When the Tucurui Dam was constructed in northern Brazil, the lake that was creat

Question

When the Tucurui Dam was constructed in northern Brazil, the lake that was created covered a large forest. Underwater logging was started. During the logging process, a tree is selected, trimmed, and anchored with ropes to prevent it from shooting to the surface like a missile when cut. Assume that a typical large tree can be approximated as a truncated cone with a base diameter of 6 ft, a top diameter of 1 ft, and a height of 75 ft. Determine the resultant vertical force that the ropes must resist when the completely submerged tree is cut. The specific gravity of the wood is approximately 0.6.

Explanation / Answer

Let A be the 20 degree side
Let B be the 30 degree side
F(Ahorz) = F(Bhorz) (equal and opposite)
(1) T(A) sin20 + T(B) sin30 = 100
(2) T(A) cos20 = T(B) cos30
(1) T(A) = (100/sin20) - (T(B) cos30/cos20)
(100/sin20) - (T(B) cos30/cos20) = T(B)cos30/cos20
T(B) = (100/sin20)/[(cos30/cos20) + (sin30/sin20)]
T(B) = 122.668
F(Bvert) = 122.668 sin30 = 61.334 lb.
F(Avert) = 100 - 61.334 = 38.666 lb.

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