Finding Acceleration of Ball on Paper

[boardbox]

Homework Statement

I have a heavy ball on a piece of paper on the floor. The paper is grabbed and moved horizontally with acceleration a. What is the acceleration of the center of the ball? The ball is assumed to not slip with respect to the paper.

Homework Equations

The Attempt at a Solution

I'm not really sure how the acceleration is supposed to interact with the ball. A thought I've had is to say that the ball doesn't translate but rotates with angular acceleration a/R since it doesn't slip.

 

[kuruman]

A thought I've had is to say that the ball doesn't translate but rotates with angular acceleration a/R since it doesn't slip.

*** Edited after reading Doc Al's Comment ***

That's almost it. See Doc Al's comment below.

 

[Doc Al]

A thought I've had is to say that the ball doesn't translate but rotates with angular acceleration a/R since it doesn't slip.

Careful. The paper exerts a force on the ball, so the ball must translate as well as rotate. Both the ball and the surface (the paper) are accelerating, but at different rates. So you can't just assume that the angular acceleration is a/R.

Hint: Assume that the paper exerts some force F on the ball. Analyze the translational and rotational dynamics of the ball using Newton's 2nd law.

 

[boardbox]

$$\Sigma$$F_ball = F_paper = ma_ball
$$\Sigma$$t = t_paper = I $$\alpha$$
t = r x F
$$\alpha$$ = a_paper/r

plug and chug

a_ball = 2a_paper/5

does that get it about right?

 

[Doc Al]

$$\Sigma$$F_ball = F_paper = ma_ball

OK.

$$\Sigma$$t = t_paper = I $$\alpha$$
t = r x F

OK.

$$\alpha$$ = a_paper/r

No. Alpha is the rotation about the center of mass--compare the acceleration of the paper with the acceleration of the center of mass.

(But you're on the right track!)

 

[boardbox]

Would it be the sum of the two accelerations over the radius?

I think rolling without slipping is $$\omega$$ = v/r so the acceleration version of that should just be the time derivative. If I have a a constant v on the paper and the ball I would expect to just sum them.

 

[Doc Al]

Would it be the sum of the two accelerations over the radius?

What if the center of mass had the same acceleration as the paper? What would be the rotational acceleration of the ball in that case?

 

[boardbox]

Zero?
You highlight sum. I'm wondering if you want magnitude of difference?

 

[Doc Al]

Zero?

Exactly. If you just added them you'd get alpha = 2a/r, which doesn't make sense.

You highlight sum. I'm wondering if you want magnitude of difference?

What you need to find alpha is the relative acceleration of paper and ball divided by r.

 

[boardbox]

I see, makes sense. Thanks for the help.