1. Set up the physical idea

A thin disc is rotating freely about its vertical central axis. An insect initially sits near the rim and then crawls along a diameter to the opposite end.

Since there is no external torque about the rotation axis, the total angular momentum of the system (disc + insect) remains constant:

$L = I\omega = \text{constant}$

So the angular speed $\omega$ depends on the total moment of inertia $I$:

$\omega = \frac{L}{I}$

Thus:

• if $I$ decreases, $\omega$ increases,
• if $I$ increases, $\omega$ decreases.

2. Write the total moment of inertia

Let:

• mass of disc $= M$
• radius of disc $= R$
• mass of insect $= m$
• insect’s distance from center at any instant $= r$

Moment of inertia of the disc about its axis is constant:

$I_{\text{disc}} = \frac{1}{2}MR^2$

Moment of inertia of the insect is:

$I_{\text{insect}} = mr^2$

Hence total moment of inertia is:

$I = \frac{1}{2}MR^2 + mr^2$

3. Track how $r$ changes during motion

The insect moves along a diameter from one end to the other.

So its distance from the center changes as:

$R \to 0 \to R$

That means:

• from the rim to the center, $r$ decreases,
• from the center to the opposite rim, $r$ increases.

Therefore $mr^2$:

• first decreases,
• then increases.

So total moment of inertia $I$:

• first decreases,
• then increases.

4. Use conservation of angular momentum

Since

$\omega = \frac{L}{I},$

when $I$ first decreases, $\omega$ first increases.

Then when $I$ increases again, $\omega$ decreases.

So the angular speed of the disc:

$\boxed{\text{first increases and then decreases}}$

5. Evaluate options

• A: continuously decreases — incorrect
• B: continuously increases — incorrect
• C: first increases and then decreases — correct
• D: remains unchanged — incorrect

6. Comparison with stored answer

Stored correct answer: C

Derived answer: C

They agree.