EncycloZine

* The most fundamental concepts in physics were derived by the great English scientist Isaac Newton in the 17th century, in the form of what are now known as Newton's three laws of motion. These three simple principles underlie all modern physics. This chapter provides an introduction to these laws.

[1.1] COORDINATE SYSTEMS, DISPLACEMENT, VELOCITY, & ACCELERATION

* The science of physics concerns the understanding of the basic rules by which our physical reality operates. This makes descriptions in physics a little difficult, since it deals with things that are so fundamental that they are hard to describe in any other terms.

The first concept in physics is that of "space", the domain in which objects exist and move. In physics, space is typically defined in "Cartesian coordinates", a framework set up to measure the location of an object along X, Y, and Z coordinates. The point where these coordinate axes come together, the "origin", is generally defined to be convenient to the problem at hand. In the metric system, spatial dimensions are generally described in meters, though for very large distances light-years are often used instead.

Let's consider a simple example, starring an adolescent mad scientist named Dexter. If Dexter's in a building, his location could be described using the first-floor entrance as an origin. He could be 19 meters beyond the entrance, 11 meters to the left of the entrance, and (assuming the building has multiple floors), 28 meters above the entrance. Dexter's location could then be given as:

    ( X, Y, Z )  =  ( 19, 11, 28 )
 

In the building example, the magnitude is the straight-line distance from the entrance to Dexter's location. The magnitude can be given by the Pythagorean theorem as:

    SQRT( X^2  + Y^2  + Z^2 )  =  SQRT( 19^2 + 11^2 + 28^2)  =  35.6 meters
 

    ( X, Y )  =  ( 19, 11 )
 

    SQRT( X^2 + Y^2 )  =  SQRT( 19^2 + 11^2 )  =  22.0 meters
 

In some cases, it is more convenient still to describe a physics problem in one dimension, for example the interactions of ball bearings colliding along a rail. If Dexter were in a hallway, his location could be given merely by a single X value, say of 19 meters. You could think of this as a vector, with the direction being only positive or negative.

It is also useful in some circumstances to consider three-dimensional physics in terms of "polar coordinates", in which the location of an object is defined by a distance from the origin of the coordinate system, along with an angle defining "azimuth" (rotation in the horizontal plane) and an angle defining "ascension" or "elevation" (rotation in the vertical plane). In two dimensions, this reduces to a length and an angle.

A full description of vector math is beyond the scope of this document. The point here is simply to introduce the concept.

The concept of location is closely related to another concept in physics, known as "displacement". This is just the vector that describes the change in location of a moving object. It doesn't say anything about how it got from place to place or how fast it did it, it just gives the difference in location.

* Given a coordinate system that defines the location or displacement of objects, we can then discuss their movement in that coordinate system, and the changes in their movement, such as speeding up, slowing down, or changing direction. The movement of objects is known as "velocity", or displacement per time. Changes in that movement are known as "acceleration", or changes in velocity per time.

In the metric system, velocity is defined in terms of "meters per second", while acceleration is given in "meters per second squared". These are also vector quantities.

For example, an object falling to the ground on the Earth accelerates at a constant rate of 9.81 meters per second. The vector direction is straight down, so we don't need to worry about vector math in this simple case. In mathematical terms:

    acceleration = 9.81
 

    velocity = acceleration * time = 9.81 * time
 

Given a constant velocity, then the displacement is the velocity times the time. This is simple to see: if an object's moving at 10 meters a second, in five seconds it covers 50 meters.

As the graph above shows, a falling object doesn't have a constant velocity, but under a constant acceleration the velocity increases in a nice straight line, and for that case we can use the average velocity times the time to give the displacement. For a straight line beginning at zero, the average velocity at any time is half the velocity at that time.

So, for constant acceleration, the displacement is:

    displacement = (velocity / 2) * time
                 = (acceleration * time / 2) * time
 	= (1/2) * acceleration * time^2
 	= (1/2) * 9.81 * time^2
 

As anyone familiar with the Coyote knows, he's incredibly unlucky, and the Roadrunner simply eats the birdseed and runs off without tripping the trigger mechanism. Of course, the Coyote, again to no surprise, just has to go fiddle with the trigger mechanism, and drops the safe on his own head.

