Chapter 9 - Rotation of Rigid Bodies - Problems - Exercises - Page 294: 9.5

(a) $\omega (t) = (0.400~rad/s) + (0.0360~rad/s^3) ~t^2$ (b) $\omega = 0.400~rad/s$ (c)$\omega = 1.30~rad/s$ $\omega_{ave} = 0.46~rad/s$ The average of the instantaneous angular velocities is $0.85 ~rad/s$. The average angular velocity is not the same as the average of the instantaneous velocities because the angular acceleration is not constant.

(a) $\theta (t) = \gamma ~t + \beta ~t^3$ $\omega (t) = \frac{d\theta }{dt} = \gamma + 3\beta ~t^2$ $\omega (t) = (0.400~rad/s) + 3~(0.0120~rad/s^3) ~t^2$ $\omega (t) = (0.400~rad/s) + (0.0360~rad/s^3) ~t^2$ (b) When t = 0: $\omega = (0.400~rad/s) + (0.0360~rad/s^3)(0)^2$ $\omega = 0.400~rad/s$ (c) When t = 5.00 s: $\omega = (0.400~rad/s) + (0.0360~rad/s^3)(5.00~s)^2$ $\omega = 1.30~rad/s$ When t = 0: $\theta = (0.400~rad/s)(0) + (0.0120~rad/s^3)(0)^2$ $\theta = 0$ When t = 5.00 s: $\theta = (0.400~rad/s)(5.00~s) + (0.0120~rad/s^3)(5.00~s)^2$ $\theta = 2.3~rad$ We can find the average angular velocity. $\omega_{ave} = \frac{\theta_2-\theta_1}{t}$ $\omega_{ave} = \frac{2.3~rad-0}{5.00~s}$ $\omega_{ave} = 0.46~rad/s$ We can find the average of the instantaneous angular velocities. $average = \frac{\omega_1+\omega_2}{2}$ $average = \frac{0.400~rad/s+1.30~rad/s}{2}$ $average = 0.85~rad/s$ The average angular velocity is not the same as the average of the instantaneous velocities because the angular acceleration is not constant. The angular velocity is not constant and it does not change linearly with time. (We can see that $\omega (t)$ includes a term with $t^2$, thus the angular acceleration is not constant.)