Constraints vs. real forces

It is confusing sometimes to realize what is a real, live force, and what is not. For instance, circular motion. You draw a free-body diagram, and sum up the forces. This is one side of the equation. The other side is your constraint on the motion, namely, that for a circular path a net force of mv^2/r is required. Sometimes, we call this "centripetal force," and that is highly misleading. It is not a real force, but a fictitious force, which basically means that if you impose the requirement that the motion is circular, an specific constraint is placed on your force balance.

When you think about it, your force balance always has a constraint, namely, that the net force has to result in mass times the acceleration required to produce the observed path. For a straight line path, zero acceleration, the force balance is zero: equilibrium. For a circular path, the required acceleration is v^2/r, so if circular motion is observed then the net force divided by mass must give v^2/r - if not, then you don't have circular motion. For generic paths, it is more complex: an observed path constrains the force balance, but it depends on the speed along the path and the local radius of curvature.

The confusion is, in my opinion, largely an unfortunate artifact of history and terminology. There is no centripetal force, it is just a boundary condition on your force balance that enforces circular motion.

Anyway: here's what I wrote to one of you earlier. We'll touch on this again in class tomorrow.

In the middle of page 336, in Equation 13-12 (F_n - m*a_g = m(-(omega^2)R)), why do we have a negative sign on the right side. I feel like gravitational acceleration (weight) and centripetal acceleration, both being directed toward the center of the earth, should have the same sign. Here, if you put them on the same side, they have opposite signs.

It is a little bit confusing because the centripetal force is not a true force in its own right, but only the *result* of all other forces. The language in the text is confusing in this regard.

What it is saying is that there are two real forces acting: the normal force acting upward, and the gravitational force acting inward. Since the box follows circular motion by virtue of being on the earth's (rotating) surface, we know that the *constraint* on the force balance in magnitude is that it must sum to mv^2/r. The constraint on direction is that it must be directed radially inward to result in a circular path.

Saying that "centripetal force points toward ..." or "the centripetal acceleration points ..." is misleading. The left side of the force balance contains all the real forces; the right side contains the constraints on the motion from the known path:

(1) straight line: constraint is that forces sum to zero
(2) circle: constraint is that forces give mv^2/r to be consistent with the path, pointing radially inward.
(3) general: what we derived a while back; sum of forces relates to the rate of change of speed and the local radius of curvature of the path.

So it is OK that they have the same sign - it says that the net force balance comes out in favor of gravity, and the net force (which we call the centripetal force, misleadingly) points in the same direction ... we stay on the surface and don't fly off.