# Homework Help: Yoyo Problem

1. Dec 13, 2013

### KiNGGeexD

Yoyo Problem:(

Question:

A cylindrical yoyo of mass m and radius R falls a height h from rest. What is its velocity after this fall?

My attempt:

I drew a diagram and at rest determined that

T-mg=ma

Where T is tension and acceleration is centripetal acceleration v^2/r

So

v= sqrt of T-gR

I also know the moment of inertia is

1/2mR^2

I'm not sure how to find the velocity after fall h?
I think it is right in front of me but I can't seem to see it :(

2. Dec 13, 2013

### voko

Conservation of energy?

3. Dec 13, 2013

### KiNGGeexD

So kinetic energy =0 at the top and kinetic energy after height h is =1/2mv^2?

So I can get potential energy with my velocity I calculated and the that would equal the kinetic energy after height h?

4. Dec 13, 2013

### KiNGGeexD

Ok I done

mgh=1/2Iω^2

Worked through to isolate for v

Since ω=v*r

And I got

v=sqrt of 4*g*h

Not sure if this is ok or not

Also can I just ignore v initial?

5. Dec 13, 2013

### TSny

Using I = (1/2)MR2 in the expression (1/2)Iω2 gives the rotational kinetic energy about the center of the yo-yo. But there is some additional kinetic energy due to the fact that the yo-yo is "falling".

6. Dec 13, 2013

### KiNGGeexD

Ah so there is angular kinetic energy and translational kinetic energy which when subtracted from the potential energy is zero?

7. Dec 13, 2013

### TSny

Yes. Both forms of KE must be included.

It is usually best to apply the conservation of energy principle in the form Ef = Ei. The way you have phrased it as taking the total KE and subtracting the PE to get zero seems to me a little dangerous even though I think you will probably get the correct answer that way.

8. Dec 13, 2013

### KiNGGeexD

Ok thanks!! I think I'm getting the hang of the physics malarkey!

Cheers friend:)

9. Dec 14, 2013

### KiNGGeexD

Would E final just be kinetic energy and E initial would be potential!
This would be the same as saying subtracting the two would be zero wouldn't it?

Potential energy is at a max when it is stationary and kinetic is at a maximum when it is moving after a time, assuming it doesn't reach the bottom of its path

10. Dec 14, 2013

### TSny

Yes, this is ok as long as you realize that you have made a choice of taking the potential energy to be zero at the final height of the yoyo. Whenever you use the formula mgh for the potential energy, you just want to be clear on where you are choosing h to be zero. If I chose h to be zero on the ground instead of at the lowest point of travel of the yoyo, then E final of the yoyo would not be all kinetic energy.

Maybe I'm belaboring a point that is already clear to you!

11. Dec 14, 2013

### KiNGGeexD

Ok I shall express the conservation of energy in the way you quoted!

12. Dec 15, 2013

### KiNGGeexD

v= square root of 2gh?

Looks nice and beat but I could have made a mistake in my maths:)
Wouldn't be the first time

13. Dec 15, 2013

### TSny

That's not the correct answer. Were you able to combine the rotational and kinetic energies into one expression?

14. Dec 15, 2013

### KiNGGeexD

Yea

mgh= 1/2mv^2 + 1/2 I ω^2

Then I solved for v?

15. Dec 15, 2013

### TSny

That's the right setup. Devil is in the details.

16. Dec 15, 2013

### KiNGGeexD

Yea sure......

mgh= mv^2+Iω^2

The halfs add to be 1

So

mgh/Iω^2= mv^2

v^2= mgh/m*Iω^2

I= 1/2MR^2

Have I made a mistake yet?

17. Dec 15, 2013

### TSny

No. Note that (1/2)(4) + (1/2)(6) = 2 + 3 = 5

But, if you said the halfs add to 1 you would get

(1/2)(4) + (1/2)(6) = 4 + 6 = 10 (wrong)

No, you can't do that (legitimately).
If you have a = b + c, it is not true in general that a/c = b. If you wanted to get b by itself, you could subtract c from both sides. But, you don't want to do that anyway.

Going back to mgh = (1/2)mv2 + (1/2)Iω2, substitute for ω in terms of v and try to simplify the right hand side.

18. Dec 15, 2013

### KiNGGeexD

Ok I will give it a bash!!!

19. Dec 15, 2013

### KiNGGeexD

Also I for some reason thought the two terms were multiplied and meant to put a subtract sign not a divide I'm am not that dumb!

20. Dec 16, 2013

### KiNGGeexD

Would it then be

v^2= (4gh)/3

Then take the square root to get v! I thought it would be neater to do it this way!

21. Dec 16, 2013

### KiNGGeexD

I ended up with the RHS as

1/2nv^2 + 1/4 mv^2

Once I substituted in omega and moment of inertia

22. Dec 16, 2013

### KiNGGeexD

Although the dimensions don't add up

23. Dec 16, 2013

### TSny

Yes, that looks right!

24. Dec 16, 2013

### TSny

Right: 1/2 mv2 + 1/4 mv2 = 3/4 mv2

25. Dec 16, 2013

### TSny

I'm not following you here.