Conservation of Energy — Question 6

Question 6

An incomplete energy audit with hard uncertainty bounds

An input meter reports $[118,122]$ J. Independent measurements give kinetic-energy increase $[28,32]$ J, potential increase $[18,22]$ J, thermal increase $[38,42]$ J, and measured outward heat $[5,7]$ J. No other reported terms exist.

1. Find the full interval for input minus all reported storage increases and output.

2. Can exact energy closure occur within these bounds?

3. If the only missing term is an outward transfer, find its allowed interval and state the limitation of that inference.

Question 6 - Solution

An incomplete energy audit with hard uncertainty bounds

(a) Combine compatible extreme bounds.

For residual $R=I-\Delta K-\Delta U-\Delta E_{\mathrm{th}}-Q_{\mathrm{out}}$, $$R_{\min}=118-(32+22+42+7)=15\ \mathrm J,$$ $$R_{\max}=122-(28+18+38+5)=33\ \mathrm J.$$ $$\boxed{R\in[15,33]\ \mathrm J}.$$

(b) Test whether zero is possible.

Exact closure would require $$R=0,\qquad0\notin[15,33].$$ Thus the reported terms cannot close the balance within the stated independent hard bounds.

(c) Interpret rather than rename the discrepancy.

If the only missing term is an outward transfer $X$, then $$R-X=0\Rightarrow\boxed{X\in[15,33]\ \mathrm J}.$$ Without that additional assumption, omitted storage, an incorrect system boundary, or invalid measurement bounds could also explain the mismatch. The interval alone does not identify its mechanism.