Introduction
In this problem, we have a string passing over a clamped pulley of mass m. The string supports a block of mass M. We need to find the force on the pulley by the clamp.

Solution
To solve this problem, we need to use the concept of free body diagrams and Newton's laws of motion.

Free Body Diagram
We draw a free body diagram of the pulley and the block. The free body diagram of the pulley is shown below:

We can see that there are two forces acting on the pulley: the tension T in the string and the force F by the clamp. The free body diagram of the block is shown below:

We can see that there are two forces acting on the block: the weight of the block Mg and the tension T in the string.

Newton's Laws of Motion
We apply Newton's second law of motion to the pulley and the block. For the pulley, we have:

F - T = ma

Where F is the force by the clamp, T is the tension in the string, m is the mass of the pulley, and a is the acceleration of the pulley.

For the block, we have:

T - Mg = Ma

Where M is the mass of the block and A is the acceleration of the block.

Force on the Pulley
We can solve the above two equations simultaneously to get the force F by the clamp. We get:

F = (M + m)g / 2

Therefore, the force on the pulley by the clamp is (M + m)g / 2.

Conclusion
In this problem, we used the concept of free body diagrams and Newton's laws of motion to find the force on the pulley by the clamp. We found that the force on the pulley by the clamp is (M + m)g / 2.