Question

What is the relationship of potential energy to conservative force?

Short answer

The conservative force equals to the negative gradient of the potential energy.

Step-by-step solution

Step 1: Definition of Concept

Potential energy: Potential energy is defined as the work done by an external agent against the action of conservative force in order to displace the object.

Explain the relationship of potential energy to a conservative force

The relation between conservative force and potential energy is given as,

$\begin{array}{c}\overrightarrow{F}=-\text{grad}U\\ =-\nabla U\end{array}$ ……………….(1.1)

Here, U is the potential energy.

Since the gradient is given as,

$\nabla =\left(\frac{\partial }{\partial x}\widehat{i}+\frac{\partial }{\partial y}\widehat{j}+\frac{\partial }{\partial z}\widehat{k}\right)$ …………………..(1.2)

From equations (1.1) and (1.2), we get,

$\begin{array}{c}\overrightarrow{F}=-\left(\frac{\partial }{\partial x}\widehat{i}+\frac{\partial }{\partial y}\widehat{j}+\frac{\partial }{\partial z}\widehat{k}\right)U\\ =-\frac{\partial U}{\partial x}\widehat{i}-\frac{\partial U}{\partial y}\widehat{j}-\frac{\partial U}{\partial z}\widehat{k}\end{array}$

Integrating equation (1.1) over small line element $d\overrightarrow{s}$,

$\begin{array}{c}\int \overrightarrow{F}\cdot d\overrightarrow{s}=-\int \left(\nabla U\right)\cdot d\overrightarrow{s}\\ =-\Delta U\end{array}$ ……………………(1.3)

Since the work done is given as,

$W=\int \overrightarrow{F}\cdot d\overrightarrow{s}$ ………………………….(1.4)

From equations (1.3) and (1.4), we get,

$W=-\Delta U$

This equation states that the work done for the conservative force is equal to the negative of the change in potential energy.