1. Identify the system and the collision point

A thin rod of:

• mass $M = 0.9\,\text{kg}$
• length $L = 1\,\text{m}$

is hinged at its upper end and initially at rest.

A particle of:

• mass $m = 0.1\,\text{kg}$
• speed $v = 80\,\text{m/s}$

hits the bottom end of the rod and sticks to it.

We need the angular speed $\omega$ of the combined system immediately after collision.

2. Use conservation of angular momentum about the hinge

During the short collision, the hinge may exert a large force, so linear momentum is not conserved.

But the angular momentum about the hinge is conserved, because the hinge force passes through the hinge and hence produces zero torque about the hinge.

So, $L_{\text{initial}} = L_{\text{final}}$

3. Initial angular momentum of the particle about the hinge

The particle strikes the bottom end of the rod, which is at distance $r = L = 1\,\text{m}$ from the hinge.

Its momentum is $p = mv = 0.1 \times 80 = 8\,\text{kg m/s}$

Since it hits perpendicularly at the rod’s bottom end, initial angular momentum about hinge is $L_{\text{initial}} = m v L = 0.1 \times 80 \times 1 = 8\,\text{kg m}^2/\text{s}$

4. Final moment of inertia after sticking

After collision, rod + particle rotate together about the hinge.

(a) Moment of inertia of the rod about one end

For a thin rod about one end, $I_{\text{rod}} = \frac{1}{3} M L^2$ $I_{\text{rod}} = \frac{1}{3}(0.9)(1)^2 = 0.3\,\text{kg m}^2$

(b) Moment of inertia of the stuck particle

The particle is at the bottom end, distance $L = 1\,\text{m}$ from hinge: $I_{\text{particle}} = mL^2 = 0.1 \times 1^2 = 0.1\,\text{kg m}^2$

(c) Total moment of inertia

$I_{\text{total}} = 0.3 + 0.1 = 0.4\,\text{kg m}^2$

5. Apply angular momentum conservation

If $\omega$ is the angular speed just after collision, then $L_{\text{final}} = I_{\text{total}}\,\omega$

So, $8 = 0.4\omega$

Hence, $\omega = \frac{8}{0.4} = 20\,\text{rad/s}$

6. Final answer

The angular speed immediately after the collision is $\boxed{20\,\text{rad/s}}$

So the required integer is: $\boxed{20}$

7. Comparison with stored correct answer

Stored correct answer = $20$

Our derived answer = $20$

Hence, they agree.