Chapter 7, Problem 1CQ

A disk rotates about an axis through its center. Point A is located on its rim and point B is located exactly halfway between the center and the rim. What is the ratio of (a) the angular velocity ω_A to that of ω_B and (b) the tangential velocity v_A to that of v_B?

To determine

The ratio of the angular velocity of point A and point B.

Answer to Problem 1CQ

The ratio of the angular velocity of point A and point B will be $1$.

Explanation of Solution

Given Info: The disc is rotating through its center. The point $$ is on the rim of the disc and point B is half way between point A and the center.

Explanation:

When a rigid body is rotating about any axis, all the points in the rigid body have the same angular speed.

Since the dick can be considered to be a rigid body, all the points in it will have equal angular velocity. So the ratio of the angular velocity of point A and point B will be $1$.

Conclusion: The ratio of the angular velocity of point A and point B will be $1$.

Since Andrea and Chunk are in the same circular platform their angular speed cannot be different so option (a) and (c) are not correct.

To determine

The ratio of the tangential velocity of point A and point B.

Answer to Problem 1CQ

The ratio of the tangential velocity of point A and point B $2$.

Explanation of Solution

Given Info: The disc is rotating through its center. The point $A$ is on the rim of the disc and point B is half way between point A and the center.

Explanation:

Tangential velocity of a rotating particle is given by,

$v_t=\omega r$

• $\omega$ is the angular speed
• $r$ is the radius

The angular speed of point A and point B is the same. The tangential velocity ratio of point A and point B depends on the ratio of their distance from the center.

$$ $\frac{v_A}{v_B}=\frac{r_A}{r_B}$

• $v_A$ is the tangential velocity of point A
• $r_A$ is the distance between point A and the center
• $v_B$ is the tangential velocity of point B
• $r_B$ is the distance between point B and the center

Since the point $A$ is on the rim of the disc and point B is half way between point A and the center,

$\frac{r_A}{r_B}=2$

Hence the ratio of tangential velocities will be,

$\frac{v_A}{v_B}=2$

Conclusion: The ratio of the tangential velocity of point A and point B $2$.