Classical Mechanics

Classical mechanics is a part of classical physics, which developed prior to the development of modern physics about a century ago. For a good lecture on classical mechanics  go to one of Walter Lewin's lectures at MIT.

Newtonian Physics


©Fernando Caracena 11 September 2012


Standing on the Shoulders of Giants

Isaac Newton (dubbed Sir by the monarch of England) made a great leap forward in physics by creatively synthesizing the works of other geniuses that came before him, to which he added his own genius-level ideas. Although he did not mention their name directly when he made the statement, 'If I have seen further it is by standing on ye sholders of Giants', it is not hard to identify these giants.

For those of you who want to see a great lecture on th esubject of Newton's Laws of Motion, see the one by Walter Lewin of MIT.

The Giants

From previous discussions, we are somewhat familiar with Galileo (see also Grokking Galileo's Physics) as the genius who developed a mathematical description of motion, a topic that was developed further by René Descartes in coordinate geometry. In addition to these two great man, Newton drew on the works of Nicolaus Copernicus and Johannes Kepler. Copernicus brought astronomy out of the middle ages by proposing that the planet Earth, and others observed in the sky such as Jupiter and Mars, all go around the Sun. Before him, astronomers believed that all the astronomical bodies orbited about our own planet. In order to enforce the idea of Earth-centric motion mathematically, ancient Greek astronomers had invented a quite complicated description of the motion of these objects (The Ptolemaic System) as a series of nested curves, called epicycles. The heliocentric (sun centered) hypothesis of Copernicus simplified the model of the solar system, which allowed a deeper understanding of planetary motion that unified the laws of planetary motion as the laws under which everything moves. Kepler was a genius of mathematical and philosophical bent who explored the Copernican system in mathematical detail using all astronomical observations that he could access. According the Wikipedia article on  Johannes Kepler "Much of Kepler’s enthusiasm for the Copernican system stemmed from his theological convictions about the connection between the physical and the spiritual; the universe itself was an image of God..." His life's work on planetary theory is summed up in Kepler's Three Laws of planetary motion:

  1. The orbit of every planet is an ellipse with the Sun at one of the two foci.
  2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Newton's Physics

Newton enlarged Galileo's physics by adding the concept of mass and inertia to the kinematics, which became the effects of forces operating on matter.

Newtonian mechanics looks like a lot of circular reasoning to those who are concerned with the pure logic of it; but it makes sense in terms of experience. First, we have the notion of a force (here symbolized by the letter, F ), which is a push or a pull that you can measure with something like a spring balance, or with more sophisticated instruments, such as a strain gauge. All vectors, which are directed quantities (such as Force), are represented as a bold symbols in the equations of physics. Aristotle argued that a force was required to keep things moving and that motion died down when the force was taken away. For Aristotle, the natural state of all Earthly things was to come to rest, motion requiring a force to sustain it. However, more perfect, celestial objects such as the sun, moon and stars, moved in circles, and in perpetual motion.

Newton's First Law of Motion--Inertia

Aristotle argued that a force was required to keep an ordinary, physical object moving and that motion died down when the force was taken away. For Aristotle, the natural state of all things was to come to rest, motion being a temporary aberration that had to be continually forced. Newton made the corner stone of his mechanics Galileo's idea that the natural state of everything is to continue doing its own thing (drifting at a constant velocity) until acted on by a force from the outside. For example, a shuffleboard puck once launched would move forever if it were not to bump up or rub against something else. Incidentally, Newton's ideas are confirmed by those of us who have traveled into outer space. If you are far away from the Earth, floating in a vacuum, and toss a ball, you will see the ball flying away from you without slowing down.

Newton made the concept of inertia, which he borrowed from Galileo, a cornerstone of his First Law of Motion: a physical object that is moving tends to move in a straight line at a constant speed, or remains at rest, if initially at rest, until acted on by an external force. Rather than rest being the final state of all things, inertia was recognized as the natural tendency of matter to preserve its state of motion. Rest being relative to the observer's frame of reference. For example, a pebble at rest on the beach, from the vantage point of the Moon, is hurling through space on a helical path. A comedian once said that inertia is Nature's way of making sure that everything does not happen all at once (in one swell foop).


On Earth, we see that the motion of physical objects winds down, because we can almost never isolate a moving body from outside forces, especially the force of friction, which is everywhere. A shuffleboard puck launched would move forever if it were not to bump up or rub against something else. A real shuffleboard in good condition will offer little friction, reducing the deceleration of the puck to almost nothing, allowing non-spinning pucks to move in a straight lines at a constant speed over its length.


