Why does this block move backwards? In the diagram given below is a mass $m_1$ placed on an inclined block of mass $m_2$. And the question is to find the distance moved by the wedge when m1 reached the lowest point. The solution was given that as there is no external force on the system the center of mass doesn't move. My question is why isn't there an external force block m1 has a force mgsinx which acts downwards, now I don't know what this force will be canceled by if I take the complete two-block system. Could someone help me with this also Why does the wedge move backward? I don't have any force acting in a backward direction because of mgsinx. Any help would be appreciated.

 A: As this is a pretty standard homework problem, I will only attempt to answer the conceptual question Why does the wedge move backwards?
Forces are responsible for changes in momentum. However, in this problem, there is only one external force acting on the two-block system, which is gravity. All the other forces are internal (the normal force of one block on the other, and so on). The external force of gravity acts only in the $y$ (vertical) direction, and there is thus no external force acting along the $x$ (horizontal) direction. As a result, the net momentum along $x$ is conserved.
I imagine the block is released from rest at the top of the wedge. In this case, the total momentum (and thus the $x-$component of the momentum as well) is zero. As the block begins to slide down the wedge, it is forced by the wedge to move along the angle $\theta$ and so it will have a velocity (and thus momentum) in the $x-$direction (in addition to the velocity in the $y$ direction).
However, we know that the net momentum of the wedge and the block in the $x$ direction must be zero as it is conserved, and so the wedge must also move in the opposite direction of the block in order to conserve the $x-$ component of the momentum.
The same argument can be used to describe the motion of the centre of mass of the system: the centre of mass will indeed "fall" as an object would under gravity, but it will not move along $x$ since there is no net force along $x$.
A: In this situation external forces are present, gravitational ones, indeed as you watch the block fall, the center of mass will move down. But there are no external forces in the horizontal direction, so the horizontal position of the center of mass has to stay the same. But if the block moves to the right the only way to maintain the horizontal position of the center of mass is to have the wedge move to the left.
Of course it is not completely trivial that a law exists that states that the center of mass has to not move in absence of external forces. This result is derived from conservation of momentum (in this case conservation of momentum along the horizontal direction). You should be able to find plenty of resources to study this demonstration if you are interested.
A: 
My question is why isn't there an external force...

So in other words, why isn't gravity considered an external force here?   This is confusing at first, because we only experience the forces caused by gravity.  But gravity itself is not a force; it is  an acceleration.
Mass m2 wants to accelerate with gravity, but the ground prevents that.  Mass m1 is likewise obstructed by m2, at first, and that's where it gets interesting.

Why does the wedge move backward?

In short, "for every action, there is an equal and opposite reaction."
Mass m2 is preventing m1 from accelerating straight down with gravity, but it can only exert a force perpendicular to the contact surface.  So m2 pushes m1 upward (reducing acceleration g) and to the right.
The equal and opposite reaction: m1 also pushes on m2 with the same magnitude but opposite direction, downward and to the left.  The downward force is absorbed by the ground, but the left force on m2 causes it to move that direction.
