Question

A man pushes a cylinder of mass $m_1$ with help of a plank of mass $m_2$ as shown. There is no slipping at any contact. the horizontal component of the force applied by the man is F. Find: (a) the accelerations of the plank and the centre of mass of the cylinder, and (b) the magnitudes and directions of frictional forces at contact points.

Answer 1

a_p = \frac{8F}{m_1 + 8m_2}, a_c = \frac{4F}{m_1 + 8m_2}, f_1 = \frac{3m_1 F}{m_1 + 8m_2} \text{ (right on cylinder)}, f_2 = \frac{m_1 F}{m_1 + 8m_2} \text{ (right on cylinder, left on plank)}

Answer 2

1. Identify the objects involved and their degrees of freedom. The plank has translational motion. The cylinder has translational and rotational motion.

2. Apply the no-slipping conditions to derive kinematic constraints between the accelerations. For the cylinder rolling on the ground, $a_c = R\alpha$. For no slipping between the cylinder and the plank, the velocity of the contact points must be equal, leading to $a_p = 2a_c$.

3. Draw free-body diagrams for each object, showing all forces acting on them. Include gravity, normal forces, and frictional forces. Assume directions for frictional forces and be consistent with Newton's third law.

4. Apply Newton's second law for translational motion ($\sum \vec{F} = m\vec{a}$) and rotational motion ($\sum \vec{\tau} = I\vec{\alpha}$) for each object. Choose a convenient axis for calculating torque (usually the center of mass).

5. Solve the system of equations formed by the equations of motion and the kinematic constraints to find the unknown accelerations and frictional forces. If the sign of a frictional force comes out to be negative, it means the assumed direction was opposite to the actual direction.