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

Well if it was 8 hours in the morning and 7 in the evening per day of the week it would be 105 divided by 2 for the number of sessions would equal to 52.5 sessions. if it was just that amount of time from that whole week, it would be 15 hours divided by 2 which would give you 7.5... Did that help?