To solve this problem, we need to analyze the forces and motions involved in the system of the plank and the cylinder. Since the cylinder does not slip on the plank, we can relate the linear acceleration of both the plank and the cylinder through their interactions. Let's break this down step by step.
Understanding the System
We have two main components: the plank of mass M and the cylinder of mass m with radius R. A horizontal force F is applied to the plank. The key point here is that the cylinder rolls without slipping on the plank, which means that the linear acceleration of the cylinder's center of mass is directly related to the linear acceleration of the plank.
Identifying Forces and Accelerations
Let’s denote the linear acceleration of the plank as a_p and the linear acceleration of the cylinder's center of mass as a_c. Since the cylinder rolls without slipping, we have:
- a_c = a_p (the linear accelerations are equal).
Applying Newton's Second Law
For the plank, the net force acting on it is the applied force F minus the frictional force f exerted by the cylinder. According to Newton's second law:
F - f = M * a_p
For the cylinder, the frictional force f is the only horizontal force acting on it, which causes its linear acceleration and angular acceleration. The equation for the cylinder becomes:
f = m * a_c
Relating Linear and Angular Accelerations
Since the cylinder rolls without slipping, we can relate the linear acceleration to the angular acceleration α of the cylinder using the equation:
a_c = R * α
From this, we can express the angular acceleration as:
α = a_c / R
Combining the Equations
Now we have three equations:
- F - f = M * a_p
- f = m * a_c
- a_c = a_p
Substituting f = m * a_p into the first equation gives:
F - m * a_p = M * a_p
Rearranging this, we find:
F = (M + m) * a_p
Thus, the linear acceleration of the plank (and the cylinder) is:
a_p = F / (M + m)
Finding Angular Acceleration
Now, substituting a_p back into the equation for angular acceleration:
α = a_p / R = (F / (M + m)) / R
This gives us the angular acceleration of the cylinder:
α = F / (R * (M + m))
Summary of Results
To summarize, we have derived the following results:
- The linear acceleration of both the plank and the cylinder is given by: a_p = F / (M + m)
- The angular acceleration of the cylinder is: α = F / (R * (M + m))
This analysis illustrates how the forces interact in a system where one object rolls over another without slipping, and it highlights the importance of understanding both linear and angular motions in mechanics.