At the risk of being cold-hearted to his distress, let's consider how long it took the safe to fall 50 meters, and how fast it was moving when it hit the poor guy. We know that:

    displacement = (1/2) * 9.81 * time^2
 

    time^2 = 2 * displacement / 9.81
 

    time = SQRT( 2 * displacement / 9.81 )
         = SQRT( 2 * 50 / 9.81 )
 = 3.19 seconds
 

Since velocity is just acceleration times the time, then the terminal velocity of the safe is:

    velocity = 9.81 * 3.19
             = 31.3 meters per second
 

* Now that we've defined basic concepts of space and motion, we can move on to the fundamental laws that govern the motion of bodies in that space.

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[1.2] GALILEAN RELATIVITY / NEWTON'S THREE LAWS OF MOTION

* One of the first questions addressed in the development of physics was: Why does some object keep moving even after the force applied to it goes away? If a cannonball is blasted out of the mouth of a cannon, it flies through the air in a smooth curve until it strikes something or is pulled down to the ground by gravity.

The answer to this question was finally unraveled by the 16th-century Italian scientist, Galileo Galilei. Galileo saw through the question by seeing that the concept of motion was strictly relative: what constituted "rest" and what constituted "motion" depended on the observer's own state of motion.

This sounds like a deep and exotic concept, but the idea is very simple. Suppose Dexter's on a ship, inside a closed cabin, sailing on a glassy smooth sea. If he doesn't look outside, there would be no way for him to know if he was moving or not. The physics of everything in the room would be exactly the same while the ship was in uniform motion as it would if the ship was sitting at the dock. This is now broadly known as the "equivalence principle".

The idea that there was an absolute state of rest was a natural mistake, since we live on a large ball of rock called the Earth that is a lot bigger than we are, or anything that we can set into motion. This means that the the Earth seems very convincingly at rest, and anything that moves over relative to it seems very convincingly in motion.

However, later generations were to realize that the Earth is itself in motion around the Sun, and the Sun is in motion around the center of our Galaxy. Such cosmic concepts of motion are nonetheless hard to grasp intuitively, and so in practice we invariably think of something sitting motionless on the ground as being "at rest" while a car driving over the ground is "in motion". In the terms of modern physics, the Earth is our default "inertial frame of reference" for motion.

It would be silly to think otherwise. It is also useful for someone interested in physics to keep in the back of his or her mind that this is a parochial point of view, and in other inertial frames of reference it is the Earth that is in motion.

* Galileo's consideration of the meaning of rest and the meaning of motion sounds a little like a word game, but it did have practical implications.

The main implication was that motion itself was less important than changes in motion, or accelerations. A force causes a change in motion of an object, and once set into motion that object will remain on that path, until another force brings it to a halt.

After Galileo, this idea was formalized by the 17th-century English genius Sir Isaac Newton in his three laws of motion:

• First Law: Objects at rest remain at rest, and objects in motion remain in motion in a straight line at constant velocity. A force must be applied to change the state of motion of an object.

• Second Law: The acceleration of an object is directly proportional to the force acting on it, and inversely proportional to its mass.

• Third Law: If two objects interact, say in a collision, the force exerted by object 1 on object 2 is matched by a force of exerted by object 2 on object 1 of the same size, but in the opposite direction.

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[1.3] THE FIRST & SECOND LAW / WORK & ENERGY

* The First Law observes that objects in motion tend to remain in motion in a straight line. The magnitude of this tendency is given by a property called "momentum", which is unsurprisingly related to the bulk or "mass" of an object, and the velocity with which it is moving. In mathematical terms, the First Law is expressed as:

    momentum  =  mass * velocity.
 

The Second Law defines force and mass. In mathematical terms, the Second Law is expressed:

    force = mass * acceleration
 

* Incidentally, mass and weight are not the same thing. A mass remains constant no matter where it is in the Universe. Weight is just a measure of the force of gravity on a mass. If Dexter goes to the Moon, he would only weigh a sixth of what he weighs on Earth, but his mass would be the same.

Modern physics recognizes that the "matter" in the Universe that constitutes its mass is made up of vast numbers of very small particles, known as "atoms", that under normal circumstances cannot be subdivided further. There are a roughly a hundred different types of atoms that have different properties. Atoms can combine to form a wide range of "molecules" that have different properties from the atoms that make them up.

