Question

A particle is acted upon by a force given by $F=-\alpha x^3-\beta x^4$ where $\alpha$ and $\beta$ are positive constants. At the point $x=0$, the particle is

• A. In stable equilibrium
• B. In unstable equilibrium
• C. In neutral equilibrium
• D. Not in equilibrium

Answer 1

The correct option is C In neutral equilibrium

Relation between potential energy and force is $F=-\frac{dU}{dx}$

$⟹$ $\frac{dU}{dx}=\alpha x^3+\beta x^4$

At $x=0$, we get $\frac{dU}{dx}{|}_{x=0}=\alpha \left(0\right)+\beta \left(0\right)=0$

Also we find $\frac{d^{2}U}{d{x}^2}=3\alpha x^2+4\beta x^3$

At $x=0$, we get $\frac{d^{2}U}{d{x}^2}{|}_{x=0}=3\alpha \left(0\right)+4\beta \left(0\right)=0$

Thus the particle is in neutral equilibrium at $x=0$.