SHM Simple Harmonic Motion, spring, pendulum.....?

Question

An apple weighs . When you hang it from the end of a long spring of force constant and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back and forth swings do not cause any appreciable change in the length of the spring.)

What is the unstretched length of the spring (i.e., without the apple attached)?


Ive gone through it all 3 times with silghtly different approaches.
I got

L = 4w/k

every time, but that is not correct. I am not sure what I am doing wrong..

Omega_bounce = sqrt(k/m) = 2*pi*f_bounce….solved for frequency of bounce.

f_pendulum = (1/2) f_bounce …. Solved for f_pendulum

omega_pendulum = 2*pi*f_pendulum = sqrt(g / L)….solved for L

Any Suggestions?

Apple WEIGHT (mg) = w

Spring constant = k

Answer 1

The period of the bounce is given by 2*π*√[m/k]. The period of the pendulum is 2*π*√[L/g]. The period of the pendulum is twice that of the bounce (freq is half), so

2*π*√[L/g] = 4*π*√[m/k]

L/g = 4*m/k

L = 4*m*g/k

L = 4*w/k

But that is with the weight attached. The unweighted length is L - m*g/k

L' = 4*m*g/k - m*g/k = 3*m*g/k

Answer 2

As far as I understand, any object moved has shm. I guess it doesn't always bounce or move back because of friction, or maybe the energy disappears some other way. I mean, isn't there a rule that for every action there is an equal and opposite reaction? Maybe they go together somehow. But then it wouldn't be motion. There were orbits in the picture I looked at. Maybe orbits.