Damped Simple Harmonic Motion — Question 7

Question 7

Dry friction produces an amplitude decrement instead of a ratio

A horizontal spring oscillator slides with constant kinetic-friction magnitude $F_c$ opposing velocity. It starts from a positive turning point $A$ and reaches a negative turning point $-B$, with $A,B>0$. Assume it slides throughout that half-cycle and does not stick at its endpoints.

1. Use work and energy to derive $A-B$.

2. For $k=100\,\mathrm{N/m}$ and $F_c=.20\,\mathrm{N}$, predict the next opposite-side amplitude from $A=.020\,\mathrm{m}$. Explain how several peaks distinguish dry friction from viscous damping, and identify the near-stop limitation.

Question 7 - Solution

Dry friction produces an amplitude decrement instead of a ratio

(a) Derivation and calculation

The sliding distance from $+A$ to $-B$ is $A+B$:$$\tfrac12kA^2-\tfrac12kB^2=F_c(A+B).$$$$(A+B)\left[\tfrac12k(A-B)-F_c\right]=0.$$$$A+B>0\Rightarrow\boxed{A-B=2F_c/k}.$$

(b) Result and interpretation

$$\Delta A=2(.200)/100=.00400\,\mathrm m,\quad\boxed{B=.0200-.00400=.0160\,\mathrm m}.$$$$\text{dry friction: }A_n-A_{n+1}=\text{constant};$$$$\text{viscous damping: }A_{n+1}/A_n=\text{constant}.$$Near rest, static friction may prevent another crossing or hold the block. The sliding derivation cannot be extended to negative amplitudes.

Physical check

The amplitude decrement has dimensions of length and is independent of the starting amplitude only while the assumed sliding half-cycle exists.