homework and exercises - Cylinder rotating without slipping on an accelerating slab

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

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!

Answer

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:

$$F_{net}=ma$$