Solve Torsional Pendulum Homework

[physiks]

Homework Statement

The same question was posted here before: https://www.physicsforums.com/showthread.php?t=73328

I'm struggling to come up with answers regarding the amplitude and energy in each case.

The Attempt at a Solution

For when the ring is dropped onto the disc when it is at rest, I can't find a way to use the maths to tell me what would happen. θ=θ₀cos(ωt+ψ) but I can't justify θ₀ not changing.

Obviously what happens to the energy follows from this (although I'm not sure how PE varies with amplitude exactly in this case so couldn't say anything quantitative).

When the ring is dropped onto the disc moving at ω_max, my initial thoughts are that the angular momentum is conserved at that instant (because the torque has no time to change the angular momentum in that instant), i.e the angular momentum of the disc a tiny bit before and of the disc plus ring a tiny bit after are the same. Then the maximum angular velocity reduces to 2ω_max/3. Using t=0,dθ/dt=ω_max,θ=0 gives θ=[√(I/c)]ω_maxsin(√c/I)t so then the amplitude rises by a factor of √(3/2). I'm really not sure about my approach here though. Also I'm not sure if I could apply θ=[√(I/c)]ω_maxsin(√c/I)t to the first case (I don't think I know anything about ω_max so couldn't).

Then the energy would obviously reduce (but I can't say anything quantitative).

If somebody could help me out that would be great, thanks!

 

[voko]

Seeing the principal equations would help.

In case 1, the basic observation is that the new pendulum is created when it is momentarily at rest, i.e., when the oscillation has reached its maximum. This by definition gives you the amplitude of the oscillation. Now, does the total energy change when the ring is added, and what does mean for the frequency?

In case 2, your main line of reasoning seems good. The angular momentum is momentarily conserved, which gives you angular velocity and thus kinetic energy which happens to be the max kinetic energy, so you can apply conservation of energy and obtain the amplitude.

 

[physiks]

Seeing the principal equations would help.

In case 1, the basic observation is that the new pendulum is created when it is momentarily at rest, i.e., when the oscillation has reached its maximum. This by definition gives you the amplitude of the oscillation. Now, does the total energy change when the ring is added, and what does mean for the frequency?

In case 2, your main line of reasoning seems good. The angular momentum is momentarily conserved, which gives you angular velocity and thus kinetic energy which happens to be the max kinetic energy, so you can apply conservation of energy and obtain the amplitude.

In terms of the frequency, the same effect occurs in each case as T=2∏√(I/c).

Case 1: so the amplitude stays the same. So the energy of the system must too be unchanged. Although it seems obvious the amplitude stays the same though, is there any physical principle which I can see this through?

Case 2: how can I relate the maximum KE to the amplitude though?

 

[voko]

Case 1: so the amplitude stays the same. So the energy of the system must too be unchanged. Although it seems obvious the amplitude stays the same though, is there any physical principle which I can see this through?

In simple harmonic motion, amplitude is directly related to max potential energy. Does the potential energy of the torsional pendulum depend on its moment of inertia?

Case 2: how can I relate the maximum KE to the amplitude though?

See above and keep in mind that total mechanical energy in SHM is constant.

 

[physiks]

In simple harmonic motion, amplitude is directly related to max potential energy. Does the potential energy of the torsional pendulum depend on its moment of inertia?



See above and keep in mind that total mechanical energy in SHM is constant.

In analogy with a mass spring system, does the PE vary with the amplitude squared? If so, why? Is there a sort of Hooke's law in this setup?

 

[voko]

This system is fully analogous to a mass-spring system. All SHM systems are mathematically equivalent.

Hooke's law for this system is given by the relation between torque and angular displacement.

 

[physiks]

This system is fully analogous to a mass-spring system. All SHM systems are mathematically equivalent.

Hooke's law for this system is given by the relation between torque and angular displacement.

Hmm I thought this but how is the PE of a pendulum of this form. Surely then it depends on -mglcostheta? I know it's true because the PE of a -kx force has to be like this. The pendulum just put me off a little.

 

[voko]

If the "pendulum" is a gravity pendulum, then it undergoes simple harmonic motion only in the "small oscillation" approximation. Large displacements result in anharmonic motion. But that is no different from a mass-spring system, which also becomes anharmonic at large displacements where Hooke's law breaks down.