Work Done on a System by an External Force — Question 10

Question 10

The location of external energy delivery matters

A particle starts with K=10 J and U=0. Along its route a smooth potential barrier reaches 12 J, then falls back to zero. A drive can add 5 J of work, either entirely before the barrier or only after it. There is no friction.

1. Can a drive located only after the barrier help the particle cross?

2. If all 5 J is added before climbing, find kinetic energy at the barrier top.

3. Explain why inserting an eventual +5 J into the initial energy balance can be causally wrong.

Question 10 - Solution

The location of external energy delivery matters

(a) Examine the energy available before the drive.

Before a downstream drive is reached, no work has yet been added: $$E=10\ \mathrm J< U_{\max}=12\ \mathrm J.$$ The particle reaches a turning point at $U=10$ J and cannot access the drive beyond the barrier.

(b) Place the transfer before it is needed.

Early input raises the total before the climb: $$E_+=10+5=15\ \mathrm J,\qquad\boxed{K_{\mathrm{top}}=15-12=3\ \mathrm J}.$$ The energy stays above the barrier, so passage is possible.

(c) Use cumulative external work up to each point.

At each location the correct balance uses cumulative work: $$K(x)+U(x)=E_i+W_{\mathrm{ext}}(\text{up to }x).$$ An eventual downstream transfer is not available at an earlier blocked point. Inserting it prematurely assumes the very passage that must be established.