Newton's Second Law of Motion

In his Second Law of Motion, Newton stated that if an unbalanced force acts on an object of mass (M), it will cause a time rate of change in the momentum of that object equal to that force--momentum, being defined as the product of an object's mass and its velocity, M v . (If there is no ambiguity in doing so, the product of two quantities is represented by placing the symbols representing them, side by side, otherwise a multiplication sign * is placed between them.) In equation form, the statement of Newton's Second Law of Motion reduces to

F =time_rate_of_change_of (M v ),                                (1a)

where "time rate of change of" represents in English what amounts to a mathematical algorithm, or procedure, performed on the momentum, not elaborated on here since the discussion is not meant to be overly mathematical, but rather to give the flavor of the mathematics (see Grokking Calculus). A prescription as “time_rate_of_change_of” in (1a) is also called a mathematical operator , or operator for short, and also known as the time derivative of differential calculus. In (1a) both F and written as bold symbols to indicate that they contain information about direction as well as magnitude, that is, they are vectors. Vectors are discussed at length below.

You may be more familiar with the mathematical statement of Newton's second law for the situation where the mass of an object does not change. In this case, only the velocity can change, (1a) reduces to the following:
                               F =M*time_rate_of_change_of (v ),                          (1b)
which is further simplified by substituting the acceleration, a , for the Time rate of change of (v ) in (1b),
                               F =M a .                                                                   (1c)

Incidentally, Newton's ideas are confirmed by those of us who have been privileged to travel into outer space. In the vacuum of space and far away from the Earth, a ball once tossed will fly off in a straight line and at a constant speed, which is only an approximation over a short path, because feeble forces of gravity are ubiquitous in space, each celestial body contributing a tug this way or that, which results in a slight curve that becomes appreciable over large distances.

Newton's Third Law

Newton's Third Law states that for every action (force) on an object, there is an opposite and equal reaction on what is producing that force. When you push on something, it pushes back on you. When two objects interact, they exert equal and opposite forces on each other. Because of this property, two objects colliding exchange equal and opposite momenta, so that the total momentum of the two does not change.

time_rate_of_change_of (M1 v1 )=-time_rate_of_change_of (M2 v2 )   (2a)


time_rate_of_change_of (M1 v1 + M2 v2)= 0.                                      (3b)

Newton's Third Law implies that the space-walking astronaut who throws a ball while drifting in outer space, will feel a recoil in the opposite direction, but that the sum of both their momenta will remain the same as a result of the ball toss.

Newtonian mechanics presents force as the agent of interaction between objects with mass. For example, connect two balls of unequal mass by a compressed spring that is oriented along their line of centers. Imagine that the balls and spring are at rest in your frame of reference. Let the spring go. If all other forces acting on the balls are negligible, they will fly apart in opposite directions at different speeds. The ball with the least mass will be going the fastest and the one with the most mass, the slowest. During their acceleration, the force propelling each ball is of the same magnitude but acting in the opposite directions. After the interaction, each ball is moving at a different speed. While the balls were interacting through the spring, they were accelerating, each in inverse proportion to its mass. The accumulated speed in each of them during their interaction was therefore inversely proportional to its mass.

Conservation of Momentum
Although the velocities acquired by two objects interacting by equal and opposite forces are not equal in magnitude, the above discussion suggests that the products of mass and velocity of the balls is; because the two velocities acquired by the interacting balls are inversely proportional to their masses. The product of the mass of anything and its velocity is called momentum. Note that the momenta of a system of particles sum as vectors. The total momentum of the two ball does not change as a result of their interaction, if no other force is acting on them from the outside. Their total momentum is conserved. Conserved quantities are of special interest in physics. Other conserved quantities are discussed in other sections.

Weight and the Force of Gravity
Weight is a concept that is sometimes used interchangeably with mass in the popular media. Weigh is really the force of gravity (usually the Earth's gravity) on an object rather than the intrinsic property of matter that mass is. Newton proposed that any two objects in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to to the square of the distance between their centers. The gravitational force on mass, m2 , produced by a mass,  m1 , at a distance |r1-r2| away is given by the following expression:

F21 = -unit_vector_pointing_from_r1_to_r2 * (G * m1 * m2)/|r1-r2|2.                     (4a)

Note that the distance is defined as

|r1-r2|=[(x1-x2)2 +(y1-y2)2 + (z1-z2)2]1/2,                 (4b)

which means that its square is just,

|r1-r2|=[(x1-x2)2 +(y1-y2)2 + (z1-z2)2]   .                   (4c)

The moon and the earth are drawn together by a tremendous force of gravity. The Earth orbits the sun because of the Sun's gravitational pull on the Earth. An apple falls to Earth from from a tree because it is attracted to the earth with a stronger force than the tugs exerted on it by other bodies in space, such as the moon or the sun. Some of these celestial objects although much more massive are so far away that their gravity is feeble.

