Conservation of Energy in a Rigid Body

[merovingian12]

Homework Statement

I'm having some trouble understanding "conceptually" the following problem (example 6.18 in Kleppner and Kolenkow.) A stick, initially upright, starts falling: find the speed of the center of mass as a function of position.

Homework Equations

Work-KE thm. for a rigid body (independent translational and rotational components)
KE for a rigid body (trans. and rot. components)

The Attempt at a Solution

The solution given in the text mostly makes sense to me, except I am a bit confused about how the initial assumptions can be justified. First of all, earlier, the text showed that the change in translational KE equals the integral of force over the displacement of the center of mass. When doing this integral, shouldn't we look at the NET force on the stick? (The text only looks at the weight and ignores the normal force pushing upward on the stick from the surface, i.e. the only work comes from gravity.) Now that I think about it, I'm not sure how the idea of work is extended to a rigid body. Specifically, what is the displacement that a given force should be integrated over? Different parts of a rigid body may have different displacements, so what should one look at? The point of contact where the force is applied?

 

[haruspex]

When doing this integral, shouldn't we look at the NET force on the stick?

The force from the ground is not moving the stick (the point of application is not yielding to the force), so it does no work.

Specifically, what is the displacement that a given force should be integrated over?

The (vectorial) displacement of the point of contact. For gravity, that should in principle be every point of the body, but for a rigid body that turns out to be the same as treating all the mass as being concentrated at the mass centre.