A solid, horizontal cylinder of mass 8.0 kg and radius 1.30 m rotates with an angular speed of 5.50 rad/s about a fixed vertical axis through its center. A 0.250 kg piece of putty is dropped vertically onto the cylinder at a point 0.900 m from the center of rotation, and sticks to the cylinder. Determine the final angular speed of the system.
To determine the final angular speed of the system, we need to apply the principle of conservation of angular momentum. Angular momentum is a conserved quantity if no external torques act on the system.
The formula for angular momentum is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The initial angular momentum of the system is the sum of the angular momenta of the cylinder and the piece of putty before the collision. The final angular momentum is the sum of the angular momenta of the cylinder and the piece of putty after the collision.
Let's calculate the initial and final angular momenta separately:
1. Initial Angular Momentum:
The angular momentum of the cylinder can be calculated using the formula Lcylinder = Icylinder * ωcylinder, where Icylinder is the moment of inertia of the cylinder and ωcylinder is the initial angular velocity of the cylinder.
The moment of inertia of a solid cylinder about its central axis is given by Icylinder = (1/2) * m * r^2, where m is the mass of the cylinder and r is the radius of the cylinder.
Plugging in the values, we have:
Icylinder = (1/2) * 8.0 kg * (1.30 m)^2
The angular momentum of the piece of putty can be calculated using the formula Lputty = Iputty * ωputty, where Iputty is the moment of inertia of the putty and ωputty is the initial angular velocity of the putty.
Since the putty is dropped vertically, it has no initial angular velocity (ωputty = 0). Therefore, the angular momentum of the putty is zero.
Now, we can calculate the initial angular momentum by adding the angular momenta of the cylinder and the putty:
Linitial = Lcylinder + Lputty
2. Final Angular Momentum:
After the collision, the putty sticks to the cylinder, resulting in a change in the moment of inertia of the system. The final moment of inertia of the system is given by Ifinal = Icylinder + Iputty.
The moment of inertia of a point mass about an axis passing through its center of mass is given by Iputty = mputty * r^2, where mputty is the mass of the putty and r is the distance from the axis of rotation to which the putty is dropped.
Plugging in the values, we have:
Iputty = 0.250 kg * (0.900 m)^2
The final angular momentum can be calculated using the formula Lfinal = Ifinal * ωfinal, where ωfinal is the final angular velocity of the system.
According to the conservation of angular momentum, the initial angular momentum must be equal to the final angular momentum, so we have:
Linitial = Lfinal
Now, we can substitute the known values into the equations and solve for ωfinal:
Lcylinder + Lputty = Ifinal * ωfinal
Solve for ωfinal to get the final angular speed of the system.