Atoms and molecules can be organized as solids, liquids, or gases. While we live on Earth completely surrounded by matter, it is actually relatively scarce in the Universe. Once we travel away from the Earth into space, the amount of matter in a volume of space becomes effectively negligible. Space empty of appreciable amounts of matter is known as a "vacuum", and most of the Universe is a vacuum.

* An example can help demonstrate what the First and Second Laws mean in practice. Suppose Dexter fires a cannonball out of a cannon. It is put into motion by the force of the explosion. If no other forces were to act on it, it would continue to go on forever in a straight line. This is referred to as "inertia".

However, the Earth's force of gravity is acting on it, and so it drops as it flies forward until it hits the ground. The force of gravity is a vector that pulls straight down while (in this example) the velocity of the cannonball is a vector pointed flat over the ground. The gravitational force doesn't change the horizontal velocity at all, it just adds a vertical velocity to it as well.

What this means that if the range of the shot is short, then if Dexter fires a cannonball on a flat trajectory out of a cannon and drops a cannonball at the same time, both cannonballs hit the ground at the same time. It falls just as fast if it is moving as it does if it dropped straight down.

If the range of the shot is long, however, it flies over the horizon. The Earth curves out from underneath it as it flies, making the distance it has to drop to hit ground longer. In fact, if Dexter fired the cannonball fast enough, the Earth would curve out underneath it so quickly that the cannonball would never hit ground, and the cannonball would orbit the Earth.

This is assuming that there is no atmosphere. The atmosphere causes resistance to the flight of the cannonball that increases with its speed and operates against the direction of its flight, slowing it down and causing it to hit the ground in time.

* A related scenario is what happens if Dexter drops an object while he's running. There is a popular misconception that the object will fall straight down to the ground at the point where he dropped it, but the First Law shows that's not the case. Ignoring drag, which is a reasonable assumption if the item is compact like a rock, it moves alongside him until it hits the ground.

This is a fact of life for aircraft dropping bombs at low altitude. The bombs, which are generally streamlined to reduce drag and allow the aircraft carrying them under its wings to fly faster, will fall forward and remain under the aircraft, which will be caught in the blast and destroyed. This is why bombs configured to be dropped at low altitude pop out small parachutes or other types of "retarders" to slow them down through drag, so the aircraft can escape.

* Closely related to the concept of force is the dual concept of "work" and "energy". Work is defined as the sum of forces applied to a mass over a distance. Energy is the capacity to do work.

For one simple example, imagine that Dexter lifts a mass against the Earth's gravity. In his immediate surroundings, the force of the Earth's gravity is for all practical purposes constant. Actually, gravity grows weaker the farther an object is away from the Earth, but for such short distances the force of gravity varies too little to notice.

Since gravity is exerting a force on the mass, he has to exert a force on the mass to lift it against gravity. If he lifts the mass to, say, three meters, the total work performed on the mass is the force times three meters. How long it takes him to lift the mass doesn't matter, nor do any side trips taken on the way up. All that is important is that it is lifted three meters.

If the mass is raised to that height, it then has a capability to do work itself by falling back down again. This is the "potential energy" of the mass, and is the same as the amount of work required to raise it that height. The potential energy of a mass in a constant is given by:

    potential_energy = G_constant * mass * height
 

    potential_energy  =  9.81 * 10 * 3  =  294 joules
 

    power  =  294 / 5  =  58.9 watts
 

    energy  =  power * time  =  5 * 58.9  =  294 joules
 

* Moving objects also store energy due to their motion. A flying brick, for example, has the capability of doing destructive work. This form of energy is known as "kinetic energy".

The kinetic energy is the sum of all the force provided over a distance needed to get the brick to a specific velocity. Consider a simple example, where a mass is accelerated from rest by a constant acceleration for a specific time interval.

As shown earlier, for a constant acceleration, the average velocity is half the final velocity V, so the total distance D traveled in the given time T is:

    D = (1/2) * V * T
 

    force = mass * acceleration = M * A
 

    energy = force * distance
 

    energy = M * A * D
           = M * A * (1/2) * V * T
 

    energy = M * (V/T) * (1/2) * V * T
           = (1/2) * M * V^2
 

    kinetic_energy  =  (1/2) * mass * velocity^2
 

By the way, since momentum is given as mass times velocity, this expression can be redefined as:

    kinetic_energy  =  (1/2) * momentum * velocity
 

     kinetic_energy  =  (1/2) * 1,000 * 27.8^2  =  385,800 joules
 

    velocity  =  SQRT( 2 * kinetic_energy / mass )
              =  SQRT( 2 * 385,800 / 0.01 )  =  8,780 meters per second
              =  31,600 KPH
 

When Dexter pulls the trigger of the dart gun, the spring returns to its normal length, exerting force on the dart to accelerate it out of the muzzle, and converting the potential energy into motion, or equivalently kinetic energy.

