In the mechanism shown in fig, the angular velocity of link AB is 5 rad/sec anticlockwise. At the instant shown, determine the angular velocity of link BC and velocity of piston C.[Figure: link AB hangs vertically from a fixed pivot A; AB = 200 mm, r…

Velocity of B: \(v_B = \omega_{AB} \times AB = 5 \times 0.2 = 1\) m/s, perpendicular to AB. With B directly below A and AB rotating anticlockwise, \(v_B\) is horizontal to the right.

Instantaneous centre (I) of BC: draw a line perpendicular to \(v_B\) at B (vertical line through B) and a line perpendicular to \(v_C\) at C (C moves vertically, so a horizontal line through C). They meet at I, directly below B at the level of C.

IB = 150 mm, IC = 450 mm.

\(\omega_{BC} = \dfrac{v_B}{IB} = \dfrac{1}{0.15} = 6.67\) rad/s (clockwise, since B above I moves to the right)

\(v_C = \omega_{BC} \times IC = 6.67 \times 0.45 = 3\) m/s (C, to the right of I, moves downward)

Vector check: \(\vec v_C = \vec v_B + \vec\omega \times \vec r_{C/B} = 1\,i + (-6.67k) \times (0.45\,i - 0.15\,j) = 1\,i - 3\,j - 1\,i = -3\,j\) m/s ✔

Answer: \(\omega_{BC} = 6.67\) rad/s clockwise; \(v_C = 3\) m/s vertically downward.