Conservation of Mechanical Energy — Question 10

Question 10

Energy fractions depend on the stated potential zero

A system with conserved mechanical energy has K=6 J and U=4 J in one convention. A second convention adds 10 J to U at every configuration.

1. Find E and the fraction K/E in each convention.

2. Does changing the convention alter any speeds or turning positions?

3. How should a claim such as half the energy is kinetic be made unambiguous?

Question 10 - Solution

Energy fractions depend on the stated potential zero

(a) Shift the total with the potential.

In the first convention, $$\boxed{E=6+4=10\ \mathrm J,\qquad K/E=.60}.$$ After shifting $U$ by 10 J, $$\boxed{\widetilde E=6+14=20\ \mathrm J,\qquad K/\widetilde E=.30}.$$

(b) Check the invariant difference.

The measurable energy gap is invariant: $$\widetilde E-\widetilde U=(E+10)-(U+10)=E-U=K.$$ Speeds and turning positions $E=U$ therefore do not change.

(c) Specify the reference and the energy being compared.

An absolute energy fraction depends on the datum: $$\frac K{E+C}\ne\frac KE\quad\text{in general}.$$ Specify the potential zero and the total being compared, or use a defined available-energy difference, whose value is invariant under a common shift.