Question

In the figure shows, pulley and spring are ideal. Find the potential energy stored in the spring (${m_1} > {m_2}$)

Answer 1

We are asked to find the potential energy stored in the spring. First, recall the formula to find potential energy stored in a spring. Draw a free body diagram of the problem. Using this diagram, find the value of displacement and use this value to find potential energy of the spring.

Complete step by step answer:
Given a figure where pulley and spring are ideal.The formula to find potential energy stored in a spring is,
$P.E = \dfrac{1}{2}k{x^2}$ (i)
where $k$ is the spring constant and $x$ is the displacement from the mean position.
Let us draw the free body diagram for the problem.

In the figure, $F$ is the restoring force of the spring, $T$ is the tension on the string and $g$ is acceleration due to gravity.
Restoring force is given by the formula,
$F = kx$ (ii)
where $k$ is the spring constant and $x$ is the displacement from the mean position.
From the figure we observe,
$T + T = F$
Putting the value of $F$ we get,
$T + T = kx$
$\Rightarrow 2T = kx$
$\Rightarrow T = \dfrac{1}{2}kx$ (iii)
From the figure we get,
$T = {m_1}g$
Putting the value of $T$ we get,
$\dfrac{1}{2}kx = {m_1}g$
$\Rightarrow x = \dfrac{{2{m_1}g}}{k}$
Now, putting this value of $x$ in equation (i) we get the potential energy as,
$P.E = \dfrac{1}{2}k{\left( {\dfrac{{2{m_1}g}}{k}} \right)^2}$
$\Rightarrow P.E = \dfrac{1}{2}k\left( {\dfrac{{4{m_1}^2{g^2}}}{{{k^2}}}} \right)$
$\therefore P.E = \dfrac{{2{m_1}^2{g^2}}}{k}$

Therefore the potential energy stored in the spring is $\dfrac{{2{m_1}^2{g^2}}}{k}$.

Note: For such types of problems, before proceeding for calculations, draw a free body diagram. A free body diagram is a diagram showing the forces and their directions acting on a given system. Here we have used the term restoring force, restoring force is the force which brings back an object to its mean position or equilibrium.