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| <title>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.</title> | | <title>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.</title> |
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| <h3 id="back-link"><a href="../../#2.8">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.8">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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. | | $2.8.37^*.$ 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. |
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| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $2.8.37^*$ | | For problem $2.8.37^*$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| <center> | | <center> |
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| <img src="2.8.37.png" | | <img src="2.8.37.png" |
| loading="lazy" width="300" /> | | loading="lazy" width="300" /> |
| <figcaption> | | <figcaption> |
| Direction of the torque of the rotational force | | Direction of the torque of the rotational force |
| </figcaption> | | </figcaption> |
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| <p> | | <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. | | 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> | | </p> |
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| <p> | | <p> |
| Thus, the momentum will make an angle with velocity $v$ equal to $\pi-\alpha$ | | Thus, the momentum will make an angle with velocity $v$ equal to $\pi-\alpha$ |
| </p> | | </p> |
| <p> | | <p> |
| Projecting onto the linear rotation speed of the wheel: | | Projecting onto the linear rotation speed of the wheel: |
| $$u=v\cos(\frac{\pi}{2}-\alpha)=v\sin\alpha\quad(1)$$ | | $$u=v\cos(\frac{\pi}{2}-\alpha)=v\sin\alpha\quad(1)$$ |
| From the drawing | | From the drawing |
| $$\sin\alpha = \frac{h}{R}$$ | | $$\sin\alpha = \frac{h}{R}$$ |
| Substituting into $(1)$ | | Substituting into $(1)$ |
| $$u=v \frac{h}{R}$$ | | $$u=v \frac{h}{R}$$ |
| From where we find the linear velocity, given that the wheel is circular | | From where we find the linear velocity, given that the wheel is circular |
| $$\boxed{\omega=\frac{u}{R}=v \frac{h}{R^2}}$$ | | $$\boxed{\omega=\frac{u}{R}=v \frac{h}{R^2}}$$ |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\omega=\frac{hv}{R^2}$$ | | $$\omega=\frac{hv}{R^2}$$ |
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