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

AI Thread Summary
The discussion centers on a challenging physics problem involving a spring, oscillations, and moment of inertia. Participants suggest starting by defining variables and deriving equations symbolically before substituting numerical values. Key energy transformations are highlighted, including the conversion of gravitational potential energy to spring potential energy as a weight falls. The conversation also touches on the relationships between kinetic energy, potential energy, and the motion of the system, leading to equations for displacement and velocity. Ultimately, the importance of maintaining a consistent energy equation throughout the analysis is emphasized.
Kratos321
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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
 

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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.
 
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.
 
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?
 
I've been thinking about your questions and I am extremely confused. Please expand?
 
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?
 
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
 
Kratos321 said:
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).
 
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