Conservation of Energy — Question 9

Question 9

Repeated partial conversion of mechanical energy

A damped mechanical system starts with 120 J of mechanical energy measured relative to its minimum accessible potential. After each complete oscillation, 75 percent of the preceding mechanical energy remains. The surrounding enclosure retains all dissipated energy as thermal; no external transfers occur.

1. Find remaining mechanical energy after three oscillations.

2. Find accumulated thermal-energy increase.

3. At the same minimum-potential crossing on successive oscillations, by what factor does speed decrease for a fixed mass?

Question 9 - Solution

Repeated partial conversion of mechanical energy

(a) Apply the retention factor repeatedly.

Apply the same fractional retention to the remaining energy each cycle: $$E_n=E_0r^n\Rightarrow\boxed{E_3=120(.75)^3=50.625\ \mathrm J}.$$

(b) Use total energy conservation.

The enclosure retains all mechanical losses: $$E_3+\Delta E_{\mathrm{th}}=E_0\Rightarrow\boxed{\Delta E_{\mathrm{th}}=120-50.625=69.375\ \mathrm J}.$$

(c) Convert an energy ratio into a speed ratio.

At the same minimum-potential crossing, $E_n=\tfrac12mv_n^2$: $$\frac{v_{n+1}^2}{v_n^2}=\frac{E_{n+1}}{E_n}=.75\Rightarrow\boxed{\frac{v_{n+1}}{v_n}=\sqrt{.75}=\frac{\sqrt3}{2}=.8660}.$$ The speed factor is the square root of the energy factor.