Rotational Motion

The formula for the ideal angle of banking ( ) for a vehicle moving at speed on a curved road with radius is: where is the acceleration due to gravity. This angle allows the vehicle to make the turn without relying on friction.

Given:
Mass of the body,
Restoring force,
Displacement,
1. To find the Force Constant (k):
The restoring force in Simple Harmonic Motion (S.H.M.) is given by Hooke's Law, (considering magnitude). The force constant is 10 N/m.
2. To find the Period of S.H.M. (T):
The period of S.H.M. is given by the formula: Using and : The period of S.H.M. is approximately 1.78 s.

The work done ( ) by a constant torque ( ) is given by the formula: where is the angular displacement in radians.
Given:
Torque,
Number of revolutions = 4
First, convert the number of revolutions into radians for the angular displacement .
One revolution corresponds to an angle of radians. Now, calculate the work done: Using : The work done during the first 4 revolutions is Joules, or approximately 50265 J. (Note: The mass and radius of the flywheel are not needed for this calculation as the torque is given directly).

Statement: The law of conservation of angular momentum states that if the net external torque acting on a system is zero, then the total angular momentum of the system remains constant (conserved).
Proof: The angular momentum ( ) of a system is related to its moment of inertia ( ) and angular velocity ( ) by .
The relationship between torque ( ) and angular momentum ( ) is that the net external torque is equal to the time rate of change of the angular momentum.
According to the law, if the net external torque on the system is zero, then .
Substituting this into the equation: This implies that is a constant with respect to time. The derivative of a constant is zero.
Therefore, .
This proves that the angular momentum of the system is conserved when no net external torque acts on it.

The power (P) in a rotational system is given by the product of torque ( ) and angular speed ( ). Given:
Torque, Nm
Angular speed, rad/s
Substituting the values: The power rating of the ceiling fan is W.