An Angular Simple Harmonic Oscillator — Question 3

Question 3

Wire radius matters to the fourth power

A circular elastic wire has supplied torsional stiffness $\kappa=G\pi a^4/(2L)$, where $a$ is wire radius, $L$ its length and $G$ a fixed material modulus. It suspends a body of fixed inertia $I$.

1. Find how the period scales with $L$ and $a$. If length is doubled, what radius factor keeps the period unchanged?

2. Estimate the fractional period change from a small $1\%$ increase in radius alone, and compare with the exact ratio.

Question 3 - Solution

Wire radius matters to the fourth power

(a) Derivation and calculation

$$\kappa=\frac{G\pi a^4}{2L},\qquad T=2\pi\sqrt{\frac{2IL}{G\pi a^4}}\propto\frac{\sqrt L}{a^2}.$$$$\frac{T_2}{T_1}=\sqrt{\frac{L_2}{L_1}}\left(\frac{a_1}{a_2}\right)^2.$$$$L_2=2L_1,\ T_2=T_1\Rightarrow\boxed{a_2/a_1=2^{1/4}=1.18921}.$$

(b) Result and interpretation

For a $1\%$ radius increase at fixed $L,I,G$,$$\boxed{T_2/T_1=(1.01)^{-2}=.980296}.$$$$\boxed{\frac{\Delta T}{T_1}\times100\%=-1.9704\%}.$$$$d\ln T=\tfrac12d\ln L-2d\ln a\Rightarrow\Delta T/T\simeq-2\%.$$

Physical check

The strong radius sensitivity follows from the fourth power in stiffness; the differential estimate agrees with the exact ratio.