Work Done on a System by an External Force — Question 7

Question 7

Zero resultant force with nonzero external work

Assume no heat transfer across the stated system boundary during the process unless otherwise specified. Two external agents pull outward on the ends of a deformable device with equal constant 10 N forces. The left end moves left 0.10 m and the right end moves right 0.10 m. Its total kinetic energy is unchanged between the initial and final configurations.

1. Find the vector sum of external forces.

2. Find the total external work.

3. If the device warms by 0.50 J and no other energy store changes except elastic energy, find its elastic-energy increase.

Question 7 - Solution

Zero resultant force with nonzero external work

(a) Opposite equal forces cancel as vectors.

Taking rightward as positive, $$\boxed{\vec F_{\mathrm{ext}}=(-10+10)\hat i=0}.$$ This constrains the resultant force, not the relative motion of the endpoints.

(b) Sum force-displacement products at both ends.

Each force acts along its own endpoint displacement: $$W_L=(-10)(-.10)=1\ \mathrm J,\qquad W_R=(10)(.10)=1\ \mathrm J.$$ $$\boxed{W_{\mathrm{ext}}=W_L+W_R=2\ \mathrm J}.$$

(c) Account for deformation and heating.

The endpoint kinetic energies are equal: $$2=\Delta K+\Delta U_{\mathrm{elastic}}+\Delta E_{\mathrm{th}}=0+\Delta U_{\mathrm{elastic}}+.50.$$ $$\boxed{\Delta U_{\mathrm{elastic}}=1.50\ \mathrm J}.$$ Zero resultant force does not imply zero work on a deformable system.