A uniform disc is rotating at a constantt speed in a vertical plane about a fixed horizontal axis passing through the centre of the disc. A piece of the disc from its rim detaches itself from the disc at the instant when it is at horizontal level with the centre of the disc and moving upward. Then about the fixed axis, the angular speed of the

Options

A. remaining disc remains unchanged
B. remaining disc decreases
C. remaining disc increases
D. broken away piece decrease initially and later

Answer

To solve the problem, we need to analyze the situation step by step:

### Step 1: Understand the Initial Conditions
A uniform disc is rotating at a constant angular speed about a fixed horizontal axis through its center. At a certain moment, a piece of the disc detaches from the rim when it is at the horizontal level with the center of the disc and moving upward.

**Hint:** Identify the initial state of the disc and the piece that detaches.

### Step 2: Determine the Velocity of the Detached Piece
When the piece detaches, it retains the same linear velocity that it had while it was part of the rotating disc. This is because, according to Newton's first law, an object in motion will continue in motion with the same speed and direction unless acted upon by an external force.

**Hint:** Remember that the linear velocity of a point on a rotating disc is given by \( v = r \omega \), where \( r \) is the radius and \( \omega \) is the angular speed.

### Step 3: Analyze Angular Momentum
The angular momentum of the system before the piece detaches can be expressed as the sum of the angular momentum of the entire disc. After the piece detaches, the angular momentum of the remaining disc plus the angular momentum of the detached piece must equal the initial angular momentum of the whole disc.

**Hint:** Angular momentum \( L \) is given by \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular speed.

### Step 4: Evaluate the Angular Speed After Detachment
Since the detached piece retains its linear velocity and moves upward, it does not exert any torque on the remaining disc. Therefore, the angular momentum of the remaining disc remains unchanged. As a result, the angular speed of the remaining disc does not change.

**Hint:** Consider the conservation of angular momentum in the absence of external torques.

### Step 5: Conclusion
The angular speed of the remaining disc remains unchanged after the piece detaches. The detached piece will move upward and its angular velocity relative to the center of the disc will change as it moves away from the center.

**Final Answer:** The angular speed of the remaining disc remains unchanged.