Is momentum conserved when all the surfaces are smooth?

Question

In general sense momentum conservation is making sense to me almost everywhere but when in physics problems scenarios where all surfaces become smooth and the objects are just gliding over one another I'm not able wrap my head around the fact the one body which is moving in a way parallel to the other body (and both bodies have absolutely no friction acting between them) makes the other body move in opposite direction.

In case of explosion and sliding off a ramp the force vectors are clear and can be defined and I'm able to visualize how conservation of momentum would take place. but in scenarios like the images I've attached with this post.

like in above question(question marked as 3 and is corssed) the block of mass 2 kg does appear to move with an accelaration of 5 m/s^2 but since the pulleys and ropes are ideal and all contact surfaces are smooth how does any amount of force can make block of mass 3 kg move?

and

And in this one specially (question marked as 6.) motion of block B and Block A is obvious but the motion of block c seems odd. There is no force acting on block c to provide it an accelaration in any direction but yet the author put an answer to the accelaration of block c as g/7?

please help...

Answer 1

The fact is, friction is just what we call the perpendicular component of the touching force between two objects. The other component is usually called normal component, and saying that there is no friction does not mean that the two objects do not exert a force on each other, it just means that this force is all normal i.e. perpendicular to their surfaces.

For example in the exercise 6 here, there is no friction between the rope and the pulley, but there can be a force perpendicular to the pulley, and in fact there is one and it pushes the pulley towards left. This is because the rope s pulling B, but this means that B is pulling the rope, but the rope then deflects towards the bottom, so it being pulled towards left, it pushes the pulley (and therefore C) towards the right direction.

Answer 2

There is a force in block C in the second diagram. If you consider the tension in the rope connecting A to B, then in the vertical section of the rope the tension is in the vertical direction (so exerts a downwards force on the pulley) and in the horizontal section the tension is in the horizontal direction (so exerts a horizontal 'rightwards' force on the pulley). The pulley is rigidly connected to block C so there is a net force on C downwards and to the right. C cannot move downwards due to the table it is sitting on, but if all the bodies are smooth and frictionless, it can move to the right.

Similarly, you should be able to see that the ropes will exert forces, in both the horizontal and vertical directions, on the pulleys in the first diagram.

In general, conservation of momentum is a direct consequence of Newton's laws, in that it can be derived from the laws. So in any closed system - i.e. one with no external forces that you don't account for - momentum will always be conserved. You could solve the equations of motion for the examples in the two diagrams you showed and confirm that momentum is conserved. But because we know that CoM is always the case, it is often simpler to use it to help solve the motions of the bodies. For example, if we calculate the acceleration of the smaller block in the first example, then we can immediately use CoM to calculate the acceleration of the 3 kg block.