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I am very confused by the following problem asked in my first year physics class:

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Please let me know if you can assist in any way! I've spent hours and hours on this question and gained absolutely nothing. Everything I do seems to lead to a contradiction one way or another.

There are other resources online I've found that mention this question, but I can't tease out a good solution from these: Force on a solid cylinder that is rolling on an accelerating block https://www.physicsforums.com/threads/a-rolling-disc-on-a-slab.594918/

Thank you!

Community
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1 Answers1

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As this is a homework question I won't give you a full solution, only point you in the right direction.

On the lower block acts a second force, $F_F$, a friction force that causes torque and the angular acceleration $\alpha$ of the cylinder:

$$F_F R=-I\alpha$$

Where $I$ is the moment of inertia of the cylinder and $R$ its radius. It carries a minus sign because it points in the opposite direction of $F$.

So the net force acting on the block is:

$$F_{net}=F-F_F$$

Now also note that for rolling without slipping, with $a$ the acceleration of the block, then:

$$a=\alpha R$$

To determine $a$ and $\alpha$ use the equations above to set up:

$$F_{net}=ma$$

Gert
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  • For the cylinder, would we not have

    F(net) = f = m*ac -> ac = f / m

    where f is the static friction and ac is the acceleration of the cylinder?

    But since it rolls without slipping we know that

    alpha*R = ac

    and

    T = Rf = Ialpha = 1/2 * m * R^2 * alpha -> alpha = ac / R = 2f/(m*R)

    giving us both

    ac = f / m and ac = 2f / m,

    a contradiction!

    (Apologies, I don't know how to make equations look nice.)

     –  Mar 21 '16 at 02:57
  • We're only concerned with $a_{block}$. That's what the question is about. Find $\alpha$ from the relevant formula. – Gert Mar 21 '16 at 03:09
  • Put another way, to find $a$ you only need to look at the forces that act on $m$. – Gert Mar 21 '16 at 03:35
  • Thank you so much Gert! I have posted my solution here: http://imgur.com/a/Ld1QT. I am still confused, however, why the cylinder has a force in the positive horizontal direction, yet does not accelerate in that direction. Is that not a violation of Newton's Second Law? –  Mar 21 '16 at 04:24
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    The cylinder does accelerate, by: $a_c=a-\alpha R$, in the stationary reference frame. Thanks! – Gert Mar 21 '16 at 12:23