PHY5200 F06

Chapter 1: Newton's Laws of Motion

Reading

Taylor 1.5 - 1.6 (today) and 1.7-2.1 (Wednesday)

Recall

Newton's First Law (the Law of Inertia)

In the absence of forces, a particle moves with constant velocity v.

Newton's Second Law (from me and Newton)

The net force F acting on a particle is always equal to the time rate of change of the momentum p = mv:

F=dp/dt

The Third Law and Conservation of Momentum

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.

Newton's Third Law

If object 1 exerts a force F21 on object 2, then object 2 always exerts a reaction force F12 on object 1 given by

F12 = -F21

In physics you will encounter many forces, and many of these are central forces. Central forces between two point-like objects point 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.

Apply the third law to a two particle system, and show that in the absence of external forces, the total momentum of the two particle system is conserved.

Multiparticle Systems

Now, consider an N-particle system, and apply the same logic to show conservation of momentum in the general case.

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.

Validity of Newton's Third Law

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.