Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving

Consider a wooden box and a cylinder of known masses m₁ and m₂, respectively,  hanging from a ceiling with the help of a massless pulley system.

The system is initially at rest and then released. What will be the velocities of the wooden box and cylinder at a specific time after the system has been released from the rest?

Here, the entire length of the rope is expressed as the combination of smaller segments attached to the wooden box and cylinder. As the system moves, both the wooden box and cylinder attain some velocities, but the entire length of the string remains constant. Therefore, the velocity expression is derived by taking the time derivative of the length of the entire rope.

Then, a free-body diagram is drawn for the cylinder, showing all the forces acting on it. Here, the integral of the net force acting on the cylinder for a given time interval t, equals the change in momentum of the cylinder. Similarly, a free-body diagram is drawn for the wooden box, and a corresponding equation is written.

Solving the above two equations simultaneously gives the cylinder and wooden box velocities. Here, it is important to note that the directions of the velocities for the wooden box and cylinder are opposite.