Show that simple harmonic motion occurs

[Doobydoo913]

Homework Statement

A wheel with moment of inertia I. A mass of M is connected to a belt and runs over the wheel with radius R and connected to a spring with a stiffness of k which is connected to the ground.

Show that if the mass is pulled down with a force of F and released the system will oscillate with SHM with a frequency

f=1/2pi sqrt k/(M+I/R^2)

Homework Equations

FI=ma Inertia force
FS = kx Spring force
TI=Ialpha Inertia torque

a=-w^2x

The Attempt at a Solution

Really confused as to the wheel being added, I don't know how to incorporate the +I/R^2 into the frequency calculation. I also don't know how to get the I/R^2 part either.

Following notes i have

a=-w^2x

F/M =-w^2x (not sure where F/M comes from can someone explain?)

-kx/M=w^2x

k/M=w^2

w= sqrt k/m

w=2 pi f sub for w

2 pi f = sqrt k/M

Im missing out the I/R^2

Need help on this one, Thankyou in advance

 

[BiGyElLoWhAt]

F/M =a

I'm sure x is supposed to be the radius.

The reason you missed out on the I/R^2 term is because you didn't include it anywhere to begin with.

I'm also having trouble interpreting the situation though. The description above isn't giving me a good visualization of how everything's connected.

 

[BiGyElLoWhAt]

Oh no, x is the displacement =P sorry about that.

 

[BiGyElLoWhAt]

Are we looking at something like this?

 

[Doobydoo913]

Im not sure how to incorporate the inertia or work it out.

 

[BiGyElLoWhAt]

Oh, I like your picture better. I thought the block was attached to the spring and it wasn't making sense.

Start out just like you did in your notes. Start with N2L, and this time incorporate the wheel.

Hint
Your first step you defined the [edit]Force on the block[un-edit] in terms of the spring, this time you have to define it in terms of the spring and the wheel.

 

[Doobydoo913]

Is there a relation between spring and wheel?

 

[BiGyElLoWhAt]

Hmmm... depends on what you mean by that. There's a relationship between a quantity used to describe the force exerted by the spring and a quantity used to describe the state of the wheel. I'm not sure if that's entirely useful though.

Why don't you explain to me step by step what the equations from your notes mean, and where the terms come from. The very first one can be derived, but it was probably given to you in class, so I will give that to you now. Walk me from step 1 to step 2... etc.

 

[BiGyElLoWhAt]

Ok so I just got home from work, and worked through this problem; as it turns out, this quantity IS useful. Sorry if that was misleading in my previous post.

 

[Doobydoo913]

I have been shown the calculations, but he can't tell me how he arrived at the solution could you explain?

 

[BiGyElLoWhAt]

Well first thing, Torque = R x F, not F x R.

Second thing, either your prof skipped a step when solving for ma or you didn't write it down. I haven't gotten through the whole procedure yet, but let's take it step by step, cool?

You know the sum of the forces on the block = mass of the block x acceleration of the block or sum(F) = ma, and therefore sum(F)/m = a.

What is the sum of the forces on the block? Keep in mind, throughout the whole system, there are 5 forces to be accounted for (not all of them DIRECTLY on the block, though). Hopefully that was a good enough hint as to clue you into what we're looking for here.

 

[Doobydoo913]

I can only think of 3 forces, and I think that is all I need for work at this level.

The force of the spring
Torque inertia of the wheel
And inertia force

Is this correct?

 

[BiGyElLoWhAt]

Yes that will work, I thought about it, and what you've got going on so far seems reasonable. I did it another way, which included tensions and weight, but they ended up dropping out anyways.

So let me ask you a question. Is there a particular aspect to this problem that you're getting lost in? Or is it just the method in general?

 

[Doobydoo913]

I think I may have cracked this

Total force to pull down = torque inertia of wheel + spring force + inertia force

I haven't got my notes on me but I've worked it through and it seems right.

Thanks for the help

 

[BiGyElLoWhAt]

No problem.