Conservation of Energy — Question 10

Question 10

A proposed isolated release must balance every energy store

An isolated device releases 100 J of chemical energy. A report claims kinetic energy increases 70 J, elastic potential increases 20 J, and thermal energy increases 15 J. No other storage changes or boundary transfers occur.

1. Check whether the reported event conserves energy.

2. If the kinetic and elastic changes are correct, find the thermal increase required for closure.

3. In a revised design, the elastic increase must remain 20 J and thermal production is at least 10 J. What is the largest possible kinetic-energy gain, and does that bound alone establish a realizable design?

Question 10 - Solution

A proposed isolated release must balance every energy store

(a) Sum signed changes, including the source decrease.

Include the source as a negative storage change: $$\boxed{\Delta E=-100+70+20+15=+5\ \mathrm J\ne0}.$$ The claimed isolated event has a 5 J bookkeeping surplus and violates the stated balance.

(b) Use the measured source and the two fixed gains.

Keeping the source, kinetic gain, and elastic gain fixed, $$-100+70+20+\Delta E_{\mathrm{th}}=0\Rightarrow\boxed{\Delta E_{\mathrm{th}}=10\ \mathrm J}.$$

(c) Apply the lower bound on the other demand.

The revised constraints imply $$\Delta K=100-20-\Delta E_{\mathrm{th}},\qquad\Delta E_{\mathrm{th}}\geq10\ \mathrm J,$$ $$\boxed{\Delta K\leq70\ \mathrm J}.$$ Equality requires the minimum thermal production. This necessary energy bound does not prove that a mechanism can satisfy the force, motion, and conversion constraints.