Question

The potential energy for a conservative force system is given by $U=a{x}^3-bx$, where $a$ and $b$ are constants. Choose from the options, the correct $x$ coordinate(s) of the equilibrium position(s).

• A. $+\sqrt{\frac{b}{2a}}$
• B. $-\sqrt{\frac{b}{2a}}$
• C. $+\sqrt{\frac{b}{3a}}$
• D. $-\sqrt{\frac{b}{3a}}$

Answer 1

Answer: (C), (D)

The correct options are
C $+\sqrt{\frac{b}{3a}}$
D $-\sqrt{\frac{b}{3a}}$

For conservative force, $F=-\frac{dU}{dx}$
For $U=a{x}^3-bx$
$F=-\frac{dU}{dx}=-3a{x}^2+b$
For equilibrium, $F=0$
$\Rightarrow 3a{x}^2-b=0$
or $x=\sqrt{\frac{b}{3a}},-\sqrt{\frac{b}{3a}}$
Thus, option (c) and (d) are correct.