Loss of Energy during Contact of Rotating Discs

When two discs with the same moment of inertia but different angular velocities are brought into contact face to face, there is a transfer of energy between the two discs. This transfer of energy is due to the conversion of kinetic energy into other forms of energy, such as heat and sound.

Initial Conditions:
- Moment of inertia of both discs is the same.
- The discs are rotating about their regular axes passing through the center and perpendicular to the plane of the disc.
- The angular velocities of the discs are ω1 and ω2 respectively.

Conservation of Angular Momentum:
According to the principle of conservation of angular momentum, the total angular momentum before and after the discs come into contact remains constant. Mathematically, this can be expressed as:

I1 * ω1 + I2 * ω2 = (I1 + I2) * ωf

where I1 and I2 are the moments of inertia of the individual discs, ω1 and ω2 are their respective angular velocities before contact, and ωf is the common angular velocity of the combined system after contact.

Conservation of Kinetic Energy:
The initial kinetic energy of the system is given by:

KE_initial = (1/2) * I1 * ω1^2 + (1/2) * I2 * ω2^2

The final kinetic energy of the system after the discs come into contact is:

KE_final = (1/2) * (I1 + I2) * ωf^2

Loss of Energy:
The loss of energy during the contact of the rotating discs can be calculated as the difference between the initial and final kinetic energies:

Loss of Energy = KE_initial - KE_final

Substituting the expressions for KE_initial and KE_final, we get:

Loss of Energy = (1/2) * I1 * ω1^2 + (1/2) * I2 * ω2^2 - (1/2) * (I1 + I2) * ωf^2

Simplifying further, we have:

Loss of Energy = (1/2) * (I1 * ω1^2 + I2 * ω2^2 - (I1 + I2) * ωf^2)

Explanation:
When the two discs come into contact, there is a transfer of angular momentum between them, resulting in a common angular velocity for the combined system. This transfer of angular momentum is accompanied by a loss of energy, which is converted into other forms such as heat and sound.

The expression for the loss of energy accounts for the initial kinetic energy of each disc and the final kinetic energy of the combined system. It represents the difference between the initial and final states of the system, taking into consideration the moments of inertia and angular velocities of the discs.

By calculating the loss of energy, we can determine the amount of energy that is converted into other forms during the contact of the rotating discs.