This example demonstrates another fundamental physical principle: energy is conserved, meaning that is cannot be either really created nor destroyed, only converted from one form to another. In the earlier example of the cannonball in this section, its kinetic energy is converted to heat due to drag, and to heat from its final impact into the ground.

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[1.4] THE THIRD LAW / MOMENTUM IN ACTION

* The Third Law describes, among other things, action and reaction. If Dexter fires a gun, the bullet flies out the barrel and the gun kicks back in the opposite direction.

Both the bullet and the gun have a certain value of momentum, and in fact it's the same value, though the directions are opposite:

    bullet_mass * bullet_velocity  =  gun_mass * gun_velocity
 

This equivalence is another way of saying that the momentum of a system is a conserved quantity. This is seen in collisions between moving masses, usually spheres. Such collisions may be "elastic" or "inelastic".

In simple discussions of momentum, it is easiest to specify it in terms of one-dimensional movements along a straight line in one direction or the other. We can consider one-dimensional examples of elastic and inelastic collisions.

* In an elastic collision, the kinetic energy of the masses is preserved. This is the case if the masses are very hard and stiff. In a head-on elastic collision between two spheres of the same size and same velocity in different directions, the two spheres bounce away from each other at their original velocity but with reversed directions, and the total momentum is conserved.

If the masses or velocities are not the same in an elastic collision, the results are little more interesting. Suppose one mass is moving twice as fast as a second mass of the same size, and so has twice the momentum. If there is an elastic head-on collision between the two, they bounce away from each other, but the second mass is now moving twice as fast as the first. This occurs because momentum is a vector quantity, and has to be conserved in direction as well as size.

Now suppose there are two masses that are moving at the same speed, but the first mass is twice as big as the second and so has twice as much momentum. If there is an elastic head-on collision between the two, they bounce away from each other, but the conservation of momentum says that the first mass will bounce away at only half its initial speed, while the second bounces away at twice its initial speed.

* In an inelastic collision, the kinetic energy of the masses is converted into heat. This happens if the masses are soft and malleable. In a head-on inelastic collision between two spheres of the same mass and same velocity in different directions, the two spheres slap into each other and come to a dead stop. Momentum is conserved once more.

Of course, if the momentum of one mass in such an inelastic collision is greater than the second, its momentum will overwhelm that of the second, and the combined mass that results from the collision will move in the direction of the first.

Suppose the first mass is twice the size of the second, but both are moving at the same velocity. Then the total momentum of the resulting particle has the same size but opposite direction as the momentum of the second particle. Since the mass of the resulting particle is three times the size of the second particle, the velocity is cut to one-third of the original.

Similarly, if the velocity of the first particle is twice that of the second, the resulting momentum is still the same. However, the resulting particle is twice the mass of the second, so the velocity is only cut in half.

* By the way, the law of reaction is very well illustrated by rocket propulsion. There is a strange popular misconception that a rocket operates by "pushing against the air", but that is not the case.

What a rocket does is throw hot gas out the exhaust at high velocity, and by conservation of momentum the rocket moves in the other direction. The air just gets in the way and causes drag. The more mass in the exhaust and the faster it is sent out, the more thrust it creates and the faster it propels the rocket. Essentially, a rocket propels itself using recoil.

A more ordinary way to think about this is to consider what happens if if Dexter and his sister Didi are are standing on slick ice next to each other, and then push each other away. Dexter don't just stand still while Didi slides away, they both slide away from the point they were at. If they're both about the same mass, they slide away in opposite directions at the same speed. If Dexter weighs only half as much as Didi, he slides away at twice the speed of Didi.

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[1.5] ELEMENTARY MECHANICS AND MACHINES / DRAG & FRICTION

* We now know enough to consider the operation of simple machines, such as levers, ramps, gears, pulleys, screw jacks, and so on. An understanding of these devices is based on elementary concepts of force, work, and power.

