Physics - W.E.P. Question with Solution | TestHub

$\mathrm{U}={\mathrm{ax}}^3-{\mathrm{bx}}^2$

$$\begin{array}{rlr}& \frac{dU}{dx}=3a{x}^2-2bx& \frac{d^{2}U}{d{x}^2}=6ax-2b\end{array}$$

For equilibrium

$$\frac{\mathrm{dU}}{\mathrm{dx}}=0\Rightarrow 3{\mathrm{ax}}^2-2\mathrm{bx}=0\Rightarrow \mathrm{x}=0,\frac{2\mathrm{b}}{3\mathrm{a}}$$

For $\mathrm{x}=\frac{2\mathrm{b}}{3\mathrm{a}},\frac{{\mathrm{d}}^2\mathrm{U}}{{\mathrm{dx}}^2}=6\mathrm{a}\times \frac{2\mathrm{b}}{3\mathrm{a}}-2\mathrm{b}=2\mathrm{b}>0$ Therefore at $\mathrm{x}=\frac{2\mathrm{b}}{3\mathrm{a}}$, potential energy is minimum. This is a condition of stable equilibrium.