Question

When an object of weight w is suspended from the center of a massless string?

Answer 1

An object suspended from its own center will not have any motion or rotation unless acted on by an outside force. In the question the idea of a massless string is introduced. The mass of the string is pointless. In this trivial example the weight w did not need to be defined as "w" as there is only one object considered.

Answer 2

I attached a picture of the diagram associated with this question.

Now,
When we check the vertical components of the tension in the rope, we will find that we have two equal components acting upwards.
These two components support the weight and each of them has a value of TcosΘ

The net force acting on the body is zero.
Fnet=Force of tension acting upwards-Force due to weight acting downwards
0 = 2TcosΘ -W
W = 2TcosΘ
T = W / 2cosΘ