How Is Kinetic Energy Distributed in a Rolling Sphere?

[Hindi]

Hi all,

I just had a question on a homework problem I was attempting, but can't find the correct solution for it yet. The answer is in the back of the book, but I wanted to know how it is derived. Anyways, here is the question:

A hollow sphere of radius .15 with I=.040 rolls without slipping up a surface inclined at 30 degrees. At a certain initial position, the sphere's total kinetic energy is 20J.

A) How much of the totak K.E. initial is rotation?

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Variables:

Radius of sphere = .15m
Moment of Inertia = .040 kg * m^2

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Attempt:
I tried this problem with setting the initial kinetic energy of the sphere-incline system to 20J. So I did:
.5MV^2 + .5I(w)^2 = 20J

This is the step that I get lost in. Should I try to find the mass of the system first or should I try a whole different approach? Any help would be appreciated! Thanks!

 

[Doc Al]

Originally posted by Hindi

.5MV^2 + .5I(w)^2 = 20J

This is the step that I get lost in. Should I try to find the mass of the system first or should I try a whole different approach?

Yes, you'll need the mass of the sphere. (Hint: what's the formula for rotational inertia of a sphere?)

Also, you'll need to apply the condition for rolling without slipping: how does ω relate to V when there is no slipping?

 

[Hindi]

Doc Al,

Thanks for the reply. One of the biggest problems I was having with finding the mass was finding the rotational inertia of a hollow sphere. In my book (Fundamentals of Physics 6th Edition), it only gives the rotational inertia of a solid sphere.

I know the rotational inertia for a solid sphere is 2/5MR^2. Will that apply to this problem as well? If that applies, then I set:
2/5MR^2=.040 kg*m^2 and find the mass that way.

Also, you'll need to apply the condition for rolling without slipping: how does ω relate to V when there is no slipping?

v=w(r) --> w=v/r

Any hints would be greatly appreciated. Thanks again!

 

[deltabourne]

The moment of inertia for a hollow sphere is 2/3*m*r^2.. find the mass, then with the equation you have (substitute in for v to find w), then plug w back into the rotational energy equation, and voila

 

[Hindi]

Hey thanks deltabourne. You gave me the answer to the missing piece of the puzzle! Found the answer to be 8.0J joules, which is correct!

 

[Hindi]

Hi all again.

Now the question wants me the find the TOTAL kinetic energy of the sphere. I keep on trying to figure this out, but I keep getting 10J, while the answer is 6J. I f anyone can point what I am doing wrong, it would be greatly appreciated!

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I found Vi=2.98m/s
Height = .75m

1/2MV^2+1/2I(W)^2 = MGH + 5/6MV^2

Subtracting 5/6MV^2 to the other side, I keep on getting 10J. What am I doing wrong?

To make things easier, I attached my work I have done so far as an image:
http://home.comcast.net/~msharma15/problem_1.jpg

 

[Doc Al]

Originally posted by Hindi
Now the question wants me the find the TOTAL kinetic energy of the sphere. I keep on trying to figure this out, but I keep getting 10J, while the answer is 6J. I f anyone can point what I am doing wrong, it would be greatly appreciated!

You'll have to state the problem more clearly. Obviously you are asked to find the total KE at some new point. What point? In your attachment you mention a point 1 m up the incline. Is that the point? Let me pretend that it is. You know the initial KE; when it raises up a height H, the KE will decrease by mgH. For 1 m up the plane, H = 1 sin(30) = 0.5 m.

 

[Hindi]

Thanks everyone...eventually solved it! Pretty easy after thinking about it.

Doc...yea it was 1m. Sorry about that!

Anyone interested in the solution, you can find it here:
http://home.comcast.net/~msharma15/problem_1.jpg