Question

Obtain expression for potential energy of a spring attached to mass m moving on a frictionless horizontal surface by graphical method.

Answer 1

To find:

Expression for potential energy of a spring attached to mass m moving on a frictionless horizontal surface by graphical method.

Solution:

First of all, let's draw the Force vs Displacement graph for spring attached to the mass.

$\therefore \: F \propto ( - x)$

$\implies \: F = - kx \: \: \: . \: . \: . \: .(k \: is \: constant)$

So, F - x graph will be straight line passing through the origin with negative slope.

Now, as per ENERGY CONSERVATION PRINCIPLE, the potential energy stored in the spring will be equal to the total work done.

$\therefore \: PE = total \: work$

$\implies \: PE = area \: under \: F - x \: graph$

$\implies \: PE = area \: of \: \Delta$

$\implies \: PE = \dfrac{1}{2} \times base \times height$

$\implies \: PE = \dfrac{1}{2} \times |F| \times x$

$\implies \: PE = \dfrac{1}{2} \times kx \times x$

$\implies \: PE = \dfrac{1}{2} k {x}^{2}$

So, potential energy in spring is ½kx².