Reaction at pin support

Not sure I understand the question.
The rope AC applies a downward force at A through its tension. That is the only way that forces at C can affect forces at A. Ropes cannot transmit torque about their endpoints or forces perpendicular to the rope.

 

You don't care.

A free body diagram can be drawn around the beam which excludes C. All that matters, as far as the equilibrium of the beam is concerned, is that the rope is in tension, which means that T_c > 0.

 

You don't care what the reaction at C is.

All you are interested in is finding the value of the distributed load w which keeps the rope in tension.

 

Looks messy.

Why don't you take moments about the pin at B? This will save you some work.

Remember, the reaction at C is not a load on the beam. The only load on the beam at point A is the tension in the rope, T_c.

 

You haven't got any reasonable answer yet that I can see. Remember, the purpose of this exercise is to find the value of W which keeps the rope in tension.

Again, for the umteenth time, RC is not a load on the beam. Like haruspex said way back, you can't push on a rope.

 

This is a superfluous calculation.

The moment calculation about point B looks good.

Then you went and spoiled it by adding the moments summed about point A.

You can write only one moment equation. Discard the moment equation about A.
Use the moment equation about B to find W, such that TC is always in tension. (TC > 0)

 

Where did this come from?

 

You might want to check that original moment equation again. There's no factors of 60 or 7 contained within it.