In a torsional pendulum, the twisted wire exerts a restoring torque on the body...but how is this possible as the wire lies on the axis of rotation..so torque exerted by it should be zero right?

That's an interesting point you’ve raised about the torsional pendulum and the role of the twisted wire. It might seem counterintuitive at first, but let’s break it down to clarify how the restoring torque is generated, even when the wire is aligned with the axis of rotation.

Understanding Torsional Pendulum Mechanics

A torsional pendulum consists of a mass suspended by a wire that can twist. When the mass is displaced from its equilibrium position, the wire twists, creating a restoring torque that acts to bring the mass back to its original position. The key to understanding this lies in the nature of torque and how it is generated in this system.

Torque Basics

Torque (\( \tau \)) is defined as the product of the force (\( F \)) applied and the distance (\( r \)) from the axis of rotation to the line of action of the force. Mathematically, it can be expressed as:

• \( \tau = r \times F \)

In the case of a torsional pendulum, the wire does indeed lie along the axis of rotation, but the crucial factor is that the wire is not simply exerting a force at the axis; it is twisted. This twist creates a tension in the wire that generates a restoring torque.

How the Restoring Torque Works

When the mass is displaced, the wire experiences a torque due to its twisted state. Here’s how it works:

• Displacement: When the mass is pulled to one side, the wire twists, creating a potential energy in the system.
• Restoring Force: The twisted wire exerts a restoring force that acts to return the mass to its equilibrium position. This force is not applied at the axis but rather at the point where the mass is attached to the wire.
• Effective Radius: Although the wire is aligned with the axis, the effective radius for the torque is determined by the distance from the axis to the point where the mass is attached. This distance is what generates the torque.

Visualizing the Concept

Imagine a door. When you push on the edge of the door, you create a torque that swings it open. If you were to push directly at the hinges, there would be no torque because the distance from the hinges (the axis of rotation) to the point of force application is zero. Similarly, in a torsional pendulum, the twisting of the wire creates a situation where the restoring force acts at a distance from the axis of rotation, thus generating torque.

Conclusion

In summary, while the wire of a torsional pendulum lies along the axis of rotation, it is the twisting action and the resulting tension that create a restoring torque. This torque is what allows the pendulum to oscillate back and forth around its equilibrium position. Understanding these principles helps clarify how systems like the torsional pendulum operate, showcasing the fascinating interplay between forces and motion.