Momentum of a mass spring system >_<

[Titandwedebil]

Homework Statement

Alright, so we have a .5kg mass, a .08 amplitude oscillation, and a 130 N/m spring constant (k). The question wants to know what the velocity of this mass spring will be when it is at MAX.

Well, actually, it asks for the MOMENTUM of the system. The formula for that is P=mass x velocity; so we need to know the velocity in order to solve for the momentum.

Homework Equations

P=m x v

The Attempt at a Solution

Well, I thought I could use T=2pi (square root) mass/k; but, the period really wouldn't be relevant,would it? What is confusing me is the fact we have to find the maximum velocity given only k, mass, and the amplitude.

 

[Mike Pemulis]

There are a couple of different ways to approach this. Which way is easier depends on what you've learned in class so far. Have you gone over potential energy, kinetic energy, and energy conservation? How about derivatives?

 

[Titandwedebil]

We have briefly gone over PE and GPE. What I know from our discussion about it is that Elastic potential energy applies to mass spring systems; and that PE=1/2kx^2 is the equation. I know we've gone over KE as well, the equation just doesn't ring a bell at the moment. Derivatives, however, we have not discussed.

 

[Mike Pemulis]

Okay, we're in business. The PE = 1/2kx^2 is the potential energy of a spring. You need the definition of kinetic energy also, so look that up. Now, the potential and kinetic energies add up to the total energy, which is constant if there is no interaction with the environment (friction, etc). So we have:

KE + PE = E_total

We want the maximum kinetic energy. What is potential energy when kinetic energy is maximum? And how can you find E_total?

 

[Titandwedebil]

Well, KE's equation is (from looking at my notes), KE=1/2mv^2, and PE should be 0 when KE is max (done conceptually in my head). Total mechanical energy, as you said, is KE + PE...so, could we combine the two equations, set all of PE to zero maybe? and then solve for V?

 

[Mike Pemulis]

Exactly. Now you need the total energy. Where does that come from?

 

[Titandwedebil]

That would have to be the amplitude, right?

 

[Mike Pemulis]

Not the amplitude itself. It's related to the amplitude, though.

Okay, I'm out of here for the day. Good luck!

 

[Titandwedebil]

Amplitude²?

I tried that, used PE+KE=TME, set PE to zero, rearranged for v (v²=A²/1/2 mass, plugged my numbers in, and ended up getting .16 meters per second for the velocity.

 

[Mike Pemulis]

Close, but not exactly. Think about what the amplitude actually is. How can you use that to obtain an energy?

 

[Titandwedebil]

Does it involve PE and KE?

 

[Mike Pemulis]

Yeah. Here's one way to look at it. We know that total energy is constant, so if we can find it at one point we know it is the same everywhere. And we also know that,

E_total = PE + KE

So if there is a certain point where we know both PE and KE, that gives E_total. Where is that point? The minimum, maximum, equilibrium, somewhere in between?

 

[Titandwedebil]

Ah! I figured it out! PE is the same as KE when KE is maxxed. :D I found the answer, w00t! Thanks so much for helping me!

Now, there's still the issue of Amplitude; which plays another role in one of my other practice problems. All I really know and all I can gather is that amplitude relates with energy and that it's the distance from equilibrium point to a crest or trough. There's a problem asking me to solve for A without giving a graph or any indication as to how high it hits on the Y axis.

 

[Mike Pemulis]

All I really know and all I can gather is that amplitude relates with energy and that it's the distance from equilibrium point to a crest or trough.

That's true. The amplitude is the distance from equilibrium to a crest or trough. Now imagine a spring with constant k and amplitude A at (for example) a crest, the maximum distance from equilibrium. What's the kinetic energy? What's the potential energy?

 

[Titandwedebil]

At the top of a crest/trough, PE would be max, and KE would be 0.

 

[Mike Pemulis]

Yup. What is this max PE?

 

[Titandwedebil]

I don't think I have enough information to plug into its equation though. If it gives me the wavelength, could I use that to solve for the amplitude? (Considering this is a spring, and not a wave?)

 

[Mike Pemulis]

No, you just need the amplitude. Let's say the amplitude is A, and the spring constant is k. You need to find a formula in terms of A and k that gives the maximum potential energy.

In order to get this, you have to realize what the amplitude actually is, physically. If you were given an oscillating spring, how would you measure the amplitude? Once you have that, it's immediately obvious.

 

[Titandwedebil]

X? PE=1/2kx^2?

 

[Mike Pemulis]

Yes! And now you can solve for the speed at the equilibrium point.

 

[Titandwedebil]

Ah! I got it! Thanks so much for all the help, Mike!