A cylinder with moment of inertia I1 rotates with an angular velocity wo about a vertical, frictionless axle. Asecond cylinder with moment of inertia I2 initially not rotating drops on to the first cylinder, since the surfaces are rough, the two eventually reach the same angular velocity then find: a) final agular velocity? b) show that kinetic energy is lost in this situation? c) the ratio of the final to the initial kinetic energy
To solve this problem, we'll use the principle of conservation of angular momentum.
a) Final Angular Velocity (ωf):
According to the principle of conservation of angular momentum, the total angular momentum before and after the two cylinders come to rest will remain constant. The initial angular momentum of the first cylinder is given by:
L1 = I1 * ωo
When the second cylinder drops onto the first cylinder, the moment of inertia becomes I1 + I2. Let ωf be the common final angular velocity of the two cylinders. Therefore, the initial angular momentum of the second cylinder can be expressed as:
L2 = (I1 + I2) * ωf
Since there is no external torque acting on the system, the initial angular momentum should be equal to the final angular momentum:
L1 + L2 = (I1 * ωo) + ((I1 + I2) * ωf)
Simplifying the equation, we get:
I1 * ωo + (I1 + I2) * ωf = I1 * ωf + (I1 + I2) * ωf
ωo = ωf
Thus, the final angular velocity (ωf) is equal to the initial angular velocity (ωo).
b) Kinetic Energy Loss:
To show that kinetic energy is lost in this situation, we need to compare the initial and final kinetic energies of the system.
The initial kinetic energy of the system is given by:
KE_initial = (1/2) * I1 * ωo^2
The final kinetic energy of the system is given by:
KE_final = (1/2) * (I1 + I2) * ωf^2
Since ωf = ωo, we can substitute it in the equation:
KE_final = (1/2) * (I1 + I2) * ωo^2
From the equations, we can see that KE_final ≤ KE_initial. This indicates that kinetic energy is lost in the process.
c) Ratio of Final to Initial Kinetic Energy:
Substituting the values, we can calculate the ratio of the final kinetic energy to the initial kinetic energy:
KE_ratio = KE_final / KE_initial
Simplifying the equation, we get:
KE_ratio = [(1/2) * (I1 + I2) * ωo^2] / [(1/2) * I1 * ωo^2]
The term ωo^2 cancels out, resulting in:
KE_ratio = (I1 + I2) / I1
This is the ratio of the final to initial kinetic energy.