# Angular momentum of sphere

1. Mar 13, 2006

### Reshma

A solid sphere of mass m and radius 'a' is rolling with a linear speed v on a flat surface without slipping. The magnitude of the angular momentum of the sphere with respect to a point along the path of the sphere on the surface is?

Moment of inertia of the sphere along any of its diameters is:${2\over 5}ma^2$

By parallel axis theorem:
Moment of inertia of the sphere rolling with respect to a point on the surface is:${2\over 5}ma^2 + ma^2 = {7\over 5}ma^2$

So the angular momentum will be:$L = {7\over 5}ma^2\omega ={7\over 5}mav$

The options given to me are:
${2\over 5}mav$
${7\over 5}mav[/tex] [itex]mav$
${3\over 2}mav$

2. Mar 13, 2006

### dextercioby

What's the center of rotation...?

Daniel.

3. Mar 13, 2006

### Reshma

The diameter of the sphere.

4. Mar 13, 2006

### dextercioby

The diameter is a line. I asked about the center, which is a point...

Daniel.

5. Mar 13, 2006

### Reshma

The ball is rolling, so the motion is linear as well as rotational. Centre...you mean circumferential motion?

6. Mar 13, 2006

### dextercioby

Yes, it has to be center if the ball, right...? So why would the angular momentum be calculated wrt to an axis around which the ball doesn't rotate...?

Daniel.

7. Mar 13, 2006

### Reshma

ok...so the ball has only circumferential momentum: So L = mav.
Thank you very much for your help .

8. Mar 13, 2006

### arunbg

We are talking of rotation and not circular motion.The locus of the points of contact of the sphere and the ground is a straight line and there is definitely an angular momentum associated with an axis through this line.
For eg, consider any point on the sphere.Even from this axis, u can still see that the point is executing pure rotation.
This shift in axis is actually useful in analysing rolling without slipping as it doesn't involve the translation of axis which is the case with an axis at the centre.
So the first answer is probably right.

Last edited: Mar 13, 2006
9. Mar 13, 2006

### Reshma

You guys are confusing me....what is the actual answer here?

10. Mar 13, 2006

### topsquark

I'm still trying to figure this line out. You seem to be saying $$a\omega=v$$. Among other problems, the units are wrong.

Come to think of it:
mav doesn't have the right units either.

-Dan

Last edited: Mar 13, 2006
11. Mar 13, 2006

### arunbg

I am pretty sure that the answer is
$$\frac{7}{5}mav$$.

Ps:Even if you were considering the "circumferential"(don't think the term exists !) angular momentum about an axis through the centre of the circular cross section of the sphere you wouldn't end up with mav.

In the eqn, 'a' stands for radius of the sphere and not acceleration.
So the dimensions do indeed match

12. Mar 13, 2006

### Staff: Mentor

Yes, your answer is correct. At any instant, the sphere can be considered to be in pure rotation about the point of contact.

You can also consider the sphere's motion to be a combination of (a) rotation about its center, plus (b) translation of its center. You'll get the same answer, of course.

13. Mar 13, 2006

### topsquark

Oops! My apologies! :surprised

-Dan

14. Mar 13, 2006

### Reshma

Thanks for the confirmation, Doc Al and Arunbg for the clarification. So my answer is indeed correct!!