How Does Angular Velocity Change with Time in a Rotating Disk?

[kappcity06]

Figure 10-21b is a graph of the angular position of the rotating disk of Fig. 10-21a.

Figure 10-21

(a) Is the angular velocity of the disk positive, negative, or zero at t = 1 s?
positive
negative
zero
(b) Is the angular velocity of the disk positive, negative, or zero at t = 2 s?
positive
negative
zero
(c) Is the angular velocity of the disk positive, negative, or zero at t = 3 s?
positive
negative
zero
(d) Is the angular acceleration positive or negative?
positive
negative

 

[andrevdh]

The angular velocity is per definition

$$\omega = \frac{d\theta}{dt}$$

which means that the angular velocity is the gradient of the tangential to the angular position versus time curve.

 

[kyle8921]

I can confirm that the graph in Figure 10-21b represents the angular position of the rotating disk shown in Figure 10-21a. Based on the graph, we can determine the angular velocity and acceleration of the disk at different time points.

(a) At t = 1 s, the angular position of the disk is increasing, indicating that the disk is rotating in the clockwise direction. Therefore, the angular velocity is positive.

(b) At t = 2 s, the angular position of the disk is decreasing, indicating that the disk is rotating in the counterclockwise direction. Therefore, the angular velocity is negative.

(c) At t = 3 s, the angular position of the disk is constant, indicating that the disk is not rotating. Therefore, the angular velocity is zero.

(d) Based on the graph, we can see that the slope of the line increases from t = 1 s to t = 2 s, indicating a positive change in angular velocity. Therefore, the angular acceleration is positive.