How can mass distribution affect the derivation of a natural frequency equation?

[knight92]

Hi I want to derive a natural frequency equation for a system. I saw this question in a book.
so they gave us an equation
frequency = (1/2pi) * ( sqrt( k/m₁+m₂+m₃/3+4m₄+Ic/r²) )

Ic = Moment of inertia of pulley about rotational axis
r = effective radius of pulley
k = spring constant

now the question says that I have to derive this equation from Newtons second law f=ma (The body is pulled down a bit and left to oscillate). I have done that no problem but I don't get why mass m3 which is the mass of the spring is divided by 3 and mass m4 which is mass of the body is multiplied by 4. Can anyone help ? I have attached an image of the setup that I drew on photoshop, I don't have a scanner so I can't scan and upload the original diagram itself sorry. In the Second image the pulley is rotated clockwise and left to oscillate/rotate 10 times to calculate moment of inertia

 

[AlephZero]

Different parts of the system are moving at different speeds.

Call the velocity of the pulley be V.
Mass m4 has Velocity 2V.
For the distributed mass of the spring along its length, the velocity varies linearly between 0 and V.
The pulley also has angular velocity V/R

Angular frequency = sqrt(strain energy / kinetic energy).

The different velocities give the different factors on the different m's in the KE.

 

[Curl]

Doesn't sound too bad, you can lump up all the masses then you just have a 2nd order ODE. Solve with MATLAB's ode45 function and determine the frequency numerically.

done