Work Done on a System by an External Force — Question 6

Question 6

Energy supplied through a moving spring support

Assume no heat transfer across the stated system boundary during the process unless otherwise specified. An ideal spring of k=100 N/m connects a clamped block to a movable support. Initially it is relaxed. The support is moved away quasistatically by 0.20 m while the block remains fixed. Choose block plus spring as the system.

1. Find the work done on the block by the spring.

2. Find the increase in spring potential energy and work by the moving-support agent.

3. Explain why force times block displacement cannot represent the total external work on this system.

Question 6 - Solution

Energy supplied through a moving spring support

(a) The block has no displacement.

The spring force acts at a stationary block contact: $$\boxed{W_{s\to b}=\int\vec F_s\cdot d\vec r_b=0},\qquad d\vec r_b=0.$$

(b) Track the moving endpoint.

The moving support changes the spring extension: $$\boxed{\Delta U_s=\tfrac12(100)(.20)^2=2\ \mathrm J}.$$ With negligible kinetic and thermal changes, $\boxed{W_{\mathrm{support}}=+2\ \mathrm J}$.

(c) Use each force at its own application point.

External work must use each contact’s own displacement: $$W_{\mathrm{ext}}=\int\vec F_c\cdot d\vec r_c+\int\vec F_a\cdot d\vec r_a=0+2\ \mathrm J.$$ The fixed clamp does no work; the moving support supplies the stored energy. Using the block displacement for both contacts would erase the actual input.