Equilibrium — Question 6

Question 6

Neutral equilibrium is different from a weak restoring force

A puck rests on a perfectly horizontal frictionless table. Ignore rotation and consider only horizontal motion. Its potential energy is independent of horizontal position.

1. Classify its equilibrium after a small displacement followed by release from rest.

2. What happens if the perturbation instead gives it a small horizontal velocity?

Question 6 - Solution

Neutral equilibrium is different from a weak restoring force

(a) Derivation and calculation

Horizontal position does not change the potential: $$U(x,y)=U_0,\qquad\vec F_\parallel=-\boldsymbol\nabla_\parallel U=\vec0.$$ After displacement and release from rest, $$\ddot{\vec r}=\vec0,\quad\dot{\vec r}(0)=\vec0 \quad\Longrightarrow\quad\vec r(t)=\vec r_{\mathrm{new}}.$$ Thus the equilibrium is $\boxed{\text{neutral with respect to position}}$.

(b) Result and interpretation

A velocity perturbation instead gives $$\boxed{\vec r(t)=\vec r_0+\vec v_0t},\qquad\vec v(t)=\vec v_0.$$ There is neither damping nor restoration: $$\frac{dK}{dt}=\vec F_\parallel\cdot\vec v=0.$$ Neutral positional equilibrium does not return a moving puck to its starting point.

Physical check

A displaced resting puck stays displaced; a moving puck continues at constant velocity.