A large circular disk of mass 2kg and radius 0.2m initially rotating at 50rad/s is coupled a smaller circular disk of mass 4kg and radius 0.1m initially rotating at 20rad/s in the same direction as large disk
A.find the final angular velocity after the disk are coupled
B. Calculate the loss of kinetic energy during this collision
A. To find the final angular velocity after the disks are coupled, we can use the principle of conservation of angular momentum.
The initial angular momentum of the system is the sum of the individual angular momentums of the two disks:
L_initial = (I1 * ω1) + (I2 * ω2)
Where:
I1 = moment of inertia of the large disk = (1/2) * 2 * (0.2)^2 = 0.02 kg.m^2
I2 = moment of inertia of the small disk = (1/2) * 4 * (0.1)^2 = 0.02 kg.m^2
ω1 = initial angular velocity of the large disk = 50 rad/s
ω2 = initial angular velocity of the small disk = 20 rad/s
So, substituting the values into the equation, we have:
L_initial = (0.02 * 50) + (0.02 * 20)
L_initial = 1 + 0.4
L_initial = 1.4 kg.m^2/s
Now, after coupling, the final angular momentum of the system is the sum of the individual angular momentums of the two disks:
L_final = (I_equivalent * ω_final)
Where:
I_equivalent = moment of inertia of the system after coupling, which can be calculated as the sum of the individual moments of inertia: I1 + I2 = 0.02 + 0.02 = 0.04 kg.m^2
ω_final = final angular velocity of the system
So, by the conservation of angular momentum:
L_initial = L_final
1.4 = 0.04 * ω_final
ω_final = 1.4 / 0.04
ω_final = 35 rad/s
Therefore, the final angular velocity after the disks are coupled is 35 rad/s.
B. To calculate the loss of kinetic energy during this collision, we need to find the initial and final total kinetic energies of the system.
The initial total kinetic energy of the system is the sum of the individual kinetic energies of the two disks:
KE_initial = (1/2) * I1 * ω1^2 + (1/2) * I2 * ω2^2
Substitute in the values:
KE_initial = (1/2) * 0.02 * 50^2 + (1/2) * 0.02 * 20^2
KE_initial = 25 + 2
KE_initial = 27 J
The final total kinetic energy of the system can be found using the final angular velocity calculated above:
KE_final = (1/2) * I_equivalent * ω_final^2
KE_final = (1/2) * 0.04 * 35^2
KE_final = 24.5 J
Therefore, the loss of kinetic energy during this collision is:
Loss = KE_initial - KE_final
Loss = 27 - 24.5
Loss = 2.5 J
Therefore, the loss of kinetic energy during this collision is 2.5 Joules.