The Force Law for Simple Harmonic Motion — Question 2

Question 2

The geometry of a spring pair sets its effective stiffness

A mass $m$ is attached either to two massless springs of constants $k$ and $3k$ in series, or to those springs in parallel. The motion is one-dimensional and both springs remain in their linear ranges. Ignore damping.

1. Derive the effective stiffness in each arrangement.

2. Find the ratio of series period to parallel period. Explain why adding the spring constants is valid in only one arrangement.

Question 2 - Solution

The geometry of a spring pair sets its effective stiffness

(a) Derivation and calculation

For series springs, the force is common and extensions add:$$x=x_1+x_2=F\left(\frac1k+\frac1{3k}\right)=\frac{4F}{3k},\quad\boxed{k_s=\frac{3k}{4}}.$$For parallel springs, the displacement is common:$$F=F_1+F_2=kx+3kx=4kx,\qquad\boxed{k_p=4k}.$$

(b) Result and interpretation

$$T_s=2\pi\sqrt{\frac{m}{k_s}},\qquad T_p=2\pi\sqrt{\frac{m}{k_p}}.$$$$\boxed{\frac{T_s}{T_p}=\sqrt{\frac{k_p}{k_s}}=\sqrt{\frac{4k}{3k/4}}=\frac4{\sqrt3}=2.3094}.$$Adding stiffnesses assumes equal extension; adding compliances assumes equal force.

Physical check

$k_s<k$ and $k_p>3k$: the series assembly is softer than either spring, while the parallel assembly is stiffer than either.