The angle is θ = 55.0°. The masses are, for the small pulley, m1 = 3.1 kg, for the traveling pulley, M2 = 5.7 kg, and for the crate, MC = 41.0 kg. What is the minimum tension with which the operator must pull on the cable (assume the cable is of negligible mass) in order to slowly raise the crate.

Please see the figure: http://i92.photobucket.com/albums/l18/bonbons06/prob93_upmass2pulleys.gif

I solved and all , but still , I need to under stanf what's going on there..

I used 2F(T) = ma + mg (in this case a = 0 )

2F(T) = (41 + 5.7) * 9.8 => F(T) = 228.83 N

and yeah I got the right answer.
but why the angel did not effect anything, I mean why we did not use it , and why also I did it right even though I made m1 like it's ignored..
thanks

Susus:

You didn't worry about the mass of pulley M1 because it doesn't move up (or down) against gravity. Hence M1*a = 0, and no force is needed to make it accelerate up or down. Now, in reality a pulley that has mass does have rotational inertia - meaning that to get it to spin as you hoist the crate requires some force. But for these types of problems you assume that the only force you're interested is iwhat it takes to holdthe crate steasy against gravity, and not to accelerate it (like you said, ma= 0). So there is no need to worry about the rotational inertia of pulley M1, and its mass can be ignored.

As for the angle of the rope into the man's hands - it does not affect the tension in the rope. It would affect how hard vertically or horizontally the man has to tug, but that's not the question here.