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 is :

Concept:

Stress- It is defined as the force acting per unit area.

 \(Stress\ =\frac{Force}{Area}\)

Strain- It is defined as deformation produced in a body in the direction of application of force divided by original dimensions of the body.

\(Strain=\frac{Change\ in\ length}{Original\ length}\)

\(Strain=∆L/L\)

where L is the original length and \(∆L\) is the change in the length of the wire.

Elastic potential energy-

• It is a type of potential energy stored by the deformation of an elastic material. For example- Energy stored in a spring when it is either compressed or stretched.
• Elastic potential energy = force × displacement

According to Hooke’s law, the force applied to stretch the spring is directly proportional to the amount of stretch. In terms of displacement, the force required to stretch the spring is directly proportional to its displacement. 

F ∝ - x

where F is the force applied on the spring and x is the displacement or amount of stretch.

The negative sign indicates displacement is in the opposite direction of the force.

hence, F = -kx,  where k is spring constant.

The elastic potential energy of stretched spring is ,

U = \(\frac{1}{2}kx^2\)

Also Elastic potential energy can be calculated as

 \(U=\ \frac{1}{2}\times Stress\times strain\times Volume\)

Calculation:

Mass of block = M 

Length of wire = L,

New length of the wire = (L + l)

Elastic potential energy = U

Using formula \(U=\ \frac{1}{2}\times Stress\times strain\times Volume\)

\(Stress=\frac{Force}{Area}=\frac{Mg}{A}\)

\(Strain=∆L/L = l/L\)

\(volume = area \times length = A\times L\)

Hence, \(U=\ \frac{1}{2}\times\frac{Mg}{A}\times\frac{l}{L}\times A\times L\)

\(U=\ \frac{1}{2}Mgl\)