A lever is one of the simplest tools. Suppose Dexter wants to lift a weight of using a long stout stick as a lever. He props the stick on top of a rock for support so that three-quarters of the stick is on his side of the rock, while the other quarter is on the other side of the rock and wedged under the weight. The rock is said to act as a "fulcrum", by the way.

If he pulls down on the lever, his end of the stick moves three times as far down as the other end of the stick under the weight moves up. The total work done on both ends of the stick is the same, but since Dexter's end of the stick moves three times farther, he only ends up applying a third of the force. This means Dexter has a "mechanical advantage" of three to one.

To balance a lever, the product of the mass and length of the lever on one side of the fulcrum must equal the product of the mass and length of the lever on the other side of the fulcrum. In this example, if the lever is a meter long and the fulcrum is 25 centimeters from one end, then 1 kilogram at the long end of the fulcrum balances 3 kilograms at the short end. If the fulcrum is 20 centimeters from one end, then 1 kilogram will balance 4 kilograms. This issue of balance comes up in other domains of physics.

* A ramp is just as simple. Suppose Dexter wants to lift a weight one meter into the back of a truck. Lifting it straight up is a good way to hurt himself, so he finds a stout plank three meters long and rolls the weight up the plank on a dolly. Once more he only does a third of the work for three times the distance, and has a mechanical advantage of three to one.

The same principle applies to gears. Two gears meshing together are basically just a set of wheels moving against each other, with the teeth ensuring they don't slip. If the driving gear is only a third as big in diameter as the gear it is driving, then the little driving gear performs three rotations for every rotation of the big driven gear. This means that the force exerted to spin the little driving gear is only a third as much as the force output by the big driven gear, and once more we get a mechanical advantage of three to one.

A pulley involves a rope looped around a pair of wheels. Pulling on the rope out of one end of the loop makes the loop shorter, but as the loop has two sides, it gets shorter at only half the rate the rope is pulled out of the loop. This means a mechanical advantage of two to one. If Dexter makes a pulley with two loops, he gets a mechanical advantage of four to one; if he makes it with three loops, and he gets a mechanical advantage of six to one.

A screw jack is simply a ramp wrapped around an axle, with the axle rotated by a handle used as a lever. If the circular motion of the handle is 30 centimeters per revolution, for example, and one turn of the screw lifts a weight by one centimeter, then the mechanical advantage is 30 to 1.

My little Toyota car has a scissors-type screw jack, in which rotating the handle turns a horizontal screw that raises a mechanical scissors arrangement. Obviously a detailed analysis of this device is a little more complicated, but still, if one rotation of the handle travels 30 centimeters to raise the jack one centimeter, I have a mechanical advantage of 30 to 1.

* The examples used in this document have generally ignored friction, which makes the math much simpler, but in the real world friction can't be ignored. The friction of an object falling through the air, or its drag, is not significant for a compact object falling a short distance, but if the object is falling a long distance that drag ultimately limits how fast the object falls to a "terminal velocity".

The actual terminal velocity depends on the smoothness and shape. A streamlined object has less drag than a blunt one, and a smooth, polished surface has less drag than a rough one.

This is because the air consists of a mix of various kinds of molecules. On the average, except for the occasional breeze or wind, these molecules are not moving at any appreciable velocity. When a moving object hits the molecules in its path, it transfers momentum to them that causes them to bounce away.

Suppose a cylinder is shot out of a cannon, end first. When the flat end surface collides with the molecules, they bounce away in the direction of the cylinder's motion, resulting in a maximum change in their momentum. Since force is a change in momentum, this means a maximum force to slow down the cylinder.

This also implies that the force slowing down the cylinder is proportional to its frontal cross-sectional area, since that increases the number of molecular collisions. A long slender cylinder fired out of a cannon has less drag than a short fat cylinder of the same mass.

Now suppose the cylinder is machined to look like a dart with a sharp point and tapered sides. As it moves through the molecules, the tapered sides tend to cause the molecules to bounce off to the sides. Relatively little momentum is transferred in the forward direction to the molecules, while the momentum transferred into the cylinder by the sideways motion of the molecules is, on the average, the same on all sides and balances out. The result is that the dart has much less drag than the cylinder.

