A thin horizontal circular disc is roating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc.

Options

A. continously decreases
B. continously increases
C. first increases and then decreases
D. remains unchanged

Answer

To solve the problem, we need to analyze the situation involving the rotating disc and the insect moving along its diameter. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
We have a thin horizontal circular disc rotating about a vertical axis through its center. An insect is initially at rest at the rim of the disc and then moves towards the center along a diameter.

**Hint:** Visualize the disc and the insect's path to understand the dynamics involved.

### Step 2: Moment of Inertia
The moment of inertia (I) of the disc about its center is given by the formula:
\[ I_{\text{disc}} = \frac{1}{2} M R^2 \]
where \( M \) is the mass of the disc and \( R \) is its radius. When the insect (mass \( m \)) moves, its contribution to the moment of inertia changes based on its distance from the axis of rotation.

**Hint:** Recall that the moment of inertia depends on the mass distribution relative to the axis of rotation.

### Step 3: Total Moment of Inertia
As the insect moves from the rim to the center, the total moment of inertia of the system (disc + insect) can be expressed as:
\[ I_{\text{total}} = I_{\text{disc}} + m x^2 \]
where \( x \) is the distance of the insect from the center. As the insect moves towards the center, \( x \) decreases.

**Hint:** Remember that the insect's position affects the total moment of inertia.

### Step 4: Conservation of Angular Momentum
Since there is no external torque acting on the system, the angular momentum (L) is conserved:
\[ L = I \omega = \text{constant} \]
where \( \omega \) is the angular speed of the disc.

**Hint:** Conservation laws are key in rotational dynamics; they help relate changes in moment of inertia to changes in angular speed.

### Step 5: Analyze Changes in Angular Speed
As the insect moves towards the center:
- The moment of inertia \( I_{\text{total}} \) decreases because \( x \) decreases.
- Since \( L \) is constant, if \( I \) decreases, \( \omega \) must increase to keep \( L \) constant.

**Hint:** Think about how a spinning figure skater pulls in their arms to spin faster.

### Step 6: Movement Beyond the Center
If the insect were to move back towards the rim after reaching the center, the moment of inertia would increase again, causing the angular speed \( \omega \) to decrease.

**Hint:** Consider the reverse scenario when the insect moves outward.

### Conclusion
Thus, during the journey of the insect from the rim to the center, the angular speed of the disc first increases and then, if the insect continues moving outward, it would decrease. Therefore, the correct answer is:
**Option C: first increases then decreases.**