A solid sphere of radius \(4\) cm and mass \(5\) kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of \(1200\) rpm. It is brought to rest in \(10\) s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are _______ and _______ respectively.

Initial angular velocity \(\omega_0 = 1200 \text{ rpm} = \dfrac{1200 \times 2\pi}{60} = 40\pi \text{ rad/s}\)

Final angular velocity \(\omega = 0\)

Time \(t = 10 \text{ s}\)

Angular acceleration \(\alpha = \dfrac{\omega - \omega_0}{t} = \dfrac{0 - 40\pi}{10} = -4\pi \text{ rad/s}^2\)

Moment of inertia of the solid sphere \(I = \dfrac{2}{5}MR^2 = \dfrac{2}{5} \times 5 \times (0.04)^2 = 0.0032 \text{ kg m}^2\)

Torque \(\tau = I|\alpha| = 0.0032 \times 4\pi = 0.0128\pi \text{ Nm}\)

Angle rotated \(\theta = \dfrac{\omega_0 + \omega}{2}t = \dfrac{40\pi + 0}{2} \times 10 = 200\pi \text{ rad}\)

Number of rotations \(N = \dfrac{\theta}{2\pi} = \dfrac{200\pi}{2\pi} = 100\)

Answer: \(0.0128\pi\) Nm, \(100\)