Lets consider the left side wedge to be wedge 1 and the right one to be wedge 2. Considering the block, the wedge 1 exerts a force N1(normal reaction) on the block, the wedge 2 a force N2 and the Earth exerts a force Mg. The wedge 1 experiences normal reaction N1 and a force mg exerted by the Earth. The wedge 2 the force N2 and the force mg.
For the block,
So, we hence N1=N2=N ( we consider N instead of N1 and N2 as the normal force for the sake of simplicity ).
Now, for wedge 1,
Since the same one counts for wedge 2 too, I ain't uploading the image for wedge 2.
Finally your solution counts as
So your solution is obviously correct to the point.
Next coming to your doubt.The cancelling out of forces etc depends on the way you consider the problem. If you consider the system of the block and wedges as a whole. You see that even though there is "Internal Movement" inside the system you considered due to "Internal forces" like normal reaction, the system as a whole isn't moving. Hence the total external force acting on the system as a whole would be zero. 499
Hence the External forces F should cancel each other. But if you consider block or wedge by wedge,like we would take for solving problems and equations( like we took here), the forces acting on a particular wedge, lets take it to be wedge 1, has to sum up to be zero since the wedge is in equilibrium. Hence, like we considered it the above problem, the force F is cancelled out by the sine component of the normal force N and vertically, the sum of the cosine component of N and the weight mg is cancelled out by the normal force N'. So, When you come into the system, you can build your distinct relations on how the the external forces cancel out each other. Saying it more simply, suppose this system is just a part of another huge problem and the movements taking place inside the system is of no concern for the the outside the system, then you just consider the external forces F to cancel each other. But if the inside of the system (i.e., the wedges and the block ) is a concern for your problem, then you take it block and wedge by wedge and solve like we did above. Please take it to your attention that as F vary the wedges and block may or may not move.And if this "may or may not"movement is of no concern for your question, you just consider the system as a whole. But if this is a matter of concern for your question, then you has to consider the inside too ( block and wedge by wedge ). 766