdot (R - r)$$

Any point of the thread, due to its inextensibility, has the same speed, therefore, the speed of the load

$$\fbox{$v_\text{rope} = v_O = \omega\cdot (R - r)$}$$

    <h4>Answer</h4>
    <p>
        $$v_\text{reel} = \omega R$$

$$v_\text{cargo} = \omega (R - r)$$

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