Question

A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 10 rev/s; 60 revolutions later, its angular speed is 15 rev/s. Calculate
(a) the angular acceleration,
(b) the time required to complete the 60 revolutions,
(c) the time required to reach the 10 rev/s angular speed, and
(d) the number of revolutions from rest until the time the disk reaches the 10 rev/s angular speed.

Answer 1

Explanation:

Given:

$\omega_0$ = 10 rev/s = $20\pi\:\text{rad/s}$

$\omega$ = 15 rev/s = $30\pi\:\text{rad/s}$

$\theta$ = 60 rev = $120\pi\:\text{rads}$

a) the angular acceleration $\alpha$ is given by

$\alpha = \dfrac{\omega^2 - \omega_0^2}{2\theta}$

$\:\:\:\:\:\:\:=\dfrac{(30\pi)^2 - (20\pi)^2}{240\pi} = 6.5\:\text{rad/s}^2$

b) $t = \dfrac{\omega - \omega_0}{\alpha} = \dfrac{30\pi - 20\pi}{6.5} = 4.8\:\text{s}$

c) $t = \dfrac{\omega - \omega_0}{\alpha}$

$=\dfrac{20\pi - 0}{6.5} = 9.7\:\text{s}$

d)$\theta = \frac{1}{2}\alpha t^2$

$\:\:\:\:\:\:\:=\frac{1}{2}(6.5\:\text{rad/s}^2)(9.7\:\text{s})^2 = 305.8\:\text{rad}$

$\:\:\:\:\:\:\:= 48.7\:\text{revs}$