Week 7: Statics

Newton’s Second Law, of course, applies only to particles – infinitesimal chunklets of mass in extended objects or elementary particles that really appear to have no finite extent. However, in week 4 we showed that it also applies to systems of particles, with the replacement of the position of the particle by the position of the center of mass of the system and the force with the total external force acting on the entire system (internal forces cancelled), and to extended objects made up of many of those infinitesimal chunklets. We could then extend Newton’s First Law to apply as well to amorphous systems such as clouds of gas or structured systems such as “rigid objects” as long as we considered being “at rest” a statement concerning the motion of their center of mass. Thus a “baseball”, made up of a truly staggering number of elementary microscopic particles, becomes a “particle” in its own right located at its center of mass.

We also learned that the force equilibrium of particles acted on by conservative force occurred at the points where the potential energy was maximum or minimum or neutral (flat), where we named maxima “unstable equilibrium points”, minima “stable equilibrium points” and flat regions “neutral equilibria”123.

However, in weeks 5 and 6 we learned enough to now be able to see that force equilibrium alone is not su cient to cause an extended object or collection of particles to be in equilibrium. We can easily arrange situations where two forces act on an object in opposite directions (so there is no net force) but along lines such that together they exert a nonzero torque on the object and hence cause it to angularly accelerate and gain kinetic energy without bound, hardly a condition one would call “equilibrium”.

The good news is that Newton’s Second Law for Rotation is su cient to imply Newton’s First Law for Rotation:

If, in an inertial reference frame, a rigid object is initially at rotational rest (not rotating), it will remain at rotational rest unless acted upon by a net external torque.

That is, ~τ = Iα~ = 0 implies ω~ = 0 and constant124. We will call the condition where ~τ = 0 and a rigid object is not rotating torque equilibrium.

We can make a baseball (initially with its center of mass at rest and not rotating) spin without exerting a net force on it that makes its center of mass move – it can be in force equilibrium but not torque equilibrium. Similarly, we can throw a baseball without imparting any rotational spin – it can be in torque equilibrium but not force equilibrium. If we want the baseball (or any rigid object) to be in a true static equilibrium, one where it is neither translating nor rotating in the future if it is at rest and not rotating initially, we need both the conditions for force equilibrium and torque equilibrium to be true.

Therefore we now define the conditions for the static equilibrium of a rigid body to be:

A rigid object is in static equilibrium when both the vector torque and the vector force acting on it are zero.

That is:

~

If F tot = 0 and ~τtot = 0, then an object initially at translational and rotational rest will remain at rest and neither accelerate nor rotate.

123Recall that neutral equilibria were generally closer to being unstable than stable, as any nonzero velocity, no matter how small, would cause a particle to move continuously across the neutral region, making no particular point stable to say the least. That same particle would oscillate, due to the restoring force that traps it between the turning points of motion that we previously learned about and at least remain in the “vicinity” of a true stable equilibrium point for small enough velocities/kinetic energies.

124Whether or not I is a scalar or a tensor form...