Energy in Simple Harmonic Motion — Question 9

Question 9

Equal amounts of extracted energy do not cause equal amplitude drops

An otherwise ideal spring oscillator starts with amplitude $A_0=.10\,\mathrm{m}$ and stiffness $k=100\,\mathrm{N/m}$. A device removes exactly $0.080\,\mathrm{J}$ at each successive equilibrium passage while sufficient kinetic energy remains. It supplies no energy between passages.

1. Find the amplitudes after the first and fifth complete extractions.

2. How many full extractions are possible? State the residual amplitude and explain why extrapolating a linear amplitude decrease is wrong.

Question 9 - Solution

Equal amounts of extracted energy do not cause equal amplitude drops

(a) Derivation and calculation

$$E_0=\tfrac12(100)(.10)^2=.500\,\mathrm J,\quad E_n=.500-.080n.$$$$\boxed{A_n=\sqrt{\frac{2(.500-.080n)}{100}}}\quad(E_n\ge0).$$$$A_1=\sqrt{.0084}=\boxed{.09165\,\mathrm m},\quad A_5=\sqrt{.0020}=\boxed{.04472\,\mathrm m}.$$

(b) Result and interpretation

$$n_{\max}=\left\lfloor\frac{.500}{.080}\right\rfloor=\boxed6.$$$$E_6=.0200\,\mathrm J,\quad A_6=\sqrt{.0004}=.0200\,\mathrm m.$$$$E_7=-.0600\,\mathrm J<0.$$A seventh full removal is impossible; at most the remaining $.0200\,\mathrm J$ can be extracted.

Physical check

Amplitude decreases as the square root of energy, so equal energy removals do not produce equal amplitude decrements.