Source: High school physics (Chinese)
Problem Sets:
Problem
A rigid body undergoes uniformly accelerated rotation about a fixed axis. The angular acceleration is $\alpha = 2.0 \text{ rad/s}^2$, and the initial angular velocity at $t=0$ is $\omega_0 = 4 \text{ rad/s}$.
- What is the angular velocity of the rigid body at $t = 3 \text{ s}$?
- How many revolutions does the rigid body turn through in the first $3 \text{ s}$?
- At $t = 3 \text{ s}$, what are the linear velocity and centripetal acceleration of a point $10 \text{ cm}$ from the axis of rotation?
[Q1] $\omega = 10 \text{ rad/s}$ [Q2] $N = 3.34 \text{ revolutions}$ [Q3] $v = 1.0 \text{ m/s}$, $a_c = 10 \text{ m/s}^2$
The motion is described by the kinematic equations for uniformly accelerated rotation. [Q1] The final angular velocity is $\omega_f = \omega_0 + \alpha t$.
$$ \omega(3\text{s}) = 4 \text{ rad/s} + (2.0 \text{ rad/s}^2)(3 \text{ s}) $$[Q2] The angular displacement is $\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$. The number of revolutions is $N = \Delta\theta / (2\pi)$.
$$ \Delta\theta = (4 \text{ rad/s})(3 \text{ s}) + \frac{1}{2}(2.0 \text{ rad/s}^2)(3 \text{ s})^2 $$[Q3] The linear velocity is $v = r\omega$ and the centripetal acceleration is $a_c = r\omega^2$. The radius is $r = 10 \text{ cm} = 0.1 \text{ m}$.
$$ v = (0.1 \text{ m})\omega(3\text{s}) $$ $$ a_c = (0.1 \text{ m})(\omega(3\text{s}))^2 $$