Question

A wire is subjected to a tensile stress. If A represents area of cross-section, L represents original length, I represent extension and Y is Young's modulus of elasticity, then elastic potential energy of the stretched wire is

A. $U = \dfrac{{2L}}{{AY}}{I^2}$
B. $U = \dfrac{{AL}}{{2Y}}{I^2}$
C. $U = \dfrac{{AY}}{{2L}}{I^2}$
D. $U = \dfrac{1}{4}\dfrac{{AY}}{L}{I^2}$

Answer 1

Hint:In order to stretch a wire, some work must be done against the restoring force. This work done is stored in the wire in the form of potential energy of wire. The energy stored per unit volume of the stretched wire is called its energy density. We will calculate the stress, strain in the wire and the young’s modulus of wire. Potential energy stored in the wire can be calculated using the stress and strain in the wire.

Complete step by step answer:
We all know that the elastic potential energy U of the stretched wire is the work done W in stretching the wire. It means that U=W.
Therefore, we will write the formula for work done.
$W = \dfrac{1}{2} \times \sigma \times \varepsilon \times V$
Where, $\sigma $ is the stress, $\varepsilon $ is the strain and V is the volume.
We will now substitute strain $\varepsilon = \dfrac{l}{L}$ and \[\sigma = Y\varepsilon \] and V= AL. Here, A is the area and L is the length and l is the extension and L is the original length.
$W = \dfrac{1}{2} \times Y\varepsilon \times \varepsilon \times V$
$ \Rightarrow W = \dfrac{1}{2} \times Y{\varepsilon ^2} \times V$
$ \Rightarrow W = \dfrac{1}{2} \times Y{\left( {\dfrac{l}{L}} \right)^2} \times AL$
$ \Rightarrow W = \dfrac{1}{2} \times Y{\left( {\dfrac{l}{L}} \right)^2} \times AL$
$ \Rightarrow W = \dfrac{{Y{I^2}A}}{{2L}}$

Hence, the correct option is C.

Note:Students should note that the stress is the force acting a wire while it is being stretched. The change in the length of the wire due to applied force, stress, is called strain. Also, Strain is not the same as the physical pressure applied on a body. Pressure is defined as force per unit area and its unit is Pascal or newton per meter square while strain is defined as the change in the length of wire per unit original length and strain has no unit.