# Can it be proven mathematically that the work done by normal force on the complete system of "curved" inclined plane and block is zero?

Using simple relative motion and Newton's laws of motion, I could prove that the work done by the normal force on a (unmounted straight inclined wedge + block) as a whole system is zero. Similarly, can it be proven mathematically that as the block slides down an unmounted curved inclined plane, the work done by the normal force on the whole system from the ground frame is zero? (The wedge is free to move on the horizontal surface and all surfaces are frictionless.)

– Dale
Commented Jul 26, 2023 at 3:20
• @Dale The normal force between the two bodies, acting during the block moves down the wedge. Commented Jul 26, 2023 at 3:48
• You don't have to prove anything. The normal force is always perpendicular to the surfaces in contact, here, and the direction of motion is always parallel to the surfaces in contact, so the normal force does no work in this situation. (Well, provided that the wedge doesn't move.) Commented Jul 26, 2023 at 5:04
• The curved inclined surface can be seen as the limit of a broken line. There're no essential differences. for mathematical rigor, all you need to do is figuring out the slope at each point on the curve. Commented Jul 26, 2023 at 5:44
• @BowlOfRed Think of a bomb (the system), with no external forces acting, exploding. Commented Jul 26, 2023 at 9:40