Two horizontal discs of different radii are free to rotate about their central vertical axes: One is given some angular velocity, the other is stationary. Their rims arc now brought in contact. There is friction between the rims. Then

Options

A. the force of friction between the rims will disappear when the discs rotate with equal angular speeds
B. the force of friction between the rims will when they have equal linear velocities
C. the angular momentum of the system will conserved
D. the rotational kinetic energy of the system will not conserved

Explanation

To solve the problem, we need to analyze the situation involving two discs with different radii, one of which is rotating while the other is stationary. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
We have two discs:
- Disc A: Rotating with an angular velocity \( \omega_1 \).
- Disc B: Stationary, with an angular velocity \( \omega_2 = 0 \).

When the rims of the two discs come into contact, friction will act at the point of contact.

**Hint:** Visualize the discs and their rotation directions. Consider the effects of friction when they touch.

### Step 2: Analyze the Effect of Friction
When the two discs touch, friction will act to oppose the relative motion between them. Since Disc A is rotating and Disc B is stationary, there will be a tendency for Disc B to start rotating in the same direction as Disc A due to the frictional force.

**Hint:** Remember that friction acts to prevent relative motion between surfaces.

### Step 3: Condition for No Relative Motion
The friction force will disappear when both discs rotate with the same angular velocity. This means that if the angular velocities of both discs become equal (\( \omega_1 = \omega_2 \)), there will be no relative motion at the point of contact, and thus no frictional force acting on them.

**Hint:** Think about when two objects moving at the same speed will not experience friction between them.

### Step 4: Conservation of Angular Momentum
In this scenario, angular momentum is not conserved because there is an external torque due to friction acting between the discs. However, the angular momentum of the system can be analyzed before and after contact.

**Hint:** Consider how external forces (like friction) affect the conservation laws.

### Step 5: Kinetic Energy Considerations
While angular momentum may not be conserved due to the frictional forces, the rotational kinetic energy of the system will not be conserved either. Some energy will be lost due to friction as heat.

**Hint:** Remember that energy can be transformed and lost in non-conservative systems.

### Conclusion
Based on the analysis:
1. The friction between the rims will disappear when the discs rotate with equal angular speeds.
2. The rotational kinetic energy of the system will not be conserved due to energy losses from friction.

**Final Answer:** Both statements regarding the disappearance of friction at equal angular speeds and the non-conservation of rotational kinetic energy are true.