Question

The potential energy of a conservative system is given by $V(x)=({{x}^{2}}-3x)\text{ joules}$ where $x$ is measured in metres. Then its equilibrium position is at 1.5 m
A. 2 m
B. 3 m
C. 1 m
D. 5 m

Answer 1

Maxima or minima of a function correspond to zero first derivative of that function. Equilibrium points occur at extrema (maxima or minima) of potential energy. Take the derivative of potential energy $V(x)$ with respect to $x$ and set it to zero. Find the value of $x$ for equilibrium position.

Formula used: $\dfrac{d}{dx}V(x)=0$

Complete step by step solution:
The equilibrium position is found by setting the first derivative of potential energy to 0.
$\begin{aligned} & \dfrac{d}{dx}V(x)=0 \\\ & \dfrac{d}{dx}({{x}^{2}}-3x)=0 \\\ & 2x-3=0 \\\ & x=1.5\text{ m} \\\ \end{aligned}$

Therefore, the correct answer is option A.

Addition information: The negative of the first derivative of potential energy with position gives force.
$F=-\dfrac{dV(x)}{dx}$

Note: Alternatively, potential energy $V(x)$ can be plotted as a function of $x$. The equilibrium points will correspond to points on the curve where the tangent is parallel to the x axis.