Determining Potential Energy Values — Question 9

Question 9

An infinite reference for a decaying force

For y at least zero, a downward force is $F_y=-A/(1+y/L)^2$, with $A=12$ N and $L=2$ m. Define U to approach zero as y approaches infinity. This force law is supplied; no gravitational inverse-square law is assumed.

1. Find U(y) satisfying the reference.

2. Find U(0), U(2 m), and work from 0 to 2 m.

3. Find the total quasistatic external work needed to move from 0 to arbitrarily large y.

Question 9 - Solution

An infinite reference for a decaying force

(a) Integrate and impose the limiting value.

Integrate the negative force: $$\frac{dU}{dy}=\frac{A}{(1+y/L)^2}\Rightarrow U=-\frac{AL}{1+y/L}+C.$$ The limit $U(\infty)=0$ fixes $C=0$: $$\boxed{U(y)=-\frac{24}{1+y/2}\ \mathrm J}.$$

(b) Evaluate the finite states.

Evaluate endpoints and use the sign convention: $$\boxed{U(0)=-24\ \mathrm J,\quad U(2)=-12\ \mathrm J},$$ $$\boxed{W_{0\to2}=U(0)-U(2)=-12\ \mathrm J}.$$

(c) Use the convergent potential rise.

Quasistatic external work equals the potential rise: $$\boxed{W_{\mathrm{ext}}=\lim_{H\to\infty}[U(H)-U(0)]=0-(-24)=24\ \mathrm J}.$$ The force decreases sufficiently rapidly for the work integral to converge despite unbounded travel distance.