A cylinder with moment of inertia I1 rotates with an angular velocity wo abouta vertical, frictionless axle. A second cylinder with moment of inertia I2 initially not rotating drops on to the first cylinder, since the surface are rough, the two eventually reach the same angular velocity then find: a) final angular velocity? B) show that kinetic energy is lost in this situation c) the ratio of the final to the initial kinetic energy

Question

A cylinder with moment of inertia I1 rotates with an angular velocity wo abouta vertical, frictionless axle. A second cylinder with moment of inertia I2 initially not rotating drops on to the first cylinder, since the surface are rough, the two eventually reach the same angular velocity then find: a) final angular velocity? B) show that kinetic energy is lost in this situation c) the ratio of the final to the initial kinetic energy

Answer 1

To solve this problem, we will use the principle of conservation of angular momentum and conservation of kinetic energy.

Let's denote:
- I1 = moment of inertia of the first cylinder
- I2 = moment of inertia of the second cylinder
- ωo = initial angular velocity of the first cylinder
- ωf = final common angular velocity of both cylinders

a) Final Angular Velocity (ωf):
Since the two cylinders eventually reach the same angular velocity, we can apply the principle of conservation of angular momentum:

I1 * ωo = (I1 + I2) * ωf

Solving for ωf, we get:

ωf = (I1 * ωo) / (I1 + I2)

b) Kinetic Energy Loss:
In this situation, friction causes energy loss. To show this, we can compare the initial and final kinetic energy.

Initial Kinetic Energy (KEi) = (1/2) * I1 * ωo^2

Final Kinetic Energy (KEf) = (1/2) * (I1 + I2) * ωf^2

The loss of kinetic energy (ΔKE) is given by:

ΔKE = KEi - KEf

Simplifying this expression, we get:

ΔKE = (1/2) * I1 * ωo^2 - (1/2) * (I1 + I2) * ωf^2

c) Ratio of Final to Initial Kinetic Energy:
To find the ratio of the final to the initial kinetic energy, we can substitute the expressions for ωf from part a) and simplify:

Ratio of Final to Initial Kinetic Energy = KEf / KEi
= [(1/2) * (I1 + I2) * ωf^2] / [(1/2) * I1 * ωo^2]
= [(I1 + I2) * (I1 * ωo) / (I1 + I2)]^2 / ωo^2
= (I1 * ωo)^2 / ωo^2
= (I1 / ωo)^2

So, the ratio of final to initial kinetic energy is equal to the square of the initial angular momentum divided by the initial angular velocity squared.

Note: In this problem, the assumption is made that the energy loss due to friction is negligible compared to the initial kinetic energy. This assumption is often valid in problems where surfaces are rough and the cylinders have significant moments of inertia.