Path Independence of Conservative Forces — Question 9

Question 9

Different loading and unloading curves

A device is stretched quasistatically from x=0 to 0.20 m. The external force required is $F_{\mathrm{load}}=100x$ N. On return it exerts a resisting magnitude described by an external holding force $F_{\mathrm{unload}}=60x$ N in the positive x direction.

1. Find external-agent work on loading and unloading.

2. Find total external work over the cycle.

3. Can the device be described by a single potential depending only on x, if no other state is allowed?

Question 9 - Solution

Different loading and unloading curves

(a) Keep the return displacement negative.

Keep the unloading limits in their physical order: $$\boxed{W_{\mathrm{load}}=\int_0^{.20}100x\,dx=2.0\ \mathrm J},$$ $$\boxed{W_{\mathrm{unload}}=\int_{.20}^{0}60x\,dx=-1.2\ \mathrm J}.$$

(b) Compute the area between the histories.

The net external work is the area between the loading histories: $$\boxed{W_{\mathrm{ext,cycle}}=2.0-1.2=0.8\ \mathrm J}.$$ Negligible kinetic changes give $W_{\mathrm{device,cycle}}=-0.8$ J.

(c) Compare identical endpoint deformation.

A single position potential would require $$W_{\mathrm{device,cycle}}=-[U(0)-U(0)]=0,$$ contrary to the measured loop. The device requires dissipation or an additional internal state beyond extension alone.