Newton's laws of motion, together with Newton's law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified explanation for a wide range of physical phenomena, most of which had been studied by scientists and philosophers before Newton, who had come to various partial, incorrect, or approximate solutions. The phenomena for which Newton's theory works well include the motion of spinning bodies; motion of bodies in fluids; projectile motion; sliding along an inclined plane; motion of a pendulum; tides of the oceans; and the orbits of planets and moons. In particular, Kepler's rules for planetary motion follow from Newton's theory, as we shall see in a later section. The law of conservation of momentum, which Newton derived as a corollary of his second and third laws, was the first conservation law to be discovered.
Despite the impressive successes of Newton's theory, it's important to realize that it is very far from the whole story. In particular, it is completely incorrect as a description of atomic or subatomic physics. When applied to an atom, a Newtonian treatment of the forces involved implies that the atom should collapse and we should not be here! To explain why there are distinct atomic orbitals for electrons, and hence why the basic processes of chemistry (such as covalent and ionic bonds) occur, one needs quantum mechanics. The inception of quantum mechanics can be traced to Max Planck's treatment of blackbody radiation in 1900, which is separated from Newton's discoveries by about two centuries. We will discuss blackbody radiation in Chapter 6.
Newtonian mechanics is inadequate to describe the dynamics of galaxies, galaxy clusters, or black holes. Newton's theory also does not correctly describe the detailed properties of the electromagnetic field. A detailed theory of the electromagnetic field was developed by James Clerk Maxwell, but Maxwell's theory (which we do not have space to describe here) is not compatible with Newtonian physics if one attempts to understand experiments involving single photons, such as the photoelectric effect. Nonetheless, on length scales varying from tiny sand particles up to the size of a solar system, and for velocities small compared to the speed of light (the speed of light is c ~ 3 x 108 meters/sec), Newtonian mechanics has proven to be an extremely good approximation. For problems near the surface of the Earth (such as modeling accelerations or collisions of cars), and for the motions of satellites, Newtonian physics is still in use today.
Newton's first law is sometimes called the law of inertia, and it states that when the net force on an object is zero, it moves in a straight line at a constant speed.
Some explanation of the term net force is needed here. A single force is represented by a vector
The meaning of this is the following. Consider your intuitive notion of applying a force; for example, pushing a rock away from you. By pushing harder or softer you change the amount of force that you exert—the magnitude of the force. You could also choose a direction in which to push the rock, so the force has a direction as well. Magnitude and direction together determine a vector. When written in the component notation, as in equation (3.2), the magnitude is determined from the Pythagorean theorem to be
The arrow from the point (0, 0, 0) to the point (Fj, F2, F3) determines the direction of the force.
Vectors add via the parallelogram rule, as explained in Chapter 2. The vector sum has the physical interpretation of applying several forces concurrently—as if you and your friend are both pushing the rock. A more familiar example, a tug-of-war, illustrates the idea of combining forces acting along a single line, but in different directions. The condition of zero net force in Newton's first law is, in this case, equivalent to the statement that the tug-of-war is a tie—both sides are pulling, but they're equally matched and (at least for a while) the rope does not move much.
In general, the net force is the vector sum of all of the forces, which corresponds physically to the statement that all of the forces are applied to the same object at the same time. Suppose that we let x (t) denote the position vector of a particle at time t. Then the statement of the first law is x(t) = x0 + vt (3.3)
where x0 and v are constant vectors representing, respectively, the position of the object at time zero, and the (constant) velocity.
Some care in the application of Equation 3.3 is advised. An object satisfying Equation 3.3 would not satisfy any similar equation in a second set of coordinates r (t), where r (t) is defined by continuously rotating x (t) about some fixed axis with some fixed angular speed. This led to the definition of an "inertial frame" to be a set of coordinates in which Newton's first law holds true.
Newton's second law is sometimes called the law of acceleration, and it states that the acceleration of an object equals the total force acting on it, divided by a constant (called the mass), which is a property of the object. The second law, in equation form, gives rise to the famous where F is force, m is mass, and a is acceleration. It is very useful to think carefully about which quantities in this equation are defined in terms of the others. In particular, most people have an intuitive notion of the meaning of the word mass, but when pressed, they often find that their intuitive notion is incomplete or incorrect.
