In a torsional pendulum, the time period is the time it takes for the pendulum to complete one full oscillation or cycle. The time period of a torsional pendulum remains unaffected by the amplitude, regardless of whether it is large or small. This behavior is due to the nature of the restoring force acting on the pendulum.
A torsional pendulum consists of a mass or object that is suspended from a wire or a rod that can twist or rotate. When the pendulum is displaced from its equilibrium position and released, a restoring torque or force is generated that acts to bring the pendulum back to its equilibrium position.
The restoring torque in a torsional pendulum is directly proportional to the angular displacement (θ) of the pendulum. According to Hooke's law for torsion, this restoring torque is given by the equation:
τ = -kθ
where τ is the restoring torque, k is the torsional constant (also known as the torsion coefficient), and θ is the angular displacement.
When the amplitude of the torsional pendulum is large, the angular displacement (θ) is also large. However, the magnitude of the restoring torque (τ) is directly proportional to θ. As a result, the larger torque counteracts the larger angular displacement, making the time period of the pendulum unchanged.
Essentially, the pendulum experiences a stronger restoring force when the amplitude is larger, but this force is required to counterbalance the larger angular displacement. Therefore, the time period remains constant regardless of the amplitude in a torsional pendulum.