In a four bar linkage, the fixed link is horizontal and has a length of 60 mm. The crank makes an angle of 30° with the fixed link and is attached to one end of the fixed link. The lengths of crank, coupler, and follower links are 20, 70 and 50 mm respectively. For the open chain configuration, the angle of coupler with respect to the horizontal is
Options:
(A) 2 degrees
(B) 32 degrees
(C) 122 degrees
(D) 152 degrees
Official Answer Key:
B
Step-by-Step Solution & Explanation
Place O₂ at origin, O₄ at (60,0). Crank pin A = (20cos30°, 20sin30°) = (17.32, 10.0) mm. B lies at intersection of circle of radius 70 mm (coupler, centered at A) and circle of radius 50 mm (follower, centered at O₄). Solving the two-circle intersection gives two solutions (open/crossed configurations); the open-chain solution gives B ≈ (76.65, 47.15) mm. Coupler angle = atan2(47.15−10.0, 76.65−17.32) = atan2(37.15, 59.33) ≈ 32°. (The other geometric solution, ≈−58°, corresponds to the crossed configuration and is not the open-chain answer.)
Verification note: Solved by direct coordinate geometry (circle–circle intersection) rather than a memorized formula.
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