Question

1. A wheel starts from rest and accelerates uniformly. As the rate of rotation changes from 20 to 50 rpm the wheel makes 40 revolutions. Find: (a) the angular acceleration; (b) the number of revolutions completed by the time the wheel reaches 20 rpm. 

Answer 1

(a) We can find the angular acceleration from the kinematic equation:

$$\omega_f^2=\omega_i^2+2\alpha\Delta \theta,$$ $$\alpha=\dfrac{\omega_f^2-\omega_i^2}{2\Delta\theta},$$ $$\alpha=\dfrac{(50\ \dfrac{rev}{min}\cdot\dfrac{1\ min}{60\ s}\cdot2\pi\ rad)^2-(20\ \dfrac{rev}{min}\cdot\dfrac{1\ min}{60\ s}\cdot2\pi\ rad)^2}{2\cdot40\ rev\cdot2\pi\ rad},$$ $$\alpha=0.045\ \dfrac{rad}{s^2}.$$

(b) We can find the number of revolutions completed by the time the wheel reaches 20 rpm from the same kinematic equation:

$$\Delta\theta=\dfrac{\omega_f^2-\omega_i^2}{2\alpha},$$ $$\Delta\theta=\dfrac{(20\ \dfrac{rev}{min}\cdot\dfrac{1\ min}{60\ s}\cdot2\pi\ rad)^2-0}{2\cdot0.045\ \dfrac{rad}{s^2}}=48.9\ rad\cdot \dfrac{1\ rev}{2\pi\ rad}=7.8\ rad.$$