Determining Potential Energy Values — Question 2

Question 2

Anchoring a potential at a nonzero position

A one-dimensional conservative force is $F_x=6-4x$ N with x in meters. A reference measurement gives U(1 m)=3 J.

1. Find U(x).

2. Find U(0) and U(3).

3. Check the force at x=2 m from your potential and find work from 0 to 3 m.

Question 2 - Solution

Anchoring a potential at a nonzero position

(a) Integrate with the correct sign.

Integrate $dU/dx=-F_x$ and then impose the reference: $$U(x)=-6x+2x^2+C,\qquad3=-6+2+C\Rightarrow C=7\ \mathrm J.$$ $$\boxed{U(x)=2x^2-6x+7\ \mathrm J}\quad(x\text{ in m}).$$

(b) Use the same constant at both states.

The two endpoint values coincide: $$\boxed{U(0)=7\ \mathrm J},\qquad\boxed{U(3)=18-18+7=7\ \mathrm J}.$$

(c) Differentiate and compare endpoints.

Differentiate to recover the force and subtract endpoints for work: $$F_x=-U'(x)=6-4x\Rightarrow\boxed{F_x(2)=-2\ \mathrm N},$$ $$\boxed{W_{0\to3}=U(0)-U(3)=0}.$$ Positive and negative work contributions cancel; the force need not vanish throughout.