14.2:
Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving
Consider a massless pulley system consisting of a wooden box and a cylinder of known masses. Determine the speed of the attached objects at a specific time after the system is released from rest.
For the analysis, establish an appropriate coordinate system to determine the length of the entire rope. The time derivative of this length equation gives the velocity equation.
Draw a free-body diagram for the cylinder, indicating its weight and the tension in the rope. Apply the principle of linear impulse and momentum to the cylinder.
Here, the initial momentum is zero, while the impulse is due to the weight of the cylinder and the tension on the rope.
Similarly, draw a free-body diagram for the wooden box, indicating all the forces acting on it. Apply the principle of impulse and momentum and substitute the known values to simplify the equation.
Now, solve the three equations simultaneously to obtain the unknown velocities.
Assuming the downward velocity as positive, the negative velocity of the wooden box indicates an upward motion of the box.
14.2:
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.