Work and Potential Energy — Question 10

Question 10

Relative extension when both ends of a spring move

An ideal spring of $k=100$ N/m initially has extension 0.20 m. Its left end moves right 0.05 m while its right end moves right 0.15 m. The moves are controlled quasistatically.

1. Find the final extension.

2. Find the spring potential-energy change and the total spring work on both ends.

3. Would translating both ends right 0.15 m instead produce the same potential change?

Question 10 - Solution

Relative extension when both ends of a spring move

(a) Subtract the two endpoint displacements.

Spring extension changes with the relative endpoint displacement: $$\Delta x=\Delta r_R-\Delta r_L=.15-.05=.10\ \mathrm m,\qquad\boxed{x_f=.20+.10=.30\ \mathrm m}.$$

(b) Use the relative configuration.

The potential change fixes the sum of spring endpoint works: $$\boxed{\Delta U_s=\tfrac12(100)(.30^2-.20^2)=2.5\ \mathrm J},\qquad\boxed{W_{s,L}+W_{s,R}=-2.5\ \mathrm J}.$$ Their separate values require the endpoint motion histories.

(c) Check rigid translation.

For equal translations, $$\Delta x=.15-.15=0\Rightarrow\boxed{\Delta U_s=0,\quad W_{s,L}+W_{s,R}=0}.$$ Rigid translation leaves the relative configuration unchanged, even though each endpoint may exchange work.