What Is Energy? — Question 6

Question 6

Hard measurement bounds can expose a missing transfer

For this introductory problem use the supplied accounting rule $\Delta E=E_{\mathrm{in}}-E_{\mathrm{out}}$, with all quantities in joules. Independent measurements give input in $[98,102]$ J, output in $[58,62]$ J and stored-energy increase in $[43,45]$ J. These are hard bounds, not standard deviations.

1. Find the energy-increase interval predicted from input and output alone.

2. Can the three observations agree if all transfers are listed?

3. If an omitted input q is the only missing term, find all possible q values compatible with the bounds.

Question 6 - Solution

Hard measurement bounds can expose a missing transfer

(a) Combine the extreme admissible readings.

The governing relation is $$\boxed{\Delta E_{\mathrm{predicted}}\in[I_{\min}-O_{\max},I_{\max}-O_{\min}]}$$

$$\Delta E_{\mathrm{predicted}}\in[98-62,102-58]=[36,44]\ \mathrm J.$$

(b) Intersect the intervals.

Yes. The common interval is $[43,44]\ \mathrm J$, so no missing transfer is compelled by these bounds. Central values give 40 J and 44 J, but a central-value mismatch alone is not proof.

(c) Solve for the omitted term.

$$q=\Delta E-E_{\mathrm{in}}+E_{\mathrm{out}}\in[43-102+58,45-98+62]=[-1,9]\ \mathrm J.$$ Requiring an input, $q\ge0$, gives $[0,9]\ \mathrm J$. Zero remains feasible.