How is mechanical energy conserved in inclined plane-block system?

Question

This is a rather standard problem where there is an inclined plane (of mass M) on a frictionless horizontal surface and a block (of mass m) slides on it. The problem asks us to calculate the speeds of both the plane and block once the block descends to the horizontal surface.

It is solved via Linear Momentum Conservation theorem (in the x-axis) and Mechanical Energy Conservation theorem.

I am confused why is the latter theorem applicable here. Mechanical energy is conserved when there is no work done by nonconservative forces. When we consider plane and block to be a system, then there are 3 external forces : mg, Mg and N (normal applied by horizontal surface on plane). mg and Mg are conservative forces but N is not. And because the centre of mass of the system goes downwards, N does negative work on the system (not 0). So, how is the Mechanical Energy conserved in this case?

Answer 1

System block and plane.
The block loses gravitational potential energy and the block and plane gain an equal amount of kinetic energy as the are no dissipative forces, eg friction.

Two forces on the block:
1 External gravitational force, $mg$, downwards due to attraction of Earth
2 Internal normal force on slope of plane due to the plane.
Positive work done by force 1 is $mgh$ where $h$ is the vertical drop of the centre of mass of the block.
Negative work is done by force 2 as there is a displacement of the force along the line of action of this force in the opposite direction to the force.

Three forces on the block:
3 External gravitational force, $Mg$, downwards due to attraction of Earth
4 Internal normal force on slope of plane due to the block - equal magnitude opposite direction to force 2 (N3L)
5 External normal force on base of plane due to the surface.
No work is done by forces 3 as the centre of mass of the plane does not move vertically.
Positive work is done by force 4 as there is a displacement of the force along the line of action of this force in the direction of the force.
No work is done by force 5 as the displacement of the force is perpendicular to the direction of the force.

The negative work done by internal normal force 2 plus the positive work done by internal normal force 4 is equal to the net work done by the two internal normal forces and is zero, as shown below.

At time $t$ the block is at position $A$ on the plane and at some later time $t+\delta t$ the block is at position $B$ on the plane.
The red vectors shown the normal force on the block due to the plane at the s(tart) and f(inish) and the blue vectors shown the same for the normal force on the plane due to the block.
For both normal forces the displacement of the forces along their line of action is $\delta s$.
The work done by the normal force on the block is $N_{\rm bp}\times(-\delta s)$ and the work done by the normal force on the plane is $N_{\rm pb}\times(+\delta s)$.
As the normal forces have the same magnitude the net work done by the two normal forces is zero.

Answer 2

Mechanical energy is conserved when there is no work done by nonconservative forces.

Mechanical energy is conserved when there is no work done by dissipative forces. However, while all dissipative forces, such as friction and air resistance/drag, are nonconservative, not all nonconservative forces, such as simple push-pull forces, are dissipative. See also the answer here: What is the difference between Non-Conservative and Dissipative?

So normal force is a non dissipative nonconservative force?

Yes.

The normal force is a push-pull force, specifically the component of a push pull force acting normal (perpendicular) to the surface. Thus it's a non-conservative force.

Although the kinetic friction force is proportional to the magnitude of the normal force, it is the kinetic friction force in the direction of displacement parallel to the surface that does dissipative work. Since the normal force is perpendicular to that direction, it doesn’t do work in that direction.

Thus the normal force is a non-conservative non-dissipative force.

Hope this helps.

Answer 3

The interaction between the block and incline does no work overall. It does positive work on the incline and negative work on the block, but the combined work by the interaction has to be 0 since each force follows the same point of contact with the same magnitude but points in opposite directions.

Therefore, the only relevant net work on the system here is that done by gravity on the block, which is already accounted for in the total mechanical energy.