# Constraint forces do no virtual work: does this always apply?

The thread title is my main question, but to give some context, I'll include a particular example that made me ask the question in the first place.

In Hand and Finch, a small block is on a frictionless inclined plane on a frictionless surface. $F_1$ is the constraint force on the block that holds it perpendicular to the inclined plane surface, $F_2$ is the force on the inclined plane that counteracts gravity (the force by the floor on which the plane sits), and $F_{g_B}$ and $F_{g_P}$ are the gravitational forces acting on the block and the plane respectively. The inclined plane can slide along the horizontal surface.

The book states that constraint forces $F_1$ and $F_2$ can do no virtual work because they act perpendicular to the directions of motion, and gravity is the only force which can do virtual work in this problem. Thus $$\delta W_{IP} = 0 \\ \delta W_{SB} = mg\sin\alpha \delta d$$ but the inclined plane moves horizontally, and that is because by Newton's third law, $-F_1$ acts on the plane. So even though $F_1$ is a constraint force, it seems to have a component parallel to the direction of motion of the inclined plane. So shouldn't there be virtual work done on the inclined plane?

• I may have answered my own question. $-F_1$ does do work on the IP due to its horizontal component. However, $F_1$ does equal and opposite work on the mass $m$, as well. In the IP reference frame the mass $m$ only accelerates in a direction perpendicular to $F_1$. However in an inertial ref. frame, since the block remains on the IP surface and the IP surface accelerates horizontally, the block also has a horizontal acceleration component in the same direction as the IP, but with a force of equal and opposite magnitude acting on it. I think this'll always be the case with constraint forces. – Striker Sep 7 '18 at 19:05