Newton defined mass as the amount of stuff a physical object contains. In everyday life we weigh things to determine the amount of stuff that they contain; but weight and mass are two different things. Weight seems to be the same as mass, but it is not. Heavy objects are harder to throw--they have more mass--but, weight is not an intrinsic property of an object. Taken to the surface of another planet or the moon, the same object changes its weight, or even becomes weightless; however, it always has the same mass. Why does weight change? The answer is that weight is really a force, the pull of some huge physical body such as the Earth on everything approaching it.
The process of weighing to determine mass is made possible because the pull of the Earth's gravity on a physical object is proportional to its mass. Galileo's famous experiment, in which he dropped to balls of different masses off the tower of Pisa demonstrated that objects of different weight fall the same way, so that when dropped from the same height at the same time, they hit he ground at the same time. This is true to the extent that you can neglect the effect of air friction, which for a cotton ball would be noticeable, but not for a cannon ball or lead shot. Through other experiments, Galileo found that all objects (for which air resistance can be neglected) fall at a constant acceleration (called the acceleration of gravity, g, which we now know to be about 9.8m s-2). Newton's Second Law of Motion (1c) indicates that the force that the Earth's gravity exerts on an object (its weight) is proportional to the mass of the object, and the above observations indicate that the constant of proportionality between mass and its weight is the acceleration of gravity, g.

The dynamics of gravity becomes kinematics

On the Earth's surface the same acceleration of all objects that do not have appreciable air resistance, regardless of their masses, results from the fact that inertial mass and gravitational mass are equal. We can show this by substituting for the Earth's gravitational pull from (4a) in the equation of motion (1c), restated here for convenience,

                                        F =M a ,                                                                   (1c)


-uz* G*Me *m/|re|2 =M a,

where -uz  is a unit vector that points downward (do not be intimidated by the notation).

The magnitude of the term on the left is evaluated using the python code fragment in listing 1, which results in a= - 9.80 m/s when rounded off to the nearest hundredth.  We recognize the value as just the acceleration of gravity on the earth's surface.

Using the Moon's mass and radius instead of that of the Earth, we find out that everything weighs about a sixth of what it does on the surface of the Earth, as has demonstrated by astronauts walking and running on the moon while loaded down by space suits weighing about 300 lbs on Earth (about 50 lbs on the moon). The astronaut David Scott also repeated Galileo's Pisa experiment using an hammer and a falcon feather (instead of different sized balls), which in the near vacuum found on the Moon, fell together to the surface of the moon. A weighing scale will still work on the Moon, if you paste a new scale on its surface, which is recalibrated to about one sixth what it would be on the surface of the Earth. On Mars's surface, gravity is about 38% that of Earth, so you would have to have still another scale for Mars.

Determining mass in a variety of environments

A more universal way of measuring mass of objects than by their weight is to employ a pan balance, which uses Archimedes' s Law of the Lever. Objects of equal weight suspended on pans of equal weight at equal distances opposite a balance point (fulcrum) on a uniform bar, will hang in equilibrium. If a little bit more weight is added to one side, that pan will dip down. As in all mechanical devices, there is a limit of precision where effects are not noticeable. A piece of lint may be inadequate to do this, but a small grain of sand may do the trick. In the case of a pan balance, known masses are stacked on one pan which is counterpoised to an object to be weighed on the other side. Through a combination of known masses large and small the user achieves a balance, and thereby determines the mass of the unknown object.
When you are in a weightless environment you cannot weigh something because there is no gravity pulling on it; however, it is still possible to determine its mass by measuring the acceleration that a known force gives it. The magnitude of the force used to accelerate it, divided by its acceleration, will then give you the object's mass.
Newtonian mechanics presents force as the agent of interaction between two things. Here is a simple thought experiment that an astronaut could do while floating weightlessly in space. Imagine that he brings two balls of unequal mass together in such a way that he compresses spring between them, which is oriented along a line between their centers. At first, everything is at rest in the astronaut's frame of reference. Then he lets go of both balls at the same time. He will then see the balls will fly apart in opposite directions along the same line as the spring alignment, but at different speeds. The ball with the least mass will be going the fastest and the one with the most mass, the slowest. During their acceleration, the force propelling each ball was of the same magnitude, but opposite in direction; yet after they separate from the spring, each ball is moving at a different speed. While the balls were interacting through the spring, they were accelerating, each in inverse proportion to its mass. The accumulated speed in each of them during their interaction was therefore inversely proportional to its mass.

Listing 1

#Gravity Python Code-----------------------------------------------------------------------------------

ipython --pylab
G= 6.67300e-11 # m3 kg-1 s-2 from Google
re=6378100 # Earth radius in meters from Google
Me= 5.97219e24 # Earth mass in kilograms from Google
print G*Me/(re*re)
# ans.:  9.796525936 m/s/s

Mm = 7.34767309e22   #Moon mass in kilograms
rm = 1737400               #Moon radius in meters
print G*Mm/(rm*rm)

# ans.:    1.62431896902 m/s (actually only 1.624319 is justified by the above numbers).

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