Challenging physics problem with spring, oscillations and moment of inertia?

[Kratos321]

Hello everyone! Here is the link to the problem:
http://i.imgur.com/2HNrQ.jpg

I don't even know where to start. Any help to get started would be greatly appreciated. Thank you so much.

Cheers

 

[haruspex]

Start by assigning variable names for the forces and any other key variables (as functions of time), then consider the spring and each mass in turn to obtain equations relating them. Work entirely with symbols for values. Don't plug the numbers in until the final step.

 

[Kratos321]

Okay, here's a start.

The spring starts out unstretched, so it has zero potential energy. Nothing is moving, so there's no kinetic energy. All the energy of the system is gravitational potential energy.

As the weight falls a distance x, some of that energy is converted to potential energy in the spring.
The gravitational potential energy decreases by mgx.
The spring potential energy increases by 1/2 kx^2.

At the weight's lowest point, there is again no motion, so all of the energy is in the spring.
This occurs 20 cm below the starting point (the highest point).
k = 2mg/x = 2 * 2kg * 9.8m/s^2 / .2m
k = 196 N/m

In between the end points, the system also has kinetic energy.
As the string moves with speed v, the pulley turns with angular speed ω = v/R.
The moment of inertia of a disk is 1/2 mR^2, so the pulley's kinetic energy is 1/4 m v^2.
Meanwhile the hanging weight has kinetic energy of 1/2 mv^2.
So the total kinetic energy of the system is 3/4 m v^2.

I'm not sure how to get from kinetic energy to oscillation period.

 

[haruspex]

Good start. If x is the spring extension at time t, what are the relationships between:
v(t) and x(t)?
KE of system and PE of system?

 

[Kratos321]

I've been thinking about your questions and I am extremely confused. Please expand?

 

[haruspex]

Let x(t) be the spring extension at time t. What differential equation relates that to v(t), the velocity of the mass?
You have determined the kinetic and potential energies of the components of the system. What equation connects these?

 

[Kratos321]

Hmm, I see. Is this right so far?

x(t) = 0.2cos(ωt)
v(t) = -0.2ωsin(ωt)

3/4 m*v^2 + mgx = 1/2*k*x^2

 

[haruspex]

x(t) = 0.2cos(ωt)
v(t) = -0.2ωsin(ωt)

I was looking for merely that dx/dt = v. Instead, you've leapt straight to the solution of the ODE. I guess that's ok, since the question effectively tells you it's SHM.

3/4 m*v^2 + mgx = 1/2*k*x^2

Not quite. The total energy should be constant: KE mass + KE pulley + PE mass.
Once you have the right energy equation, you can use your equations for x(t) and v(t).