## Mechanics.Today the academic discipline of mechanics includes two components: kinematics and dynamics. Kinematics is the "geometry of motion" using only the concepts of position, velocity and acceleration. Dynamics introduces the concepts of mass and force. The so-called "scientific revolution" of the 16th and 17th centuries put the study of mechanics on a firm mathematical foundation, with models that were capable of making predictions that could be experimentally tested. ## Copernicus and Kepler model the planetary system.The Copernican heliocentric solar system model was drastically simplified (removing all epicycles) and put on a mathematically sound basis with Kepler's three laws of planetary motion: - Each planet moves in an elliptical orbit with the sun at one focus of the ellipse.
- The radius vector of a planet sweeps equal areas in equal times.
- The square of the orbital period of a planet is equal to the cube of its mean orbital radius.
## Galileo finds laws of terrestrial motion.Galileo studied terrestrial motion of bodies with constant acceleration, which included falling bodies and projectiles. He defined the concept of acceleration, and found three laws of constantly accelerated motion: **d =**(t/2)**(v**_{o}+ v)**d = v**t + (1/2)_{o}**a**t^{2}- v
^{2}= v_{o}^{2}+ 2**a •d**
## Newtonian dynamics.Newton looked at motion in a more general way. His great accomplishment was to unify the understanding of motion to include both terrestrial (earthly) and celestial (heavenly) motions. He introduced (rather, refined and clarified) two concepts: mass and force. We state his three laws using somewhat modern language. - A body at rest remains at rest unless a force acts upon it. An object in a state of uniform motion remains in that state of motion unless an external force acts upon it.
- When a body is acted upon by an external force, it accelerates. The relation between the force and the acceleration is
**F =**m**a**. - When body A exerts a force on body B, then B exerts and equal and oppositely directed force on A.
We can state the content of these laws in a more compact way as equations: 1st and second law: 3rd law: Force and acceleration are vector quantities, so we indicate that fact with boldface symbols. Mass is a scalar quantity. The compact notation is nice, but we should always do more than memorize equations. Me must have clearly in mind (a) what every symbol means, and (b) any qualifications of limitations on the equations. I.e., we need to know the context in which the equations apply. Here it's absolutely crucial that we think, as we read the first equation: "The These laws seem innocent enough, but contain a powerful amount of information and insight. Philosophers have discussed the implications of ## Newton's gravitational law.Galileo's work showed that bodies near the earth have constant acceleration directed toward the earth. The climate of thought about heavenly and earthly laws was changing, and people now asked questions such as "What keeps the planets moving?" and "What keeps the moon in its orbit?" Asking such questions supposes that there's an underlying reason similar to the reasons for projectile motion. Of course, no one had reasons for either, but many were beginning to suspect that there were unifying principles and laws that might embrace both earthly and celestial phenomena.
Newton pondered the question of what kept the moon in its orbit, and the answer did Imagine an impossibly high mountain on the earth, as shown in the figure. A cannonball fired from the top of the mountain would fall to earth, but if it were fired with higher velocity, it would have a greater range. If one could somehow increase the firing velocity enough, it would fall into a circular orbit completely around the earth. The moon is falling in such a near-circular orbit. If the firing velocity of the cannonball is still larger, it could move in an elliptical orbit. This diagram illustrated the idea that all of these motions are due to the same cause, an acceleration downward, that is, always directed toward the center of the earth. Aristotle had considered motion toward the earth to be "natural", requiring no force. For Aristotle, straight line motion would be "unnatural" and require a force. Newton is turning this on its head, saying that straight line motion requires no force, while all other paths do require a force. He further says that that force is always in the direction of the acceleration, and related to the acceleration by F M and m are the masses of the two bodies, and R is the distance of their separation. G is a "universal gravitational constant. The force each body experiences is directed toward the other body. It's size is F G is the universal gravitational constant. - G = 6.67300 × 10
