Question

A wheel rotates clockwise about its central axis with an angular momentum of $\mathbf{600}\mathbf{}\mathbf{kg}\mathbf{.}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{s}$. At time t=0, a torque of magnitude 50 N.mis applied to the wheel to reverse the rotation. At what time tis the angular speed zero?

Short answer

The time at which angular speed is zero is 12s.

Step-by-step solution

Step 1: Given Data

$$\begin{gathered} \mathrm{L}=600\mathrm{kg}.{\mathrm{m}}^2/\mathrm{s} \\ \mathrm{\tau }=50\mathrm{N}.\mathrm{m} \end{gathered}$$

Determining the concept

Torque is equal to the time rate of change of linear momentum. Using this formula, find the time at which the momentum is zero. Zero momentum implies zero angular velocity.

Formula is as follow:

$\mathrm{\tau }=\frac{\mathrm{L}}{\mathrm{t}}$

where,data-custom-editor="chemistry" $\mathrm{\tau }$is torque, L is angular momentum and t is time.

Determining the time at which angular speed is zero

To find the time, use the following formula:

$$\begin{gathered} \mathrm{\tau }=\frac{∆\mathrm{L}}{∆\mathrm{t}} \\ ∆\mathrm{t}=\frac{∆\mathrm{L}}{\mathrm{\tau }} \\ ∆\mathrm{t}=\frac{{\mathrm{L}}_{\mathrm{f}}-{\mathrm{L}}_{\mathrm{i}}}{-50} \\ ∆\mathrm{t}=\frac{0-600}{-50} \\ ∆\mathrm{t}=12\mathrm{s} \end{gathered}$$

Zero angular momentum means the angular velocity is zero. Hence, atdata-custom-editor="chemistry" $∆\mathrm{t}=12\mathrm{s}$, the angular velocity would be zero.

Hence, the time at which angular speed is zero is 12 s.

Therefore, angular velocity would be equal to zero when angular momentum is zero. Using Newton’s second law of motion in the angular form, the time at which angular momentum is zero can be found.