Determining Potential Energy Values — Question 8

Question 8

When an infinite reference is unavailable

An idealized force is $F_y=-5$ N at every positive height y, extending indefinitely. Someone proposes defining U=0 at y approaching infinity.

1. Find U(H)-U(0) for a finite height H.

2. Does this difference approach a finite limit as H grows?

3. Give a valid finite reference and the work from y=2 to y=5 m.

Question 8 - Solution

When an infinite reference is unavailable

(a) Integrate over a finite interval first.

For finite $H$, integrate before taking any limit: $$\boxed{U(H)-U(0)=-\int_0^H(-5)dy=5H\ \mathrm J}\quad(H\text{ in m}).$$

(b) Test the proposed limiting assignment.

The proposed reference fails because $$\lim_{H\to\infty}[U(H)-U(0)]=\lim_{H\to\infty}5H=+\infty.$$ No finite constant yields finite potential at finite height and zero at infinity for this constant-force model.

(c) Choose a permissible datum.

A valid finite reference is $U(0)=0$: $$U(y)=5y,\qquad\boxed{W_{2\to5}=U(2)-U(5)=10-25=-15\ \mathrm J}.$$ Finite potential differences remain well defined.