Determine the maximum weight W of the block that can be susp

Determine the maximum weight W of the block that can be suspended in the position shown if cords AB and CAD can each support a maximum tension of 80 lb. Also, what is the angle theta for equilibrium?

Solution

cord AB will more tension than CAD.

so lets assume Tab = 80lb

now balancing forces on pulley,

Tab(- sin20i + cos20j) - Tacdj + Tcd(sin@i - cos@j) = 0

-27.36i + 75.175j + Tacd(sin@i - (1+cos@)j = 0

Tacd sin@ = 27.36

Tacd (1 + cos@) = 75.175

1 + cos@ / sin@ = 75.175 / 27.36

(1 + cos@)^2 = (2.75 sin@)^2

1 + 2cos@ + cos^2@ = 7.55 - 7.55cos^2@

8.55 cos^2 @ + 2 cos@ - 6.55 = 0

cos@ = 131/171

@ = 40 deg .......Ans

W = Tacd = 27.36 / sin40 = 42.56 lb ....Ans