Question

The potential energy of a particle varies with distance $x$ as shown in the graph. The force acting on the particle is zero at:
A) C
B) D
C) B and C
D) A and D

Answer 1

Hint: The variation of the potential energy of some particle with distance is given by a curve. The force acting on this particle can be obtained as the change in the potential energy of the particle with respect to the distance which is the first derivative of the potential energy. Now the first derivative will be zero for the maximum and minimum potential energy.

Formula used:
The force acting on a particle is given by, $F = - \dfrac{{dU}}{{dx}}$ where $U$ is the potential energy and $x$ is the distance of the particle.

Complete step by step solution:
The above graph depicts the variation of the potential energy of the particle as the distance changes. Points A, B, C and D represent the potential energies at different distances.
Express the relation for the force acting on the particle.
The potential energy $U\left( x \right)$ of the given particle is represented as a function of its distance $x$ .
Then the force acting on the particle can be expressed as $F = - \dfrac{{dU}}{{dx}}$ .
We have to determine the points at which the force acting on the particle is zero.
i.e., $F = - \dfrac{{dU}}{{dx}} = 0$ .
$\dfrac{{dU}}{{dx}} = 0 \Rightarrow U = {U_{\min }}$ or $U = {U_{\max }}$
So force will be zero when the potential energy has a minimum value or a maximum value.
From the above graph we see that at point B the potential energy of the particle has a maximum value and at point C, the potential energy of the particle has a minimum value.
So we can conclude that at points B and C the force acting on the particle is zero.

Hence the correct option is C.

Note: The first derivative of the potential energy of the particle with respect to the distance actually represents the slope of the curve. So when we say that at points B and C, the force is zero, we essentially mean that the slope of the curve is zero at those points. The slopes at B and C will be horizontal tangent lines.