Question

A ceiling fan is rotating at the rate of 3.5 rps and its moment of inertia is $1.25 kg{m^2}$. If the current is switched off, the fan comes to rest in 5.5 s. The torque acting on the fan due to friction is:
(A) $2.5 Nm$
(B) $5 Nm$
(C) $7.5 Nm$
(D) $10 Nm$

Answer 1

Hint:In this problem,we are going to apply the concept of rotational motion and the torque is the product of moment of inertia and angular acceleration whereas angular acceleration is the ratio of change in angular velocity to time. So, after applying the formula of torque and angular acceleration, we can arrive at the correct result.

Complete step by step answer:
A ceiling fan is rotating at the rate of 3.5 revolutions per second which means the fan makes 3.5 revolutions in one second. The moment of inertia (I) is$1.25kg{m^2}$ and it is defined as the sum of the products of the masses of all the particles constituting the fan and the square of their distances from the axis of rotation of the fan. When the fan is switched off, the fan comes to rest in time t is 5.5 s and final angular velocity ω is zero.
In one revolution, the total angle covered by the fan is $2\pi $radians.
Hence, the initial angular velocity,
$\left( {{\omega _0}} \right)$ of the fan in 3.5 revolutions = $3.5 \times 2\pi = 7\pi $
Angular acceleration$\left( \alpha \right)$ is defined as the rate of change of angular velocity i.e., difference of final and initial angular velocity divided by time.
$\alpha = \dfrac{{\omega - {\omega _0}}}{t}$
$\Rightarrow\alpha = \dfrac{{0 - 7\pi }}{{5.5}}$
$\Rightarrow\alpha = \dfrac{{ - 7 \times 3.14}}{{5.5}}\left[ {\pi = 3.14} \right]$
$\Rightarrow\alpha = - 3.99 = - 4rad/s$
The torque$\left( \tau \right)$is defined as the turning effect of the force and it is numerically equal to the product of moment of inertia and angular acceleration.
$\tau = I\alpha $
$\Rightarrow\tau = 1.25 \times \left( { - 4} \right)$
$\therefore\tau = - 5Nm$ [negative sign indicates negative acceleration i.e., retardation because the fan comes to rest]
Magnitude of torque = 5 Nm.
Therefore, option B is correct.

Note: The value of final angular velocity is zero because the fan comes to rest after the given time and negative sign of torque indicates retardation. Sometimes students get confused between angular acceleration and linear acceleration, so always remember that the concept of angular acceleration applies only when the body has rotational motion while the concept of linear acceleration applies when the body has linear motion.