No, the magnitude of the force exerted by the string on block 1 is not equal to the weight of block 2. The reason is as follows.

One important thing to know is that the tension T in the string is the same on both sides of the pulley, since the pulley is massless and frictionless. This tension is equal to the magnitudes T₁ and T₂ of the forces exerted by the string on blocks 1 and 2, respectively. Thus, T = T₁ = T₂.

Now, if T were equal to the weight W₂ of block 2, the net force on this block would be zero and, by Newton's second law, block 2 would not accelerate. However, when the blocks are moving, they are accelerating. Thus, the string tension T, and the magnitude of the force exerted by the string on block 1, cannot be equal to W₂.