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GATE 2022 ME (ME1) – Question 31

Engineering Mechanics · Impulse and momentum, energy formulations · 1 mark · Numerical answer

The plane of the figure represents a horizontal plane. A thin rigid rod at rest is pivoted without friction about a fixed vertical axis passing through O. Its mass moment of inertia is equal to 0.1 kg·cm$^2$ about O. A point mass of 0.001 kg hits it normally at 200 cm/s at the location shown, and sticks to it. Immediately after the impact, the angular velocity of the rod is ___________ rad/s (in integer).

a horizontal rod pivoted at O, with the point mass moving at 200 cm/s perpendicular to the rod and hitting it at 10 cm from O.

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Correct answer: 10

Explanation

The angular momentum about O is conserved: $mvr = (I_O + mr^2)\omega$. The left side is $0.001 \times 200 \times 10 = 2$ kg cm²/s. The moment of inertia after sticking is $0.1 + 0.001 \times 10^2 = 0.2$ kg cm². So $\omega = \frac{2}{0.2} = 10$ rad/s.