Question

A light spring with spring constant $$1 200 N/m$$ is hung from an elevated support. From its lower end hangs a second light spring, which has spring constant $$1 800 N/m$$. An object of mass $$1.50 kg$$ is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.

Answer 1

The same force makes both light springs stretch.
(a) The hanging mass moves down by
$$x=x_{1}+x_{2}=\dfrac{mg}{k_{1}}+\dfrac{mg}{k_{2}}=mg\left ( \dfrac{1}{k_{1}}+\dfrac{1}{k_{2}} \right )$$
$$=(1.5kg)(9.8m/s^{2})\left ( \dfrac{1}{1200N/m}+\dfrac{1}{1800N/m} \right )$$
$$=2.04\times 10^{-2}m$$
(b) We define the effective spring constant as
$$k=\dfrac{F}{x}=\dfrac{mg}{mg(1/k_{1}+1/k_{2})}=\left ( \dfrac{1}{k_{1}}+\dfrac{1}{k_{2}} \right )^{-1}$$
$$=\left ( \dfrac{1}{1200N/m}+\dfrac{1}{1800N/m} \right )=720N/m$$