Loss of Energy when Two Discs are Brought into Contact

When two discs of the same moment of inertia are rotating about their respective axes passing through the center and perpendicular to the plane of the disc, and then brought into contact face to face, there will be a loss of energy during this process. Let's analyze this situation step by step.

Moment of Inertia of the Discs
The moment of inertia of a disc rotating about an axis passing through its center and perpendicular to its plane is given by the formula:

I = (1/2) * M * R^2

Where:
- I is the moment of inertia
- M is the mass of the disc
- R is the radius of the disc

Initial Angular Velocities of the Discs
Let the initial angular velocity of the first disc be w1, and the initial angular velocity of the second disc be w2.

Angular Momentum of the Discs
The angular momentum of each disc is given by the product of its moment of inertia and angular velocity:

L1 = I1 * w1
L2 = I2 * w2

Total Angular Momentum before Contact
Before the discs come into contact, the total angular momentum of the system is equal to the sum of the angular momenta of the individual discs:

L_total = L1 + L2

Conservation of Angular Momentum
When the two discs come into contact, angular momentum is conserved. This means that the total angular momentum before contact should be equal to the total angular momentum after contact.

L_total_before = L_total_after

Angular Velocity after Contact
Let the angular velocity of the discs after contact be wf.

L_total_before = L_total_after
(I1 * w1) + (I2 * w2) = (I1 + I2) * wf

Loss of Energy
To calculate the loss of energy during this process, we need to consider the change in kinetic energy. The initial kinetic energy of the system is given by the sum of the kinetic energies of the individual discs:

KE_initial = (1/2) * I1 * w1^2 + (1/2) * I2 * w2^2

The final kinetic energy of the system is given by the kinetic energy of the merged discs:

KE_final = (1/2) * (I1 + I2) * wf^2

The loss of energy is the difference between the initial and final kinetic energies:

Loss of Energy = KE_initial - KE_final

Conclusion
In conclusion, when two discs of the same moment of inertia are brought into contact face to face, there will be a loss of energy during this process. This loss of energy can be calculated by considering the change in kinetic energy of the system before and after contact. The loss of energy is a result of the redistribution of angular momentum and the change in angular velocity after the discs come into contact.