Taylor 1.5 (today) and 1.6 - 1.7 (Wednesday)
In the absence of forces, a particle moves with constant velocity v.
The net force F acting on a particle is always equal to the time rate of change of the momentum p = mv:
While the first law is used to define what an inertial reference frame, and the second law tells us how an object reacts when subject to force, Newton's third law tells us something about the forces exerted between interacting objects. It is used in proving the conservation of momentum, and is useful in the solution of many problems. In fact, the third law can be derived from conservation of momentum, assuming that forces are central.
If object 1 exerts a force F21 on object 2, then object 2 always exerts a reaction force F12 on object 1 given by
In physics you will encounter many forces, and many of these are central forces. Central forces act between two point-like objects along the line connecting the two objects. This is not a requirement of the third law, but it does make it easier to see the validity and consequences.
As an example, consider a system of two particles, 1 and 2, acting on each other by central forces F12 and F21 = -F12, and acted on by external forces F1ext and F2ext. The net force acting on particle 1 is
and the net force acting on particle 2 is
The net force acting on the system of particles 1 and 2 is
by virtue of the third law. With the second law, we can relate the forces to changes in momentum,
and
The momentum of the system is the sum of the momenta of the individual parts
therefore, by the rules for differentiating vectors, the rate of change of the momentum of the system is the sum of the rates of change of the momenta of the parts
again using the third law to cancel the inter-particle forces. The importance of this result is that it says, irrespective of the interactions going on between the two particles, if the external force is zero, the total momentum of the system is conserved (its time rate of change is zero). We will now generalize this to systems of arbitrary size.
Now, consider an N-particle system, and apply the same reasoning to show conservation of momentum in the general case. We'll use the Greek letters α and β as subscripts denoting particular particles in the system, α, β ∈ {1, 2, ..., N}.
The net force acting on particle α is
where the sum is for all particles β not equal to the particle of interest α.
The net force acting on the system of particles is the sum over all values of α
where the sum of all the external forces on the particles is called the total external force, Fext. The double sum can be evaluated with some careful reasoning. First, the sum over β ≠ α can be written as the sum over β < α plus the sum over β > α,
Next, notice that the double sum ∑α∑β>α is the same as the doube sum ∑β∑α<β. This can be verified by arranging the terms in the sum in a matrix (rows corresponding to α and columns to β say), and verifying that the same set of terms appear. The change in the sums is equivalent to changing from summing over the terms in a row, then adding up all the tow sums, to summing over the terms in a column, then adding up the column sums. Then, since α and β are just dummy indices, they can be switched in the sum, yielding
Finally, according to the third law Fβα = -Fαβ, such that the double sum vanishes:
and we find that the net force is equal to the net external force on the system, all the internal forces canceling out,
The third law relates the rate of change of a particle's momentum to the net force on the particle,
The momentum of the system is the sum of the momenta of the individual parts
therefore, by the rules for differentiating vectors, the rate of change of the momentum of the system is the sum of the rates of change of the momenta of the parts
This generalizes the result for two particles determined above, and can be interpreted as stating that if the net external force on an arbitrary system of particles is zero, the rate of change of the total momentum of the system equals the net external force applied to the system. This leads to a statement of the general principle of conservation of momentum.
If the net external force Fnet on an N-particle system is zero, the system's total momentum P is constant.
Is Newton's third law valid always? Implicit in Newton's third law is a concept of simultaneity -- the two forces F12 and F21 must apply at the same time -- that is incompatible with special relativity.
We can also learn something by considering a case of non-central forces, such as the magnetic force between two charges moving at right angles to each other. Each moving charge produces a magnetic field that causes a Lorentz force on the other charge. That force must be perpendicular to the direction of motion of the charge, and lie in the plane defined by the motion of the two charges. Therefore, the two forces cannot be in opposite directions, as required by the third law.
The resolution to this paradox is beyond the scope of this course, but it involves the realization that the magnetic field itself carries momentum, and that the force is between each particle and the magnetic field, not the particles themselves. Then the third law works correctly for each particle, and the momentum of the particles plus magnetic field is conserved.
© 2007 Robert Harr