Question

A solid cylinder of mass m and length l and area of cross section A is placed vertically on the ground. If Young's modulus is Y then what is the strain energy stored in the cylinder?

Answer 1

Given that,

Mass = m

Length = l

Cross section = A

Young's modulus = Y

We need to calculate the change in length

Using formula of young modulus

$Y=\dfrac{\dfrac{F}{A}}{\dfrac{\Delta l}{l}}$

$Y=\dfrac{mgl}{\Delta l\times A}$

$\Delta l=\dfrac{mgl}{YA}$

We know that,

The initial energy $E_{i}= mgl$

The final energy $E_{f}= mg(l+\Delta l)$

We need to calculate the strain energy stored in the cylinder

Using formula for change in energy

$\Delta E=E_{f}-E_{i}$

Put the value into the formula

$\Delta E=mg(l+\Delta l)-mgl$

$\Delta E=mg\Delta l$

$\Delta E=mg\times\dfrac{mgl}{YA}$

$\Delta E=\dfrac{(mg)^2l}{YA}$

Hence, The strain energy stored in the cylinder is $\dfrac{(mg)^2l}{YA}$