When hung from an ideal spring with spring constant k = 1.5 N/m, it bounces up and down with some frequency ω, if you stop the bouncing and let it swing from side to side through a small angle, the frequency of the pendulum is half the bounce frequency ω/2. What is the length of the un-stretched spring l = ? Note ω should not be part of your answer.

L = ¼ k g / m

Explanation:

This is an interesting exercise, in the first case the spring bounces under its own weight and in the second it oscillates under its own weight.

The first case angular velocity, spring mass system is

w₁² = k / m

The second case the angular velocity is

w₂² = L / g

They tell us

w₂ = ½ w₁

Let's replace and calculate

√ (L / g) = ½ √ (k / m)

L / g = ¼ k / m

L = ¼ k g / m