A wheel of radius R can rotate freely around its axis. Drive belts moving at a speed v are pressed against the side surface of the wheel at a distance h from the axis of rotation. Determine the steady-state angular velocity of the wheel if its contact with the drive belt occurs only along the rim.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

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    <h3> Statement </h3>
    <p>
        $2.8.37^*.$ A wheel of radius $R$ can rotate freely around its axis. Drive belts moving at a speed $v$ are pressed against the side surface of the wheel at a distance $h$ from the axis of rotation. Determine the steady-state angular velocity of the wheel if its contact with the drive belt occurs only along the rim.

    <h3>Solution</h3>
    <p>

In order for the wheel to move it is necessary that a moment of force associated with the frictional force from drive belts is transmitted, which is directed along the axis of rotation.

  <p>
  Thus, the momentum will make an angle with velocity $v$ equal to $\pi-\alpha$
    </p>
  <p>
      Projecting onto the linear rotation speed of the wheel:
      $$u=v\cos(\frac{\pi}{2}-\alpha)=v\sin\alpha\quad(1)$$
      From the drawing
      $$\sin\alpha = \frac{h}{R}$$
      Substituting into $(1)$
      $$u=v \frac{h}{R}$$
      From where we find the linear velocity, given that the wheel is circular
      $$\boxed{\omega=\frac{u}{R}=v \frac{h}{R^2}}$$
    </p>


    <h4>Answer</h4>
    <p>
        $$\omega=\frac{hv}{R^2}$$
    </p>

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