Are the Spring and Pendulum Systems Oscillating with the Same Frequency?

[SpaceExplorer]

A weight W is suspended from a rigid support by a hard spring with stiffness constant 'K'. The spring is allowed to have only vertical motion. A simple pendulum of length 'l' with a bob of mass 'm' (mg<<W) is suspended from the weight W and is set oscillating in a horizontal direction. After sometime, the spring is observed to be exhibiting vertical oscillations of high amplitude. Are the 'Spring system' and 'Pendulum system' oscillating with the same frequency? And what should be the function 'f', if K=f(W,l)?

 

[Simon Bridge]

That's a neat problem - have you had a go tackling it yet?
Have you, say, drawn free body diagrams for the two masses?

What is the context the problem is set?

I'm picturing a slinky coil attached to a piston-head (W) which moves up and down in a cylinder in the ceiling.
The end of the piston sticks out of a hole in the cylinder and there is a hook on the end.
The pendulum swings from the hook.
That about right?

 

[SpaceExplorer]

I've tried. But there's so much approximation I needed to use to derive the final relation, (using energy consideration), I'm no longer satisfied with or sure about my answer. So better I thought if it be run by experts. I found both systems to be oscillating with the same freq. and K=w/l.
Yes, your physical description seems right.

 

[Simon Bridge]

OK. You need a way to be confident in your solution (or to tell if you've made a mistake)?
Does it make sense that the two parts should have the same frequency in this situation?
Hint: resonance

 

[SpaceExplorer]

Yes! That should be it, the question did say about the 'high amplitude' of the resultant oscillations, so there's a high possibility that the composite system is showing resonance, i.e the driven system may undergo oscillations at the same frequency as that of the driving system. Thanks for the help. It was great relearning resonance in the context of this problem.

 

[Simon Bridge]

When you see coupled oscillators and "big amplitude" think of "resonance" and "normal modes".
Since there is no external driving force, the two oscillators will exchange energy periodically.