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

the motion of a pendulum is a classic example of mechanical energy conservation. a pendulum consists of a mass (known as a bob ) attached by a string to a pivot point. as the pendulum moves it sweeps out a circular arc, moving back and forth in a periodic fashion.