A wheel is at rest. Its angular velocity increases uniformly and becomes 60 rad/sec after 5 sec. The total angular displacement is

Correct Option: 150 rad

To find the total angular displacement of the wheel, we can use the equations of rotational kinematics under uniform angular acceleration.

1. Identify the given values:
- Initial angular velocity (${\omega }_0$) = 0 rad/sec (since the wheel starts from rest)
- Final angular velocity ($\omega$) = 60 rad/sec
- Time interval ($t$) = 5 seconds

2. Calculate the angular acceleration ($\alpha$):
Angular acceleration is the rate of change of angular velocity:
$\alpha =\frac{\omega -{\omega }_0}{t}$
Substituting the given values:
$\alpha =\frac{60-0}{5}=12{\text{rad/sec}}^2$

3. Calculate the total angular displacement ($\theta$):
Using the second equation of rotational kinematics:
$\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$
Substituting the values:
$\theta =\left(0\right)\left(5\right)+\frac{1}{2}\left(12\right){\left(5\right)}^2$
$\theta =0+6\times 25$
$\theta =150\text{rad}$

Alternative Method (Using Average Angular Velocity):
Since the acceleration is uniform, the average angular velocity (${\omega }_{\text{avg}}$) is given by:
${\omega }_{\text{avg}}=\frac{{\omega }_0+\omega }{2}=\frac{0+60}{2}=30\text{rad/sec}$
The angular displacement is then:
$\theta ={\omega }_{\text{avg}}\times t=30\times 5=150\text{rad}$