Determining Potential Energy Values — Question 7

Question 7

Reference shifts versus calibration errors

Two instruments report $U_A(x)=2x^2+5$ J and $U_B(x)=2x^2-7$ J. A third reports $U_C(x)=3x^2+5$ J. All use the same position coordinate in meters.

1. Can A and B describe the same interaction? Find their offset.

2. Compare their predicted works from x=1 to x=2 m.

3. Can a reference shift reconcile A and C? Give a force-based check.

Question 7 - Solution

Reference shifts versus calibration errors

(a) Compare the full functions.

Compare the full functions: $$\boxed{U_B-U_A=(2x^2-7)-(2x^2+5)=-12\ \mathrm J}.$$ The constant offset allows the same interaction with different reference zeros.

(b) Subtract endpoints within each convention.

Each instrument predicts the same endpoint work: $$\boxed{W_A=7-13=-6\ \mathrm J},\qquad\boxed{W_B=-5-1=-6\ \mathrm J}.$$ Both also give $F=-4x$ N.

(c) A changing discrepancy is not a zero shift.

Instrument C differs in a position-dependent way: $$U_C-U_A=x^2,\qquad F_C=-6x\ne-4x=F_A.$$ No additive constant reconciles their derivatives, so A and C cannot represent the same force calibration.