Linear and Angular Velocity - Questions 9 Help

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The angular velocity refers rate at which an point rotates about the axis of rotation. It is typically expressed in radians/second, revolutions/minute (rpm), or degrees/second. The angular velocity can be related to the linear velocity and the distance the point is from the axis of rotation. See How to Think About This Situation for more details.

There are four similar versions of this question. Here is one of the versions:

Version 1:
In the World Peace Lab, a bucket filled with peas is whirled in a horizontal circle with varying radii and linear velocities. The circle radius (R) and linear velocity (v) for Case A and Case B are shown. How does the angular velocity of the two buckets compare to one another?

The angular velocity in Case A is ______ the angular velocity in Case B.
the same as
two times greater than
four times greater than
two times less than
four times less than

Angular velocity is the rate at which a point on the turntable rotates about its axis. This rate is measured as a change in the angular position divided by a change in time, Δθ/Δt. The linear velocity refers to a distance traveled per unit of time. It is sometimes referred to as the tangential velocity for an object moving in a circle.

These two velocity quantities are related. The linear velocity (v) for an object rotating in a circle of radius R is related to the angular velocity (ω) and the radius of the circle (R). The equation relating these quantities is

v = ω*R.

Since this question pertains to angular velocity, it is often useful to re-arrange the equation to the form of

ω = v/R.

This form of the equation leads to the claim that the angular velocity is directly proportional to the linear velocity and inversely proportional to the radius. In this question, one of the buckets has twice the linear velocity of the other bucket. This would lead to twice the angular velocity. But the same bucket completes a circular arc with twice the radius. Since angular velocity and radius are inversely proportional, twice the radius would also lead to one-half the angular velocity. So here are two factors that have opposite effects upon the angular velocity. When combined, these two factors and their effects offset each other and both objects end up with the same angular velocity.

One way to conceptualize this is to think of angular velocity as the rate at which the angular position changes (Δθ/Δt). An object that is traveling twice as fast will change the angular position at twice the rate. However since the same object is traveling along a circular path with twice the radius and twice the circuference, it must have twice the linear speed to keep up with the slower object that is traveling along a circular path that has one-half the circumference. Each object would complete a full 360-degree (2•π radians) revolution in the same amount of time.

Try these links to The Physics Classroom Tutorial for more help with understanding the concept of angular position and displacement:

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