Acceleration of the Plank and Center of Mass of the Cylinder:

To find the acceleration of the plank and the center of mass of the cylinder, we can apply Newton's second law of motion.

Let's consider the forces acting on the system:

1. Force applied by the man (F): This force is applied horizontally to push the cylinder and generates motion.

2. Weight of the cylinder (m1g): This force acts vertically downwards due to the gravitational pull on the cylinder.

3. Normal force (N1): This force acts perpendicular to the contact surface between the cylinder and the plank.

4. Frictional force (f1): This force acts parallel to the contact surface between the cylinder and the plank.

5. Weight of the plank (m2g): This force acts vertically downwards due to the gravitational pull on the plank.

6. Normal force (N2): This force acts perpendicular to the contact surface between the plank and the ground.

7. Frictional force (f2): This force acts parallel to the contact surface between the plank and the ground.

Now, let's analyze the forces acting on the cylinder:

1. In the horizontal direction, the force applied by the man (F) is the only force causing acceleration. Therefore, we can write the equation:

F - f1 = m1 * a

2. In the vertical direction, the weight of the cylinder (m1g) and the normal force (N1) must balance each other. Therefore, we can write the equation:

N1 - m1g = 0

Solving these two equations simultaneously will give us the acceleration of the plank (a) and the center of mass of the cylinder.

Frictional Forces at Contact Points:

To find the magnitudes and directions of the frictional forces at the contact points, we need to analyze the forces acting on the plank and the cylinder separately.

For the plank:

1. In the horizontal direction, the force applied by the man (F) and the frictional force between the plank and the ground (f2) must balance each other to prevent slipping. Therefore, we can write the equation:

F - f2 = 0

2. In the vertical direction, the weight of the plank (m2g) and the normal force (N2) must balance each other. Therefore, we can write the equation:

N2 - m2g = 0

For the cylinder:

1. The frictional force between the cylinder and the plank (f1) and the weight of the cylinder (m1g) must balance each other to prevent slipping. Therefore, we can write the equation:

f1 - m1g = 0

Solving these equations will give us the magnitudes and directions of the frictional forces at the contact points.

In conclusion, by applying Newton's second law and analyzing the forces acting on the system, we can find the acceleration of the plank and the center of mass of the cylinder. Additionally, by analyzing the forces acting on the plank and the cylinder separately, we can determine the magnitudes and directions of the frictional forces at the contact points.