A rough surface similarly has faces at a microscopic level that can cause the molecules to bounce forward. Smoothing and polishing the surface of the dart reduces these losses.

These same principles apply to falling objects. There are actually a number of other, subtler considerations in designing an object to reduce drag, but they are well beyond the scope of this discussion.

Since momentum is directly proportional to velocity, no matter how smooth or streamlined an object is, drag forces increases linearly with speed. Double the speed, and you double the force of drag. At some speed, the force of drag matches the force of gravity. There is no longer a net force on the object, and the object cannot accelerate to a higher speed. It has reached its "terminal velocity".

Of course, this does not apply to objects falling in a vacuum. I emphasize this because we live at the bottom of an atmosphere, and it affects our preconceptions on how things fall. If the Earth had no atmosphere, a gram of feathers would fall at the same rate as a gram of steel. That's not too hard to swallow, but a gram of feathers would also fall at the same rate as a tonne of feathers, or a tonne of steel.

In fact, everything would fall to earth at the same rate, increasing in speed by 9.81 meters per second squared, no matter how heavy it was or what its shape was. Heavy objects do not fall faster than light ones in a vacuum. It is drag that makes this simple fact somewhat counterintuitive.

* Similarly, if Dexter tries to shove an object sitting on the ground, he has to deal with a force of friction proportional to the object's weight and to the roughness between the two surfaces. Interestingly, though it's obvious from experience once pointed out, the force of friction is greater for an immobile object than for one that's moving. It takes more effort to get something moving than it does to keep it moving. The friction force that resists putting an object in motion is known as "static friction", and the lower friction force that resists keeping an object in motion is known as "sliding friction".

Of course, the friction force can be reduced by making the sliding surface smoother, for example by using a lubricant. Using rollers or a dolly also greatly reduces the force of friction in moving the object, though there are still various friction forces at work, for example due to the weight of the object on the axles of the dolly.

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[1.6] A BROADER VIEW: FORCES AND FIELDS

* We have defined force, and discussed gravitational forces a little. Gravity is actually only one of four fundamental forces in the Universe:

• gravitational
• electromagnetic
• strong nuclear
• weak nuclear

All other forces are just indirect manifestations of these four basic forces. The only ones we actually deal with directly are the gravitational and electromagnetic forces. The strong nuclear force holds the cores of atoms together, while the weak force only shows up in certain processes of atomic disintegration, and neither will be mentioned further in this document.

The gravitational force is the most familiar of these four forces, since we deal with it all the time. By the way, gravity is the weakest of all the four forces, and is about forty orders of magnitude (one followed by 40 zeroes) times weaker than the electromagnetic force. To visualize just how weak a force gravity is, consider that when you pick up a cup of coffee, you are overcoming the gravitational attraction of the entire Earth on that cup.

However, gravity is the only one that is effective over long distances, and so predominates over the entire Universe.

Gravity is an attractive force between two or more masses, such as the Earth and the Moon. The size of this force is proportional to the size of the two masses, and inversely proportional to the square of the distance between them:

    gravitational_force  =  constant * mass1 * mass2 / distance^2
 

If Dexter travels to other planets, he will experience different gravitational forces and have a different weight. Since the vast majority of us never get very far off the ground, much less travel to other planets, in our practical experience the force of gravity is a constant, with an acceleration of 9.81 meters per second, as discussed earlier.

* Physicists use the concept of a "field" to describe forces like gravity. This can be considered as a three-dimensional map of the direction and magnitude of the force at any point in space.

Two-dimensional representations of fields usually use lines to map such fields, with the density of the lines giving the strength of the field and arrows attached to the lines to give their directions. Leveraging off this abstraction, physicists often refer to "lines of force" or "lines of flux" in descriptions of fields and field interactions, though in classical physics the fields are actually continuous and the lines are just a convenient abstraction.

In the special case of living on the surface of the Earth, the gravitational field is uniform. No matter where we place a mass in the three dimensions of our immediate surroundings, the direction of the force is straight down and the magnitude is the same. Potential energy in this uniform field, as mentioned in the previous section, is directly proportional to the height of a mass.

In most analyses of gravitational forces in the greater universe, the gravitational field is regarded as acting between simple points of matter that have no extent. This is actually a very good approximation for nearly all practical purposes.

