# Three Newton's Laws

## Newton's three laws -- what do they mean?

Newton's laws provide a prescription for calculating the path of a body moving under the influence of external forces. In this knol, we explain the laws on the high school level, without the use of advanced mathematics and technical jargon. This presentation differs from most textbooks in focusing on the meaning of the physical concepts, rather than analyzing Newton's formulation. If you ever wondered what "action equals reaction" was supposed to mean, this knol is for you.

## Explaining the Explanation

Essential to understanding Newton's work is a grasp of the experimental basis behind his discoveries. The previous knol describes the sky as it was perceived before the year 1600. It discusses, with numerous animated illustrations, the apparent motion of the stars and planets, the controversy about "what is orbiting what," as well as the roles of latitude and longitude. It describes data collected by Tycho de Brahe and summarized by Kepler in his three laws. (Pink border that means icon is clickable, leading to the previous knol "The Sky Before the Telescope";

## Newton's World - The Sacred and the Secular

Theology and theories explaining experimental data were not clearly separated in Newton's time. The controversy which Galileo's discoveries stirred up with the Catholic church are but one small illustration. The transition from the medieval belief in crystaline spheres emitting celestial music to the post- Newtonian world of scientific astronomy, guided by his theories, was slow but unrelenting during the 14th to 17th centuries.

Newton was born in 1648, just a few months after Galileo Galieli died, and 126 years after Ferdinand Magellan's crew completed the first circumnavigation of the globe. As he was growing up , Newton knew that the Earth is round, that gravity, "pointing down", pulls things to the center of the Earth. He knew that Galileo had aimed his telescope at the sky and observed "four new stars" clearly orbiting Jupiter, not the Earth. These objects are now known as Jupiter's moons. Galileo had also observed the surface of our moon, and saw it was not unlike surface of the Earth. It was not a perfect sphere made of "heavenly etheral substance" but rather a stony desert covered with craters. The time was ripe to bridge the divide which in the past had separated the secular and the sacred spheres, and to show that same laws are valid everywhere " as on the Earth, so in the heavens".

While the telescope was in use in Newton's time, and Newton himself made important improvements in it, data on the motion of the planets had already been gathered and organized before the telescope. Roughly a half century before Newton was born, Johannes Kepler and Tycho de Brahe had observed both novas (new stars appearing in the heavens) and comets, which passed without hindrance through the hypothetical "crystal sphere" (1557). Their accomplishments and lives are discussed in more detail in "The Sky Before the Telescope." knol. Their work culminated in Kepler's laws, the understanding of which is essential for the next step, which was Newton's. Newton made a successful synthesis of what was then known and wrote an equation which was able to produce the orbits of the planets. Today, we call such equations "the equations of motion."

As in Dumas' "The Three Musketeers," there are actually four, not three, laws needed to explain the motion of planets. The "three laws" of textbook fame and the "law of universal gravity" have to work together to produce the equation of motion. On the other hand, the "First Law" is a consequence of the second. so we take one away and add another, we end up with three laws anyway. Before plunging into Newton's theoretical contributions, we need to look at another important area of exploration essential to understanding his work, the state of physical theory in his time.

If you already understand the basic concepts of static forces, you may skip the following section and go directly to the section on dynamic forces.

## Static Forces and the Equations for Equilibrium

The motion of the planets had been observed to be periodic: their movement repeats, without apparent changes. This contrasts with observations on the Earth, where a moving mass tends to slow down and eventually stop. When all motion has come to a stop, the system is said to be in equilibrium. These observed differences had led to the ancient belief that the laws governing the heavens were different that those on earth. Detailed new observations of the heavens and explorations of forces had begun to erode this belief, however.

The concept of force was less clearly defined in Newton's time than it is today. Scholars of the Middle Ages discussed "vis viva," a living force, which brings stationary objects into motion, and "vis morte," a dead force, which created tension, or pressure, but did not give birth to new motion. Humans had, of course, experienced both since the advent of the human race.

