Question

Two points are located on a rigid wheel that is rotating with decreasing angular velocity about a fixed axis. Point A is located in the rim of the wheel and pint B is halfway between the rim and the axis. Which one of the following statements concerning this situation is true?
a.the angular velocity at point A is greater than that of point B.
b.Both points have the same centripetal acceleration.
c.Both points have the same tangential acceleration.
d.Both points have the same instantaneous angular velocity.
e.Each second, point A turns through a greater angle than point B.

Answer 1

d.Both points have the same instantaneous angular velocity.

Explanation:

Let's analyze each option one by one:

a.the angular velocity at point A is greater than that of point B.  --> FALSE. In fact, for a rigid body (such as the rigid wheel in this problem), all the points along the body rotates with the same angular velocity. This is because the angular velocity is defined as

$\omega=\frac{\theta}{t}$

where $\theta$ is the angular displacement and t is the time taken. Therefore, all the points rotate by the same angle $\theta$ in the same time t.

b.Both points have the same centripetal acceleration.  --> FALSE. The centripetal acceleration of a point along the disk is given by

$a_c=\omega^2 r$

where $\omega$ is the angular velocity and r is the distance of the point from the axis of rotation. Here A is located farther away from the axis with respect to B, so it will have a greater centripetal acceleration.

c.Both points have the same tangential acceleration.  --> FALSE. The tangential acceleration is given by

$a_t = \alpha r$

where $\alpha$ is the angular acceleration. $\alpha$ is the same for every point for a rigid body, but $r$ is not (A is located farther away), so A and B do not have same tangential acceleration.

d.Both points have the same instantaneous angular velocity.  --> TRUE. This is true for what we said at point A).

e.Each second, point A turns through a greater angle than point B. --> FALSE. The two points cover the same angular displacement in the same time, since they have same angular velocity.

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