Conservation of energy (rotation+translation)

[mbrmbrg]

Homework Statement

A small solid sphere, with radius 0.25 cm and mass 0.68 g rolls without slipping on the inside of a large fixed hemisphere with radius 23 cm and a vertical axis of symmetry. The sphere starts at the top from rest. The moment of inertia of a sphere is I = 2/5 MR².
(a) What is its kinetic energy at the bottom?
(b) What fraction of its kinetic energy at the bottom is associated with rotation about an axis through its center of mass?

Homework Equations

E=U+K=constant

U=mgh

$$K=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2$$

The Attempt at a Solution

For part (a) only:
I started to write up the energy conservation law in great and glorious detail, but then I realized that since K_i=0 and U_f=0, and the question only wants to know K at bottom, I could say
$$mgh=K$$
$$(.00068kg)(9.81m/s^2)(0.23m)=K$$
K=0.00153J

But the correct answer is 0.152J, a rather striking difference. Where did I go wrong?

 

[Doc Al]

Your method and answer seem correct to me. (Could be a typo in the answer key or the problem statement.)

 

[mbrmbrg]

Thanks! (the night before the final. Who to trust? My professor's answer key or Doc Al? You got it!)

 

[vanesch]

If you really want to nitpick you can say that the height over which the small sphere is travelling, is not 2R but 2(R-r). It won't make much difference given that R >> r.

 

[mbrmbrg]

That actually occurred to me briefly; but to make a difference of 1e2!

 

[Doc Al]

If you really want to nitpick you can say that the height over which the small sphere is travelling, is not 2R but 2(R-r). It won't make much difference given that R >> r.

Since its a hemisphere, the height traveled would be R, not 2R. And using a height of R-r, instead of R, would make the KE even smaller--so that won't help!

 

[vanesch]

Since its a hemisphere, the height traveled would be R, not 2R. And using a height of R-r, instead of R, would make the KE even smaller--so that won't help!

There's clearly a problem with the orders of magnitude, but I thought the problem was that the book said something of 152 instead of 153...

And concerning the hemisphere, you don't know how it is oriented

 

[Doc Al]

There's clearly a problem with the orders of magnitude, but I thought the problem was that the book said something of 152 instead of 153...

I wouldn't worry about the third significant figure in the answer, considering that the data given only has two significant figures.

And concerning the hemisphere, you don't know how it is oriented

Lots of luck rolling that ball down the inside of a bowl with a horizontal axis of symmetry! And the problem did state that the axis of symmetry is vertical.

 

[vanesch]

Lots of luck rolling that ball down the inside of a bowl with a horizontal axis of symmetry! And the problem did state that the axis of symmetry is vertical.

Damn ! I'm not going to talk me out of this one