Question

If a wheel starts at an angular velocity of 7.5 rad/s and it comes to rest in 22 seconds, what is the wheel’s angular acceleration? What is the angular displacement of the wheel?

Answer 1

Given:

The initial angular speed is,

$\omega_0=7.5\text{ rad/s}$

The final angular speed is,

$\omega=0$

The time taken to stop is,

$t=22\text{ s}$

To find:

the wheel’s angular acceleration and the angular displacement of the wheel

Explanation:

From kinematics laws for rotational motion, we get,

$\begin{gathered} \omega=\omega_0+\alpha t \\ \end{gathered}$

here, the angular acceleration is,

$\begin{gathered} \alpha=\frac{\omega-\omega_0}{t} \\ =\frac{0-7.5}{22} \\ =-0.34\text{ rad/s}^2 \end{gathered}$

Now,

$\begin{gathered} \omega^2-\omega_0^2=2\alpha\theta \\ \theta=\frac{\omega^2-\omega_0^2}{2\alpha} \end{gathered}$

The angular displacement is,

$\begin{gathered} \theta=\frac{0^2-7.5^2}{2\times(-0.34)} \\ \theta=82.7\text{ rad} \end{gathered}$

Hence, the angular acceleration is

$-3.4\text{ rad/s}^2$

The angular displacement is 82.7 rad.