Question

Chapter 16 A mechanism consisting of two rods BC and CD as shown in the figure. Rod BC rotates about the pin B uniformly with a constant angular velocity w, whereas the end of rod CD is connected to the block A. At the instant = 60°, X = 6 m and w = 3 rad/s, calculate: (a) The velocity of block A (b) The acceleration of block A.

Answer 1

Sol. Given values : x=6m., w - Brad/s. Diagram (schematic). assume assume col raw an imaginary vector from B' to 'D. direction BD Now BD = BC + CD Bb = $0 + OD 0 - © Equating LHS of and to 50 +OD = BC + CD using complex equation notation (et) @ becomes To = xelo txea - OD Con=x) (-ix for dirn When 0= 60, the mechanism resembles equilateral - triangle, BO = 0 e / o x 2 xe() - OD = 8 CHAP 7 sao) equating real components cos (60) + cos (-x) =) cosd = -1 a = 60° C negative direction)

from equation 50 = xe otxex - od Bo is the displacement of piston (from the plane on which В pin is resting : 50 = xétio txearoo (differentiate with Brespect to time) dt de Co as oo is constant) AO at - here to - w (given) where as da is unknown (let = wel as v exists in vertical direction real component must be equal to Zero even at 0=60°, x=-60 dt 10

VE X Cicoso - Sino) (w) + x (ticosd-sira) (Wal by HARMA part @xsino w-X-sina to. anwes Wa=-3 rad/s. - v = (6m) (icosto ) (3) + (6m) (-icos 60°) (-3). Tv = (180 m/s) i ( direction indicated by ;) (a) answer as we differentiate v with respect to time again we get acceleration a' . La = dv = x (eo) (wir + dw i) + xeld (War" ( 14 d Wх 1-97 - from W = constant . dw = = 0 (constant w) . as acceleration is in vertical direction, real part=0 dt

X SIM taking real components -Xw roso y sin 0 (0) 4 - Xwa cos x - x sinad was so dwe (sin6o) = -9 - 9 & g. u dwe - I dwa = -6 J3-rad/ ² (downward dirn) idt ota using dwa in acceleration equation a z C6m) (isin 6o -q rad (7) + 6m) -isin 60) (-9 rad/s) u tom ( i costo) ( - 6 vs rads? a= 6 x 1 1 x - 658 m/h (a = 1853 i m/ answer (b)
tThis method is by using the complex notation of (cos(X) + i sin(X)) where the real part represents horizontal direction and vertical component is represented by the coefficient of the complex term 'i'.