Reading a Potential Energy Curve — Question 5

Question 5

A flat region does not stop a moving particle

Assume one-dimensional classical motion with $K=E-U(x)$ and $F_x=-dU/dx$, with no dissipation. A 1 kg particle moves right with E=6 J. Its potential is U=4 J for $1\le x\le3$ m, then rises linearly to U=8 J at x=4 m.

1. Find force and speed throughout the flat region.

2. Find the turning coordinate on the rising segment.

3. What would happen to a particle placed at rest strictly inside the flat interval with no disturbance?

Question 5 - Solution

A flat region does not stop a moving particle

(a) A zero slope means zero force.

On the flat segment, $$\boxed{F=-U'=0},\quad K=6-4=2\ \mathrm J\Rightarrow\boxed{v=\sqrt{2(2)/1}=2\ \mathrm{m/s}}.$$ Zero force means coasting, not stopping.

(b) Intersect the rising potential with E.

On the rising segment, $$U(x)=4+4(x-3),\qquad U(x_t)=6\Rightarrow\boxed{x_t=3.5\ \mathrm m}.$$ The force $F=-4$ N reverses the rightward motion at that point.

(c) Change the initial energy consistently.

A separately prepared particle at rest on the plateau has $$E=U=4\ \mathrm J,\qquad F=0\Rightarrow a=0.$$ It remains at its chosen location. Small position shifts within the flat interval lead to other equilibria without restoring force.