They recognized different static forces: the force needed to move an object or lift a weight, the force needed to buttress a cathedral wall or arch, and the force of springs. Boyle titled his 'gas law' article on gas resisting compression "The spring of air"
They also knew dynamic forces, the force of friction and the force needed to get objects to move. Newton's contemporaries Robert Hooke and Robert Boyle, who also belonged to that delightful club, the Royal Society, which met in London on every full moon, were interested in all kinds of forces. They eventually derived equations and explanations for determining the force of a spring when extended (Hooke's law) and Boyle's law for gases. Even ordinary folks knew about magnets, which exercised force on iron and other magnetic materials. They understood that the force got weaker when the distance between the magnet and the iron was increased. Static forces depend on the position of the objects, but not on their state of motion. The science dealing with equilibrium, statics or mechanostatics, was important for the building of structures, arches, and domes. Based more on experience and intuition in those days, rather than on more elaborate calculations of today, the theory and the equations needed for such calculations were nevertheless known.

In this applet, called a spring-pendulum, heavy bobs can be hung on elastic springs. The amount of friction can be adjusted. system which exhibits the two types of behavior we mentioned above, motion as seen in the sky and motion as seen on Earth. When the friction constant b is set to zero, the mass keeps moving in a wavy, periodic motion forever, like the planets. When you introduce friction , the amplitude decreases and after a few oscillations the motion stops and the system is in equilibrium. This second behavior illustrates the situation common on earth. If we know the parameters of the system, that is the strength of the spring and the weight of the bob, we can calculate its equilibrium position by solving a simple algebraic equation with one unknown. Of course, we can measure that position, too. Here we have both the theory and the experiment. Use the applet in lieu of building the experiment. Click on the applet and try different parameters for the system, different spring constants, different masses. Drag the bob away from the equilibrium point using the mouse, and then click start. Damped spring pendulum At this stage, we are not attempting to calculate the pattern of the bobbling motion which the mass makes before it settles into equilibrium. That motion is the dynamics of the system. However, the same spring and mass will always tend to the same equilibrium and this, the statics of the system, we can easily calculate. The equation to solve it in this case is the balance of two forces. The spring pulls the mass up; gravity pulls it down. An algebraic equation describing this simple pendulum looks like this:

```            K * y      +      m *g         = 0
force of spring + weight of mass m  = 0
```
There is one unknown, a number y, the extension of the spring. You may call this balance "action equals reaction," the forces being equal in magnitude, and opposite in sign

( F.spring = - F.weight).

However, it it is better to write this equation as "the forces are balanced", i.e. their sum is zero. ( F.spring + F.weight = 0 ):
This can also be described by sayin: Momentum is conserved, as we will see later and even more later.

#### At equilibrium, the sum of all forces acting on a mass is zero.

Vector algebra

Warning: Two paragraphs of math concepts ahead! We promised no higher math, but we must mention vectors. The motion of the mass in the spring-pendulum applet was just up and down in one dimension, along a vertical line. Position was described by one number (the elevation above the floor, or the extension of the spring) and so was velocity. In the applet at the left, motions are confined to a plane, or two dimensions. The position is given by two numbers, and forces have a direction. For our purposes, we can say that a vector is a pair of numbers. When we consider motion in three dimensional space, then positions, velocities, and forces are given by a triplet of numbers. The concept of vectors involves more than this, but considering a vector as n-tuple of numbers, known as the components of a vector, (a pair, a triplet,..) is sufficient here for our uses. One vector equation is actually three equations, one for each component. Think of it as a shorthand way of grouping together the numbers which have similar physical meanings.

### Three forces in an equlibrium

Equilibrium of Three Forces The static equilibrium applet at the right shows a more complex system, a system of three bobs, constrained by ropes and pulleys so as to move together in one plane. Playing with the applet will illustrate clearly what is explained in words below. The positions of the three masses considered as a unit are called the "configuration of the system." When we get serious about the study of any system, we need to select a "frame of reference," which is a way to assign numbers to different positions of masses, i.e. different configurations of the system. Here the configuration is determined by the position of the knot where three ropes are joined. The knot moves in the vertical plane and its position determines the heights or elevations of the three masses above a selected horizontal plane, e.g. the floor of room. The forces acting at the knot are "vectors," meaning they have a magnitude and a direction. These 2 components, magnitude and direction, are shown by the arrows. Note that a longer arrow means a greater magnitude.

