Question

The potential energy for a conservative force system is given by $U=a{x}^2-bx$. Where a and b are constants find out
1) The expression of force
2) Equilibrium position
3) Potential energy at equilibrium

Answer 1

Given that

The potential energy for a conservative force system is given by $U=a{x}^2-bx$

Force , $F=-\frac{dU}{dx}=b-2ax$

Equilibrium position is where $F=0$

Hence at $x=\frac{b}{2a}$ , the Equilibrium occurs

Potential energy at Equilibrium, $U_{eq}=a(\frac{b}{2a}{)}^2-b(\frac{b}{2a})=-\frac{b^2}{4a}$