Determining Potential Energy Values — Question 3

Question 3

Joining two potential formulas at a force boundary

For $0\le x\le1$ m, $F_x=3$ N. For $1\le x\le3$ m, $F_x=-2$ N. Set U(0)=4 J; the force remains finite at the join.

1. Find U(x) on the first interval.

2. Find a continuous U(x) on the second interval.

3. Find U(3) and explain why setting an independent zero at x=1 would be inconsistent.

Question 3 - Solution

Joining two potential formulas at a force boundary

(a) Integrate from the reference.

On the first interval, $$\boxed{U(x)=U(0)-\int_0^x3\,ds=4-3x},\quad0\leq x\leq1.$$ The joining value is $U(1)=1$ J.

(b) Carry the joining value into the next interval.

Carry that value into the second integral: $$\boxed{U(x)=1-\int_1^x(-2)ds=1+2(x-1)=2x-1},\quad1\leq x\leq3.$$

(c) Preserve the physical continuity.

At the final point, $$\boxed{U(3)=2(3)-1=5\ \mathrm J},\qquad U(1^-)=U(1^+)=1\ \mathrm J.$$ A bounded force contributes vanishing work over an infinitesimal interval. Resetting the zero at the join would introduce an unphysical jump and corrupt cross-boundary work.