6.1 Potential energy $\left( {\Delta U} \right)$ is negative of the work done by conservative forces.

  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 $\left( {\Delta U} \right)$
    7.0 Work energy theorem
    8.0 Power
    9.0 Types of equilibrium
    10.0 Work done by a distributed mass

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6.1 Potential energy $\left( {\Delta U} \right)$ is negative of the work done by conservative forces.

Potential energy $\left( {\Delta U} \right)$ is negative of the work done by conservative forces. Suppose a particle goes from point ? to ? by any of the path.

The work done by the conservative forces is $W_{CF}$. As we know relation between potential energy, force and displacement we can write,,$$dU = - \overrightarrow F .d\overrightarrow r $$ Integrating the above equation from point $A$ to $B$ we get, $$\begin{equation} \begin{aligned} \int\limits_A^B {dU} = - \int\limits_A^B {\overrightarrow F } .d\overrightarrow r \\ {U_B} - {U_A} = - {W_{CF}} \\ \Delta U = - {W_{CF}} \\\end{aligned} \end{equation} $$

Note:

1. Change in potential energy depends only on the initial and final positions.
2. Work done by conservative forces $(W_{CF})$ also depends only on the initial and final positions. So, the work done by conservative forces $(W_{CF})$ and the change in potential energy will be always same between point $A$ and point $B$ irrespective of the path followed.

Defining absolute potential energy about any point

The absolute potential energy at any point is with respect to the reference at infinity. So, the potential energy at infinity is assumed to be zero $\left( {{U_\infty } = 0} \right)$.

Therefore, the absolute potential energy at any point is defined as the work done by conservative forces $(W_{CF})$ in bringing an object from infinity to that point. Mathematically, $$dU = - \overrightarrow F .d\overrightarrow r $$ Integrating the above equation from $\infty $ to any point $A$ we get, $$\begin{equation} \begin{aligned} \int\limits_\infty ^A {dU} = - \int\limits_\infty ^A {\overrightarrow F } .d\overrightarrow r \\ {U_A} - {U_\infty } = - {W_{CF}} \\ {U_A} - 0 = - {W_{CF}} \\ {U_A} = - {W_{CF}} \\\end{aligned} \end{equation} $$