Finding the maximum kinetic energy of a harmonic oscillator

[apbuiii]

Homework Statement

A harmonic oscillator with a vertical mass on a string has a hanging mass of 2m and a spring constant of K. It oscillates with an amplitude of Z. When its position is at a distance Z/2 of the equilibrium point, its potential energy is U_i. What is the maximum kinetic energy K_m?

Homework Equations

E=1/2mV² + 1/2mA²

The Attempt at a Solution

I realized that the max kinetic energy must be equal to the to the max potential energy because total energy is constant. I just don't know where to put the equations to make it fit.

Can someone show me the set up? The answer give was K_m = 4U_i

Thanks in advance!

 

[technician]

Can you see that ω^2 = k/2m ?
Also max velocity = ωZ
So you can get an expression for the max KE
The velocity at a displacement x is given by v^2 = ω^2(A^2-x^2)
can you use this to get the KE at a displacement Z/2
If you combine this lot you should get the answer
Hope this helps... I will follow this up if you have trouble

 

[apbuiii]

I don't think I used your method, but I did end up figuring it out. Thanks for your help though.