A wheel accelerates uniformly from rest to a speed of 300 rpm in 1 second. How many revolutions will be made by wheel during this interval?

Concept:

In rotational motion under uniform angular acceleration, the angular displacement is given by:

\( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \)

Where,

• \( \omega_0 \) = initial angular velocity
• \( \alpha \) = angular acceleration
• \( t \) = time
• \( \theta \) = angular displacement (in radians)

Given:

Initial angular velocity, \( \omega_0 = 0 \) (wheel starts from rest)

Final angular velocity, \( \omega = 300~\text{rpm} = \frac{300 \times 2\pi}{60} = 10\pi~\text{rad/s} \)

Time, \( t = 1~\text{s} \)

Calculation:

Angular acceleration, \( \alpha = \frac{\omega - \omega_0}{t} = \frac{10\pi}{1} = 10\pi~\text{rad/s}^2 \)

Now, using the equation for angular displacement:

\( \theta = \frac{1}{2} \alpha t^2 = \frac{1}{2} \times 10\pi \times 1^2 = 5\pi~\text{radians} \)

Convert radians to revolutions:

\( \theta = \frac{5\pi}{2\pi} = 2.5~\text{revolutions} \)