The concept of vectors and their algebra is the subject for a separate knol, one on linear algebra and geometry. However, this applet has an option ("Parallelogram of forces") which shows how, at equilibrium, the sum of two forces is equal in size, and opposite in direction, to the third force.  As in the case of pendulum in the first applet, we can find the equilibrium point. (Note that we cannot as yet describe the process by which it settles to that equilibrium.) In still more complex systems, we may need several equations to determine the position of several masses in a three dimensional space. In such cases, we have several algebraic equations, for several unknowns. By solving the equations and finding those unknowns, we can determine the equilibrium configuration of the system.

#### The Catenary Curve

In even more complex cases, not even several algebraical equations are sufficient. A classical example is this question: "What is the shape of a rope suspended by its ends, as illustrated at the right?"

Here, the unknown is not a single number, but a curve. That curve, called a catenary , can be found as the solution of a different type equation, a differential equation". While Newton was looking for an equation which would have as its solution a curve, the orbit  of a planet, Gottfried Wilhelm Leibniz in France formulated an equation for catenary and developed differential calculus independetly of Newton, when solving that problem. . For for more on catenary click here .

It is not difficult to get an intuitive feel for differential calculus: Imagine the hanging rope replaced by a chain, made up of many links. You can see the question of the shape as a problem of the static equilibrium of many masses. There would be a large number of algebraic equations, which would have as their solution the elevations of the links. Here catenary is compared with parabola in red . The elevations of all the links define the curve. That curve, that catenary is shown here in blue, compared with a red parabola. (If you enter the search term "catenary' into a search engine, you will find other interesting properties it has.)

Thus we see that differential equations can be imagined as a large number of algebraic equations. While Leibniz and his colleagues to used the tools of differential calculus to find the static equilibrium of complex mechanical systems, Newton used these same mathematical methods to find the dynamics of a simple system, the motion of a single mass in a field of force.

## Dynamic Forces and The Equation of Motion

Our discussion so far has concentrated on static forces. The force of a spring depends only on the position of the spring, and remains constant as long as the spring is extended. This is a static force. Force of gravity, usually called weight, is also static; it depends on distance of the two masses, but not on their relative speed. Now we will add dynamic forces to the equation. Both ordinary folks and scholars were, of course, familiar with dynamic forces: Centrifugal force, the pull on a rope holding an mass as it is swung around in a circle, and the force of friction. Centrifugal force is a special instance of inertia. While inertia was known before, Newton gave to force of inertia exact form, making it a well-defined physical quantity.

We mentioned the force of friction when discussing the spring-pendulum applet. It is a dynamic force because it depends on velocity. Friction is zero when velocity is zero. In other words, when an object is not moving, there is no friction. How friction depends on velocity is a complex story. In some cases, friction is proportional to velocity. An example of this is an object pulled slowly through a fluid (Stokes law). In other cases it depends on velocity squared, or just the sign of the velocity. Fortunately for us in this knol, we do not need to consider friction in our discussion. Newton does not talk about frictional force in his three laws because he was thinking about planets, and friction was seen to be so very small that it could be ignored. Planets move through a vacuum. Nevertheless, they do slow down slightly over milennia. It is not a coincidence that moon is always facing the Earth with the same side; its rotation around its axis has slowed so that it now makes only one rotation per orbit. However, these are other stories. Newton focused on the other dynamic force, the force of inertia. Recall that weight is a force; it is the force of gravity acting on a mass. Weight is proportional to mass; doubling the mass doubles the weight. In a similar way, the force of inertia is proportional to the mass of a body. It also is proportional to the object's change in velocity, that is to its acceleration. Acceleration, in the mathematical sense, can increase or decrease the velocity; it can be positive or negative. Force is required to get an object going faster, to get it to slow down, to stop, or to change its direction. An object with greater mass requires more force to get it to perform these actions than does a object with a smaller mass. Doubling the mass requires doubling of the force to get comparable acceleration. If you encounter the textbook statement that "Force is mass multiplied by acceleration," remember that this refers only to this kind of force, the force needed to overcome inertia. There are many forces, as we have already discussed. The force of inertia is just one additional force, a dynamic force, which Newton added to the collection of known forces. It was known qualitatively before Newton. Newton gave it numerical values. (Since force, like velocity and acceleration, is a vector, it has both direction and magnitude, and so is described by several numbers.) We can calculate the motion of any moving body with this general rule: The sum of all forces (dynamic and static) acting on each mass is zero. Each moving mass in a system has its own equation of motion. This is not an algebraic equation, but a differential equation. For the simple spring-pendulum system discussed above, that equation looks like this:

```
K * y              - m * y''     = 0
The force of the spring -  inertia     = 0
```
Here y, the extension of the spring, describes the position of the mass. The symbol y'' describes the acceleration (the change of velocity with time). The symbol y', not used in this equation, would describe its velocity (the change of position with time).

