A large circular disk of mass 2kg and radius 0.2m initially rotating at 50rad/s is coupled a smaller circular disk of mass &radius 0.1m initially rotating at 20rad/s in the same direction as large disk
A.find the commen angular velocity after the disk are coupled
B.calculate the loss of kinetic energy during this collision
A. To find the common angular velocity after the disks are coupled, we can use the principle of conservation of angular momentum.
The initial angular momentum of the system before coupling can be calculated as the sum of the individual angular momenta of the two disks:
L_initial = I1 * ω1 + I2 * ω2
Where I1 and I2 are the moments of inertia of the large and small disks, respectively. Since both disks are rotating in the same direction, we can take the positive direction as the original direction of rotation.
I1 = 0.5 * m1 * r1^2
I2 = 0.5 * m2 * r2^2
Plugging in the values:
I1 = 0.5 * 2 * 0.2^2 = 0.02 kgm^2
I2 = 0.5 * 1 * 0.1^2 = 0.005 kgm^2
Now we can calculate the initial angular momentum:
L_initial = 0.02 * 50 + 0.005 * 20
L_initial = 1 + 0.1
L_initial = 1.1 kgm^2/s
After the disks are coupled, the total moment of inertia of the system will be the sum of the moments of inertia of the two disks:
I_total = I1 + I2
I_total = 0.02 + 0.005
I_total = 0.025 kgm^2
Using the principle of conservation of angular momentum, the final angular momentum of the system will be equal to the initial angular momentum:
L_final = I_total * ω_final
Therefore, we can solve for the common angular velocity after the disks are coupled:
ω_final = L_initial / I_total
ω_final = 1.1 / 0.025
ω_final = 44 rad/s
B. To calculate the loss of kinetic energy during this collision, we can first calculate the initial kinetic energy of the system:
KE_initial = 0.5 * I1 * ω1^2 + 0.5 * I2 * ω2^2
Plugging in the values:
KE_initial = 0.5 * 0.02 * 50^2 + 0.5 * 0.005 * 20^2
KE_initial = 25 + 2.5
KE_initial = 27.5 J
The final kinetic energy of the system can be calculated as:
KE_final = 0.5 * I_total * ω_final^2
KE_final = 0.5 * 0.025 * 44^2
KE_final = 24.2 J
Therefore, the loss of kinetic energy during this collision is the initial kinetic energy minus the final kinetic energy:
Loss of KE = KE_initial - KE_final
Loss of KE = 27.5 - 24.2
Loss of KE = 3.3 J
So the loss of kinetic energy during this collision is 3.3 Joules.