A solid, horizontal cylinder of mass 12.0 kg and radius 2.30 m rotates with an angular speed of 6.70 rad/s about a fixed vertical axis through its center. A 1.60 kg piece of putty is dropped vertically onto the cylinder at a point 0.600 m from the center of rotation and sticks to the cylinder. Determine the final angular speed of the system
momentum is conserved
initial momentum=final momentum
I*wi = (I+1/2 massputty*r^2)wf
solve for wf.
Look up the moment of inertia I for the cylinder.
To determine the final angular speed of the system, we need to apply the principle of conservation of angular momentum. The angular momentum before and after the putty is dropped should be equal.
The angular momentum is given by the formula:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Before the putty is dropped, the angular momentum of the cylinder is:
L_cylinder = I_cylinder * ω_cylinder
The moment of inertia of a solid cylinder rotating about its central axis is given by:
I_cylinder = (1/2) * m_cylinder * r_cylinder^2
Substituting the given values, we have:
I_cylinder = (1/2) * 12.0 kg * (2.30 m)^2
Now, we can calculate the angular momentum of the cylinder before the putty is dropped.
L_cylinder = I_cylinder * ω_cylinder
Next, after the putty is dropped, it sticks to the cylinder and rotates with it. The combined system of the cylinder and the putty can be treated as a single object.
The moment of inertia of the system (cylinder + putty) is given by:
I_system = I_cylinder + I_putty
The moment of inertia of a point mass rotating about an axis is given by:
I_point = m_point * r_point^2
Substituting the given values, we have:
I_putty = 1.60 kg * (0.600 m)^2
Now, we can calculate the total moment of inertia of the system.
I_system = I_cylinder + I_putty
After the putty is dropped and sticks to the cylinder, the final angular momentum of the system is:
L_system = I_system * ω_final
According to the conservation of angular momentum:
L_cylinder = L_system
So, we can equate the initial angular momentum of the cylinder to the final angular momentum of the system:
I_cylinder * ω_cylinder = I_system * ω_final
Now, we can solve for ω_final:
ω_final = (I_cylinder * ω_cylinder) / I_system
Substitute the values we found earlier and calculate the final angular speed of the system.