Work and Potential Energy — Question 9

Question 9

Bounds on a potential change from a work sensor

Work by a conservative interaction from A to B is known only to lie in $[11,15]$ J. A reference assignment gives $U_A=4$ J exactly.

1. Find the tight interval for $U_B$.

2. If a second instrument instead reports $U_B\in[-9,-7]$ J, are the instruments compatible?

3. Can you conclude the exact work by taking the midpoint of the overlapping interval?

Question 9 - Solution

Bounds on a potential change from a work sensor

(a) Reverse the endpoints when subtracting.

Subtract the bounded work, reversing the endpoint order: $$U_B=4-W_{AB}\Rightarrow\boxed{U_B\in[4-15,4-11]=[-11,-7]\ \mathrm J}.$$

(b) Intersect the allowed states.

Intersect the independent constraints: $$[-11,-7]\cap[-9,-7]=[-9,-7]\ \mathrm J.$$ They are compatible and imply $\boxed{W_{AB}\in[11,13]\ \mathrm J}$.

(c) Do not invent a probability model.

The midpoint is only one allowed value: $$W_{\mathrm{midpoint}}=(11+13)/2=12\ \mathrm J,\qquad W_{AB}\in[11,13]\ \mathrm J.$$ Hard bounds supply no probability distribution or exact central estimate.