To determine the acceleration of both the cylinder and the plank when a force F is applied, we need to analyze the forces acting on each object and how they interact with one another. Given that the cylinder rolls without slipping on the plank, we can apply Newton's second law and the rotational dynamics of the cylinder. Let's break this down step by step.
Understanding the System
We have two objects: a cylinder of mass m and radius R, and a plank of mass M. The force F is applied to the plank, which is on a smooth surface. The friction between the cylinder and the plank is sufficient to prevent slipping, meaning the cylinder will roll as it moves with the plank.
Forces Acting on the Plank
When the force F is applied to the plank, it experiences an acceleration due to this force. The frictional force between the plank and the cylinder will also act on the plank, opposing the motion of the cylinder. Let's denote the frictional force as f.
- The net force acting on the plank is: F - f = M * a_p, where a_p is the acceleration of the plank.
Forces Acting on the Cylinder
For the cylinder, the frictional force f is the only horizontal force acting on it. This frictional force causes the cylinder to accelerate and also causes it to roll. The equations for the cylinder can be expressed as follows:
- The net force acting on the cylinder is: f = m * a_c, where a_c is the acceleration of the cylinder's center of mass.
- Since the cylinder rolls without slipping, the relationship between the linear acceleration of the cylinder and its angular acceleration is given by: a_c = R * α, where α is the angular acceleration.
Relating the Accelerations
Since the cylinder rolls without slipping, the acceleration of the cylinder's center of mass (a_c) is equal to the acceleration of the plank (a_p). Therefore, we can set a_c = a_p = a. Now we can rewrite the equations:
- For the plank: F - f = M * a
- For the cylinder: f = m * a
Solving the Equations
Now we have two equations with two unknowns (f and a). We can substitute the expression for f from the second equation into the first equation:
Substituting gives us:
F - m * a = M * a
Rearranging this equation leads to:
F = (M + m) * a
From this, we can solve for the acceleration a:
a = F / (M + m)
Finding the Frictional Force
Now that we have the acceleration, we can find the frictional force f using the equation for the cylinder:
f = m * a = m * (F / (M + m))
Final Results
In summary, the acceleration of both the cylinder and the plank is given by:
a = F / (M + m)
The frictional force acting on the cylinder can be calculated as:
f = m * (F / (M + m))
This analysis shows how the forces interact in a system where friction plays a crucial role in preventing slipping, allowing us to find the accelerations of both the cylinder and the plank effectively.
