A uniform solid cylinder of mass M and radius R is at rest on a slab of mass m, which in turn rests on a horizontal, frictionless table. If a horizontal force is applied to the slab, it accelerates and the cylinder rolls without slipping. Find the acceleration of the slab in terms of M, R, m, and F.

To find the acceleration of the slab, we can consider the torque produced by the applied force on the system.

First, let's consider the forces acting on the system. The applied force F acts on the slab, and we assume it doesn't apply any torque on the slab or the cylinder. The normal force N acts upward on the slab due to the weight of the cylinder, and the weight W acts downward on the system.

For the cylinder, there are three forces acting on it: the weight W, the normal force N, and the frictional force f. The weight W acts downward at the center of mass of the cylinder, the normal force N acts upward at the center of mass, and the frictional force f acts in the direction opposite to the motion of the cylinder.

Since the cylinder rolls without slipping, the acceleration a of the slab is related to the angular acceleration alpha of the cylinder and the radius R of the cylinder by a = alpha * R. And the torque produced by the applied force F is equal to the torque due to the frictional force f.

Now, let's analyze the torques. The torque due to the applied force F is zero since it acts at the same height as the center of mass of the slab. The torque due to the frictional force f is equal to the product of the frictional force f and the radius R, which is -f * R, as it acts in the opposite direction.

Since the torque produced by the applied force and the torque due to the frictional force are equal, we have F * R = f * R. Also, we have the relationship f = mu * N, where mu is the coefficient of friction.

Now, let's connect the mass and normal force of the cylinder and the slab. The weight acting on the slab is m * g, and the weight of the cylinder is M * g. Since the slab is not accelerating vertically, the normal force N is given by N = m * g + M * g.

Substituting f = mu * N and N = m * g + M * g into the equation F * R = f * R, we have F * R = mu * (m * g + M * g).

Finally, solving for the acceleration a of the slab, we have a = (F * R) / (m + M).

Therefore, the acceleration of the slab in terms of M, R, m, and F is a = (F * R) / (m + M).