Elastic Potential Energy Stored in an Extended Wire

When a block of mass M is suspended by a long wire of length L, the length of the wire becomes (L + l). The elastic potential energy stored in the extended wire can be explained as follows:

1. Introduction:
When the block is suspended, it stretches the wire due to the force of gravity acting on it. This stretching of the wire results in the storage of elastic potential energy.

2. Definition:
Elastic potential energy is the energy stored in a stretched or compressed elastic object. It is given by the formula E = (1/2) k x^2, where E is the elastic potential energy, k is the spring constant, and x is the displacement of the object from its equilibrium position.

3. Wire as an Elastic Object:
In this scenario, the wire can be considered as an elastic object. When the block is suspended, it exerts a force on the wire, causing it to stretch. The wire behaves like a spring, with its own spring constant.

4. Spring Constant of the Wire:
The spring constant of the wire can be determined by Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement. Mathematically, F = -kx, where F is the force exerted by the wire, k is the spring constant, and x is the displacement of the wire.

5. Calculation of Elastic Potential Energy:
To calculate the elastic potential energy stored in the extended wire, we need to determine the displacement of the wire from its equilibrium position. In this case, the displacement is (L + l) - L = l.

Using the formula for elastic potential energy, E = (1/2) k x^2, and substituting the values of k and x, we can calculate the energy stored in the wire.

6. Significance of Elastic Potential Energy:
The elastic potential energy stored in the extended wire represents the work done to stretch the wire against its restoring force. This energy can be released when the wire returns to its equilibrium position, causing the block to oscillate.

Conclusion:
When a block of mass M is suspended by a long wire, the wire stretches and stores elastic potential energy. This energy can be calculated using the formula E = (1/2) k x^2, where k is the spring constant of the wire and x is the displacement of the wire from its equilibrium position. The elastic potential energy represents the work done to stretch the wire and can be released during oscillations.