A wheel of radius r rolls to the right without slipping on a...

A wheel of radius $r$ rolls to the right without slipping on a horizontal road. Its axle moves at a constant speed $v_{\text {axle }}$ (a) Find the velocities of points $A, B,$ and $C$ with respect to the axle. Express your answers in terms of $v_{\text {axle }}$ and $r,$ as needed. [Hint: In the reference frame of the axle, the wheel is rotating in place at a constant angular speed $\omega .]$ (b) Find the velocities of points $A, B,$ and $C$ with respect to the road. (c) Comment on the velocity of point $C$ with respect to the road.

A wheel of radius $r$ rolls to the right without slipping on a horizontal road. Its axle moves at a constant speed $v_{\text {axle }}$ (a) Find the velocities of points $A, B,$ and $C$ with respect to the axle. Express your answers in terms of $v_{\text {axle }}$ and $r,$ as needed. [Hint: In the reference frame of the axle, the wheel is rotating in place at a constant angular speed $\omega .]$ (b) Find the velocities of points $A, B,$ and $C$ with respect to the road.
(c) Comment on the velocity of point $C$ with respect to the road.

Key Concepts

Rolling Without Slipping

This concept refers to the situation where a wheel rolls on a surface such that the point of contact with the surface has zero velocity relative to the surface at each instant. It implies a fixed relationship between the linear speed of the axle (or center of mass) and the angular speed, given by v = ?r, and ensures that there is no skidding or slipping between the wheel and the ground.

Angular Velocity

Angular velocity (?) is a measure of how fast the wheel rotates around its center. In rolling motion, it directly relates to the linear speed via the equation v = ?r. The knowledge of angular velocity is essential for determining the velocities of specific points on the wheel relative to any reference frame, as each point's velocity is a result of both rotation and translation.

Relative Motion

Relative motion examines how velocities are perceived from different reference frames. In this problem, it is crucial to analyze the movement of points on the wheel with respect to both the moving axle and the stationary road. The principle of relative motion involves vector addition of the translational velocity of the axle and the rotational velocity of points on the wheel.

Reference Frames

The use of different reference frames, such as the axle frame and the road (inertial) frame, is fundamental to problem solving in rotational dynamics. By moving to the axle frame, the wheel appears to be rotating in place, simplifying the calculation of rotational contributions. Transforming back to the inertial frame (the road) requires adding the axle’s translational velocity to these relative motions.

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