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Chapter 16: Problem 54

The crate is transported on a platform which rests on rollers, each having a radius \(r\). If the rollers do not slip, determine their angular velocity if the platform moves forward with a velocity \(\mathbf{v}\).

Short Answer

Expert verified
The angular velocity \( \omega \) of the rollers can be calculated by the equation \( \omega = \frac{v}{r} \), where \( v \) is the linear velocity and \( r \) is the radius of the rollers.

Step by step solution

01

Understanding given Information and Required

From the exercise, it is known that the platform moves with a velocity \( v \), the radius of each roller is \( r \) and the rollers do not slip. The objective is to find the angular velocity \( \omega \) of the rollers.
02

Apply the Relation between Translational and Rotational Motion

For no-slip conditions, the relation between the linear velocity of the platform and the angular velocity of the roller can be represented as \( v = r\omega \). This equation states that the linear speed of the platform is equal to the product of the radius of the roller and its angular velocity. Hence, we can rearrange this equation to isolate \( \omega \) by dividing both sides by \( r \). This gives us \( \omega = \frac{v}{r} \). This equation is used to calculate the angular velocity of the rollers.
03

Calculate the Angular Velocity

With the given knowledge of the linear velocity \( v \) and the radius of the roller \( r \), one can substitute these values into the equation \( \omega = \frac{v}{r} \) to calculate the required angular velocity \( \omega \).

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