In elementary courses, it is often emphasized that mass is a quantity that is invariant with respect to a body's location; this is as opposed to weight, which would be different on the Earth versus the Moon. This is because the weight of an object A is roughly speaking the magnitude of the gravitational force between that object and a much larger object P, which could be a planet. Therefore, the weight is GmAmp/r2, where G is the gravitational constant, mA is the mass of A, mP is the mass of the planet, and r is the distance between the center of the object and the center of the planet.
The above statement that mass should be invariant under change of location gives a property of mass, but we still haven't defined it. In the above expression for the weight, what is mA? How does one calculate it, given an object? There are at least two definitions of mass in common usage, both in terms of Newton's second law. The gravitational mass is determined using scales and the local force of gravity; two objects at the same height above planet P are said to have the same gravitational mass if they have the same amount of attraction to planet P. The inertial mass is found by applying a known force to an unknown mass, measuring the acceleration, and then defining m to be F/a. Interestingly, astronauts measure inertial mass when in a "weightless" situation (meaning that they can no longer rely on the Earth's gravity to measure gravitational mass).
Newton was already well aware that the proportionality between inertia and gravitational attraction is an independent empirical fact, not something that follows from the first principles of his theory. Newton also noted that this proportionality does not apply to forces in general, citing as an example the force of magnetism, which is not proportional to the mass of the attracted body. Is the proportionality of inertial mass to gravitational mass then an accident? Whether it is or not, Einstein's general theory of relativity conceives of gravitational motion as inertial motion in curved spacetime. In such a theory, inertial mass and gravitational mass are not just accidentally proportional but they are the same concept.
One could envision a third definition of mass: if a bar of pure gold contains N = 1023 atoms of gold, then its "atomic mass" is defined to be N times the number of protons in a single atom (79 for gold) times a constant, mp, added to N times the number of neutrons per atom (118 for gold) times a constant, mn, added to N times the number of electrons per nucleus (equal to the number of protons for a stable atom) times a constant, me. Now there are three undetermined constants: m„, m and m . We then define p' n e the atomic mass similarly for two more stable elements and perform experiments that determine the constants mp, mn and me by setting the atomic mass equal to the gravitational-inertial mass. One may now extend the definition of atomic mass to all other elements and composite substances. Are the gravitational and inertial masses, now believed to be identical, also related to the atomic mass that we have defined here? Is there a proof of this? I will leave this as a question to be explored by the reader!
Newton's third law, sometimes called the law of reciprocal actions, states that for every force, there is a reaction force, equal in magnitude and in the opposite direction.
Consider a system of n particles, with positions oc1, ... , xn. Let Fj denote the force that particle i exerts on particle j. Newton's third law, given above, then states that
This is because multiplying all of the components of a vector by the constant -1 represents a vector of the same length, but in the opposite direction.
There is much confusion that stems from the difference between this and the so-called strong form of Newton's third law (thus making the above the weak form of the law). The weak form asserts the existence of a reciprocal force that is equal in magnitude and opposite in direction, while the strong form asserts that, additionally, the reciprocal force acts along the line joining the two particles. The strong form is satisfied by electrostatic forces and by gravity, but not by all types of forces that exist in nature. In particular, the strong form is not satisfied by the Lorentz force, which is the force exerted on a particle in the presence of a magnetic field. The Lorentz force on particle i from particle j (assuming both particles have electric charge Q) is given by
where vt is the velocity of the ith particle and B. is the magnetic field generated by the jth particle. In Equation 3.5, the symbol x denotes a new mathematical operation that we have not yet studied in this book: the cross product, also called vector product.
This is as good a place as any to introduce the cross product. This is a product that takes two vectors A and B, and gives you back a third vector, denoted A x B which is guaranteed to be perpendicular to the original two. (Thus, it can't exist in two dimensions, where there are only two perpendicular directions!) Of course, given that A x B is perpendicular to A and B just specifies the line that it lies on, but doesn't tell us much about its direction within that line or its length. The direction is easy to describe. Assume your right hand is stretched out flat, with the thumb extended. If the index finger on your right hand points along A, while B comes straight out of your palm, then A x B is in the direction of your thumb. The length is given by where A and B denote the lengths of the vectors with the same name, and 0 is the angle from |A| to |B|.
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