GATE 2024 ME – Question 54

Theory of Machines · Displacement, velocity and acceleration analysis of plane mechanisms · 2 marks · Numerical answer

At the instant when OP is vertical and AP is horizontal, the link OD is rotating counter clockwise at a constant rate $\omega = 7$ rad/s. Pin P on link OD slides in the slot BC of link ABC which is hinged at A, and causes a clockwise rotation of the link ABC. The magnitude of angular velocity of link ABC for this instant is _________________ rad/s (*rounded off to 2 decimal places*).

Link OD pivoted at O rotates counter-clockwise. A is 150 mm to the right of O and 150 mm above it. P is on OD at the height of A, so OP is 150 mm and AP is horizontal. The slot BC of link ABC passes through P at 60° to the horizontal.

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Correct answer: 12.06 to 12.18

Explanation

Here $OP = 150$ mm, so P moves with $v_P = \omega \times OP = 7 \times 0.15 = 1.05$ m/s, horizontally to the left (OD turns counter-clockwise and P is above O). The velocity of P is the sum of the velocity of the point of ABC that is under P and the sliding velocity along the slot, which is at 60°. The point of ABC under P is 0.15 m to the left of A, so its velocity is vertical. Resolving horizontally, the slide speed is $s\cos 60^\circ = 1.05$, so $s = 2.1$ m/s. The vertical velocity of that point is then $2.1\sin 60^\circ = 1.819$ m/s. So $\omega_{ABC} = \frac{1.819}{0.15} = 12.12$ rad/s (clockwise).