The Force Law for Simple Harmonic Motion — Question 5

Question 5

Added masses can separate unknown stiffness from hidden inertia

A spring oscillator has unknown spring constant $k$ and unknown moving fixture mass $M$. With an added mass $m_1=0.20\,\mathrm{kg}$ its squared period is $T_1^2=0.30\,\mathrm{s^2}$. With $m_2=0.50\,\mathrm{kg}$ it is $T_2^2=0.60\,\mathrm{s^2}$. The spring mass is negligible.

1. Infer $k$ and $M$ from the two measurements.

2. Predict the squared period with no added mass. Why would plotting $T$ rather than $T^2$ against added mass conceal the simple linear relation?

Question 5 - Solution

Added masses can separate unknown stiffness from hidden inertia

(a) Derivation and calculation

$$T^2=\frac{4\pi^2}{k}(M+m)=s\,m+b.$$$$s=\frac{0.60-0.30}{0.50-0.20}=1.00\,\mathrm{s^2/kg},\quad\boxed{k=\frac{4\pi^2}{s}=39.4784\,\mathrm{N/m}}.$$$$M=\frac{T_1^2}{s}-m_1=0.30-0.20=\boxed{0.100\,\mathrm{kg}}.$$

(b) Result and interpretation

$$\boxed{T_0^2=sM=0.100\,\mathrm{s^2}},\qquad T_0=0.31623\,\mathrm s.$$$$T(m)=\sqrt{s(M+m)},\quad\frac{dT}{dm}=\frac{s}{2T},\quad\frac{d^2T}{dm^2}=-\frac{s^2}{4T^3}<0.$$The $T$ graph is curved; the $T^2$ graph separates stiffness (slope) from fixture mass (intercept).

Physical check

Back-substitution gives $T^2=0.30$ and $0.60\,\mathrm{s^2}$ for the two stated added masses.