Question

Figure $\mathbf{\text{10-22\hspace{0.17em}b}}$is a graph of the angular position of the rotating disk of Fig.$\mathbf{\text{10-22\hspace{0.17em}a}}$. Is the angular velocity of the disk positive, negative, or zero at (a)$\mathbf{t}\mathbf{\text{=1s}}$, (b)$\mathbf{t}\mathbf{\text{=2s}}$, and (c)$\mathbf{t}\mathbf{\text{=3s}}$? (d) Is the angular acceleration positive or negative?

Short answer

1. Angular velocities of the disk, at point$\mathrm{t}=1\text{\hspace{0.17em}s}$is positive.
2. Angular velocities of the disk, at point$\mathrm{t}=2\text{\hspace{0.17em}s}$is zero.
3. Angular velocities of the disk, at point$\mathrm{t}=3\text{s}$is negative.
4. The angular acceleration is negative.

Step-by-step solution

Step 1: Given data

Graph of angular position vs time.

Determining the concept

Find the sign of the slope of the graph which represents the sign of angular velocity. Conclude about the angular acceleration from the nature of the graph.

(a) Determining if the angular velocity of the disk is positive, negative, or zero at  t= 1 s

The given graph is an angular position vs time graph, so the slope of the graph is angular velocity.

At point $\mathrm{t}=1$, it is positive because the slope is positive.

(b) Determining if the angular velocity of the disk is positive, negative, or zero at t= 2 s 

At point $\mathrm{t}=2$, it is zero because the slope is zero.

(c) Determining if the angular velocity of the disk is positive, negative, or zero at t= 3 s

At point $\mathrm{t}=3$, it is negative, because slope is negative.

(d) Determining if the angular acceleration is positive or negative

The given fact that the original function is concave down, its second derivative must be negative.