Question

how many revolutions does the object make during the first 4 s? express your answer with the appropriate units. view available hint(s) provide feedback 1 of 1 ω (rad/s) 20 10 0 0 1 2 3 4 t (s)

Explanation:

Step1: Calculate angular - displacement for the first 2 s

The angular - velocity $\omega$ as a function of time $t$ is linear from $t = 0$ to $t=2\ s$. The average angular - velocity in the first 2 s, $\omega_{avg1}=\frac{0 + 20}{2}=10\ rad/s$. Using the formula $\theta_1=\omega_{avg1}t_1$, with $t_1 = 2\ s$, we get $\theta_1=10\times2 = 20\ rad$.

Step2: Calculate angular - displacement for the next 2 s

From $t = 2\ s$ to $t = 4\ s$, the angular - velocity is constant, $\omega_2=20\ rad/s$. Using the formula $\theta_2=\omega_2t_2$, with $t_2 = 2\ s$, we get $\theta_2=20\times2=40\ rad$.

Step3: Calculate total angular - displacement

The total angular - displacement $\theta=\theta_1+\theta_2=20 + 40=60\ rad$.

Step4: Convert angular - displacement to revolutions

Since 1 revolution $=2\pi\ rad$, the number of revolutions $n=\frac{\theta}{2\pi}$. Substituting $\theta = 60\ rad$, we get $n=\frac{60}{2\pi}=\frac{30}{\pi}\approx9.55$ revolutions.

Answer:

$\frac{30}{\pi}\text{ revolutions}\approx9.55\text{ revolutions}$