^{-11} m^{3} kg^{-1} s^{-2}The force on each body of an interacting pair has the same size, and the force on each is directed toward the other body. They are therefore equal size and opposite direction, as required by Newton's third law. Quite a number of people at the time suspected that there was such a gravitational force, and even supposed it might have an inverse-square dependence on distance (1/r Still, many critics complained that Newton still hadn't explained gravity, but had only given a law for it. Newton's response was classic "I feign [invent] no hypotheses." He meant that he was not going to speculate on a cause by inventing fictitious "reasons", for it was enough to have a correct set of laws. Other critics were bothered by the very notion that bodies could exert forces on each other with no medium intervening between them. They called this force an "occult force", for it seemed mystical or magical. It was, in fact, occult, for "occult" means "hidden". The French philosopher Rene Descartes had a competing theory. It explained all forces between planets as due to an intervening medium, an "ether" substance, and he even postulated properties of this ether to account for both gravitational and magnetic force interactions. You seldom see it mentioned in textbooks, for it died an early death. It was a philosopher's grandiose speculative theory of everything, but lacking the detail and discipline to make any firm and testable predictions. If I described it in detail here, you'd think it a rather crackpot theory. Even the philosopher Voltaire, who was also French, could not swallow Descartes' theory, prefering Newton's instead. Descartes was trying to do physics in the same way Aristotle had, with much speculation and very little contact with the real world of hard data. But other natural philosophers had progressed beyond that method, and Descartes' notions were not at all influential. Christian Huygens chided Descartes, saying that if Descartes had only read and understood the works of his contemporaries he wouldn't have made such mistakes. Newton also believed that space was filled with an "ether" substance, but he didn't develop a detailed theory of it, nor did he make any use of that idea in formulating his mechanics! We will see the ether idea revived in the 19th century, to the point where it becaume the object of laboratory experiments and much theorizing, only to die again in the early 20th century when Albert Einstin developed relativity theory, and, like Newton, Einstein did not use the ether idea. It played no role in his theory. This fact showed that the ether idea simply wasn't necessary to understand the physical phenomena, and never had been necessary. But more of that later. ## Circular motion.As an example of the power of the Newtonian mechanics, we consider the problem of circular motion. The acceleration of a body is the change in its velocity divided by the time duraton of that change,
or direction, or both,
requires acceleration.
A body moving in a curved path is accelerating, even if its speed is constant in size.
Therefore, from Newton's law, we know that the net force on it is non-zero.The diagram shows a a body moving with constant speed, .
We give them distinguishing subscripts because they have different directions, even
though they have the same size.
During that time the body has moved through angle V_{2}α. At the
right we show a vector diagram of the relation between these velocities and their vector difference, .Δ VNow consider the limiting case as the time interval gets very small, approaching zero. The angle approaches zero also. The diagrams have two similar, very skinny triangles. We can write:
So: and:
But
so we can write:
which becomes
This is the We can associate this acceleration with the inward (radial) component of whatever net force
happens to be acting on the body (of mass In this way Newton answered an important and difficult question: "How can a gravitational force from the sun, acting on a planet, be the cause of constant speed circular motion of that planet?" The general idea was not new, but Newton put it on a sound mathematical footing. Newton's gravitational law expressed the size of the gravitational force between any two bodies. So, considering the gravitational force of the sun on the earth, given by Newton's law: F since this is the only force acting on the earth, and is radial (toward the sun), we use our previously derived result, equating: F So, v For a circular orbit, its circumference is related to its radius by c = 2 π R The time for one revolution (the period, T) of a circular orbit is related to its speed by v = c / T Combining the last three equations gives: T Observe the important idea here. We have used Newton's laws (much as Newton did) to ## Work, kinetic energy, impulse, momentum.Prof. Heidi Hileman has a nice web page, Energy and Conservation in Physics, about the history of the development of ideas about energy. Further progress was slow. The history of formulation of concepts of energy and momentum is complex, with many persons struggling with the problem. People recognized that the quantity mv To see how these concepts arise from the raw materials of Galileo and Newton's mechanics, let's play with the equations a bit. Suppose a force acts on a body of mass m during which time the mass is displaced by amount (m/2)v In modern textbooks this is called the "work − kinetic energy" theorem, where the (m/2)v Suppose a force acts on a body of mass m for a time Δt. Using Newton's F =
m (v −
v/Δt, which may be rearranged to:_{o})
v −
mv = Δ(m_{o}v) .The quantity m
Or, "The net force on a body equals the rate of change of its momentum." Newton's version used calculus, and expressed the right side as the time derivative of the momentum. You may say "What's the big deal about momentum and kinetic energy? Both depend on velocity and mass. How are they different. Or are they?" They are In physics courses, typically a chapter is devoted to energy, and another to momentum. Some problems of mechanics yield to the application of one or the other of these concepts. But some problems, especially those where two bodies interact, require the simultaneous application of both concepts to arrive at answers. - —Donald E. Simanek, Feb, 2005.
Go to the next chapter, The assymetric Atwood machine. An example of linear and rotational kinematics. |