Physics > Work Energy and Power > 3.0 Spring Force
Work Energy and Power
1.0 Introduction
2.0 Work done by a constant force
3.0 Spring Force
4.0 Conservative & Non-conservative forces
5.0 Kinetic Energy $(K)$
6.0 Potential energy $\left( {\Delta U} \right)$
6.1 Potential energy $\left( {\Delta U} \right)$ is negative of the work done by conservative forces.
6.2 Types of potential energy
6.3 Law of conservation of mechanical energy
7.0 Work energy theorem
8.0 Power
9.0 Types of equilibrium
10.0 Work done by a distributed mass
3.1 Work done by a spring force
6.2 Types of potential energy
6.3 Law of conservation of mechanical energy
When the spring is elongated by a distance $x$ we get, $$\begin{equation} \begin{aligned} {F_{Ext}} = kx ...(i) \\ {F_T} = - kx ...(ii) \\\end{aligned} \end{equation} $$
Work done by the spring force is, $$\begin{equation} \begin{aligned} {W_S} = \int\limits_0^x {{{\overrightarrow F }_T}.d\overrightarrow x } \\ {W_S} = \int\limits_0^x {\left( { - kx} \right)dx} \\ {W_S} = - \int\limits_0^x {kxdx} \\ {W_S} = - \frac{k}{2}\left[ {{x^2}} \right]_0^2 \\ {W_S} = - \frac{1}{2}k{x^2}...(iii) \\\end{aligned} \end{equation} $$
As we know that the potential energy stored in the spring will be negative of the work done by the conservative forces. Therefore, $$\Delta U = \frac{1}{2}k{x^2}...(iv)$$
Note: Whether the spring is elongated or compressed the potential energy stored in both the situation is $\Delta U = \frac{1}{2}k{x^2}$
