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I was doing this problem, and I had trouble drawing the free body diagram of the pulley.

(The pulley is massless, frictionless.)

The problem

I had trouble understanding how both the ends of the string will apply a tension $T$ along the string on the pulley, and also, why doesn't the string that is in contact with the pulley apply either a contact force (as the string and the pulley are in contact), or a tension force (as the string is stretched)?

If I just assumed that this is how the string applies tension on the pulley, I was able to solve the problem, but I fail to understand why this is the case.

PS:While this is technically a homework problem, I did this problem, but just had a doubt in drawing the free body diagram of the pulley.

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how both the ends of the string will apply a tension T along the string on the pulley

Because the pulley is massless and so there should be same tension on both sides to make the net force 0 ('zero').

why doesn't the string that is in contact with the pulley apply either a contact force (as the string and the pulley are in contact), or a tension force (as the string is stretched)?

Umm... you can understand this one by reading this answer,the explanation is good.

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The tension acts in both directions at every point in the string. It pulls the wall at point C and the sliding mass to the right, and the top and bottom of the pulley to the left. If the pulley had mass, the static friction between the string and pulley would cause the pulley to rotate. That would require that the tension at the top to be somewhat greater than at the bottom, and the force F would be greater than the sum of the two in order to accelerate the pulley.

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