Consider the following setup (see illustration above): a mass m is connected to a circular section of a rail by means of a rod (with negligible mass) of length r, where r is the radius of the rail. The connection point P of the rod can move frictionless along the rail but is mounted such that the rod is always perpendicular to the rail, i.e. the mass is always at the center of the rail circle.
The rail is fixed to the ground and the mass subject to gravity. Initially, the rod is vertical (i.e. the weight of the mass fully supported by the rail). Now we push the rod along the rail until it is horizontal. The question is what work has to be done to do this? Let's see. The component of the gravitational force acting along the rail is given by
[tex]F(\theta) = m*g*sin(\theta)[/tex]
where [itex]\theta[/itex] is the angle from the vertical.
The work associated with moving the rod through an angle [itex]\pi/2[/itex] is then
[tex]W = \int F(s) ds = r*\int_0^\frac{\pi}{2} F(\theta) d\theta = m*g*r*\int_0^\frac{\pi}{2}sin(\theta) d\theta = m*g*r[/tex]
This means the work required to move the rod from the vertical to the horizontal corresponds to the work of lifting the mass trough a height difference r. However, the mass m has always stayed in the same location, so no work against gravity was done at all.
How is this paradox resolved?