Visualizing this from a field perspective also helps explain why gravity obeys an "inverse square" law, decreasing in strength with the square of the distance. The gravitational force of the Earth can be visualized as lines of force radiating with uniform density in all directions, with the direction of the lines pointing into the Earth.

Now let's draw an invisible shell around the Earth at some radius. All the lines of flux pass through this shell. Since the surface area of a sphere is 4 * PI * radius^2, then if we draw a second shell at double the radius, the surface area of this shell is four times that of the first. The same number of lines of flux pass through this outer shell as they do the inner shell, but the surface area has been squared, meaning the density and so the strength of the gravitational field has been attenuated by the same factor.

* Electromagnetic forces have similarities and differences to gravitational forces. Gravitational forces occur between masses. Mass is irrelevant to electrical forces, they only rely on a property known as the "electrical charge" the objects involved.

"Charge" is a fundamental concept that can only be explained in terms of itself. An electric field works on charges. Charges are what an electrical field works on. However, charges are tangible, and can be measured. Physics defines a certain fundamental unit of charge that cannot be subdivided further. This charge is very small and is usually measured in terms of 6,240,000,000,000,000,000 of these basic charges. This amount is known as a "coulomb".

The basic law of electrical force is very similar to that of gravity. The electrical force is proportional to the product of the charges of each object and inversely proportional to the square of the distance between them:

    electrical_force  =  constant * charge1 * charge2 / distance^2
 

One major difference between gravitational forces and electrical forces is that charges have polarity: they can be positive or negative. The force between unlike charges is attractive, pulling the charges together. The force between like charges is repulsive, driving them apart. In contrast, as far as anyone's ever seen, gravity is always attractive.

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[1.7] FOOTNOTE: EXPONENTIAL NOTATION & METRIC PREFIXES

* This chapter has introduced basic metric units of physical measurement, such as meters, joules, watts, and so on. The following chapters will expand on this system of measurements, and to make proper use of it you will need to understand two things: "exponential notation" and "metric prefixes".

Physics deals with quantities that may vary over a wide range, leading to very large or very small values, for example:

    1,470,000,000,000
    0.000000000000592
 

    1,470,000,000,000  =  1.47 * 1,000,000,000,000 
                       =  1.47 * 10^12
 	      =  1.47E12
 
    0.000000000000592  =  5.92 * 0.000000000001
                       =  5.92 * 10^-12
 	      =  5.92E-12
 

    1E-6  =  10^-6   =          0.000001
    1E-5  =  10^-5   =          0.00001
    1E-4  =  10^-4   =          0.0001
    1E-3  =  10^-3   =          0.001
    1E-2  =  10^-2   =          0.01
    1E-1  =  10^-1   =          0.1
    1E0   =  10^0    =          1 (by convention)
    1E1   =  10^1    =         10
    1E2   =  10^2    =        100
    1E3   =  10^3    =      1,000
    1E4   =  10^4    =     10,000
    1E5   =  10^5    =    100,000
    1E6   =  10^6    =  1,000,000
 

Physicists go even farther in simplifying such matters of scale using metric prefixes that specify a multiplying factor for a particular unit of measurement. For example, a "millimeter" is a thousandth of a meter, a "centimeter" is a hundredth of a meter, and a "kilometer" is a thousand meters. The standard metric prefixes include:

    exo   (E)  =  1E18   = 10^18
    peta  (P)  =  1E15   = 10^15
    tera  (T)  =  1E12   = 10^12
    giga  (G)  =  1E9    = 10^9
    mega  (M)  =  1E6    = 10^6
    kilo  (k)  =  1E3    = 10^3
    hecto (h)  =  1E2    = 10^2
    deka  (d)  =  1E1    = 10
 
    centi (c)  =  1E-2   = 10^-2
    milli (m)  =  1E-3   = 10^-3
    micro (mu) =  1E-6   = 10^-6
    nano  (n)  =  1E-9   = 10^-9
    pico  (p)  =  1E-12  = 10^-12
    femto (f)  =  1E-15  = 10^-15
    atto  (a)  =  1E-18  = 10^-18
 

Please remember that physical calculations have to performed with consistent units. In the metric system, by default the basic units of calculations are meters, kilograms, and seconds. If units with different scales are used, such as kilometers, grams, or hours, the calculations must be rescaled accordingly. This can be a surprisingly confusing and troublesome issue.

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