Question

the figure shows the angular - velocity - versus - time graph for a particle moving in a circle. (figure 1) part a how many revolutions does the object make during the first 4 s? express your answer with the appropriate units. view available hint(s)

Explanation:

Step1: Find area under angular - velocity graph

The angular displacement $\theta$ is given by the area under the $\omega - t$ graph. The graph has a triangular part from $t = 0$ to $t=2\ s$ and a rectangular part from $t = 2$ to $t = 4\ s$.
For the triangular part: The area of a triangle $A_1=\frac{1}{2}\times base\times height$. Here, base $b = 2\ s$ and height $h=20\ rad/s$. So $A_1=\frac{1}{2}\times2\times20 = 20\ rad$.
For the rectangular part: The area of a rectangle $A_2=length\times width$. Here, length $l=(4 - 2)\ s=2\ s$ and width $w = 20\ rad/s$. So $A_2=2\times20=40\ rad$.
The total angular displacement $\theta=A_1 + A_2=20 + 40=60\ rad$.

Step2: Convert angular displacement to revolutions

We know that $1$ revolution $=2\pi\ rad$. Let $n$ be the number of revolutions. Then $n=\frac{\theta}{2\pi}$. Substituting $\theta = 60\ rad$, we get $n=\frac{60}{2\pi}=\frac{30}{\pi}\approx9.55$ revolutions.

Answer:

$9.55$ revolutions