A block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths 2 m and 3 m. The hoist weighs 350 N. The ropes, fastened at different heights, make angles of 50° and 38° with the horizontal. Find the tension in each rope and the magnitude of each tension.

276 N and 225 N

Explanation:

Draw a free body diagram.  There are three forces on the hoist:

Weight force 350 N pulling down,

Tension force T₁ pulling up and left 50° from the horizontal,

Tension force T₂ pulling up and right 38° from the horizontal.

Sum of forces in the x direction:

∑F = ma

T₂ cos 38° − T₁ cos 50° = 0

T₂ cos 38° = T₁ cos 50°

T₂ = T₁ cos 50° / cos 38°

Sum of forces in the y direction:

∑F = ma

T₂ sin 38° + T₁ sin 50° − 350 = 0

T₂ sin 38° + T₁ sin 50° = 350

Substitute:

(T₁ cos 50° / cos 38°) sin 38° + T₁ sin 50° = 350

T₁ cos 50° tan 38° + T₁ sin 50° = 350

T₁ (cos 50° tan 38° + sin 50°) = 350

T₁ = 350 / (cos 50° tan 38° + sin 50°)

T₁ = 276 N

T₂ = T₁ cos 50° / cos 38°

T₂ = 225 N

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