Damped Spring Applet

Instead of focusing on the static condition of equilibrium, we are now interested in the motion of the suspended bob before it reaches equilibrium. Let us look at it in the simplest possible terms. The bob moves up and down with a certain velocity before coming to rest. The velocity is changing and the force of inertia is resisting that change. The sum of the force of inertia and the force of the spring equals zero. That equation can be solved, of course. The damped spring applet here illustrates the solution which these explanations and equations describe. When the frictional or damping parameter is zero, the solution is a periodic curve is called a sinusoid.

### The Motion of the Planets

Newton was interested in a slightly more complex motion than the oscillation of a pendulum, namely in the motion of a planet orbiting the sun. It was clear to Newton, as to many others before him, that the moon, the closest of all heavenly bodies, is orbiting the Earth. The Earth attracts the moon, as it does everything else, and the moon is kept in its circular orbit by centrifugal force. Here we have the dynamic balance of two forces, the force of gravity and centrifugal force, which is a special case of the force of inertia. [5-8] Newton was able to verify that Earth's gravity at the moon's orbit is weaker than on the surface of the Earth. That calculation is easy for a circular orbit, but the orbits of the planets were known not to be circular.

In this applet, you may select a date to see the position of planets, choose animation or you single step by << and >> buttons.
This is how planet orbits look from Earth. When observed from the 'firmamnet' (imaginary sphere to which the stars are atatched) Orbiys are ellipses, with the sun in one focus. When Halley and others approximated the orbits as circles with the sun at the center, they saw that centrifugal force was just right to balance gravity, provided that gravity was getting weaker with inverse square of their distance from the sun. The tantalizing problem of calculating that motion exactly was the problem Halley challenged Newton to solve. Newton described his solution in his Principia. These are the basic conceptual underpinnings of Classical aka Newtonian mechanics.

### Solving the equations of motions

You can use the two applets described below to see how Newton's equations are solved and obtain the trajectory of a moving body. You can now be a prime mover: Create a planet with a stroke of your mouse. The direction and speed of that stroke define the velocity of new planet; position and velocity determine the orbit. This applet demonstrates how an orbit is generated by a differential equation, once the position and velocity of an object is given. If the initial speed is too small, the planet will crush into the star; if it is too large, exeeding escape velocity, the planet will fly away, never to return.
Great advances in any field are usually prepared by decades of work by many others. That was the case with Newton's breakthrough as well. Kepler, for example, was thinking about gravity causing the curvature of the planetary orbits and wondered what law of gravity would be correct. Newton tried several ideas and found the one which worked. That is stated in his fourth law, usually called the "law of universal gravitation": Gravity gets weaker as the "inverse square" of the distance. It is illustrated here. Simply stated, if the distance between bodies is doubled, the attraction is reduced 4 times. For the more mathematically inclined: Mathematically stated, if the distance is r and gravity is g, then at distance 2*r gravity will be g/4. In more complex mathematical terms: Gravity changes as 1/(r*r). Or, r^(-2) = r to the power of -2. Here -2 is the exponent (- means inverse and 2 means square).

The inverse square law applies to the attenuation of many other physical quantities, such as electric charges and the intensity of light or sound.

You can effortlessly check Newton's calculations by means of this applet. Only the exponent -2 will produce ellipses.
In another visualization, the picture on the right shows how equations of motion are solved. The blue vector is velocity and red vector is the force of gravity, proportional to acceleration. When solving the equation of motion, the computer proceeds in small time steps. Acceleration is used to update velocity and velocity is used to update position. In each time step a new red vector is calculated using the law of gravity.

You now have an outline of the concepts and a taste of the mathematics in Newton's discoveries. What remains is to be aware of the different, and sometimes confusing, terminology some textbooks use and of the limitations of our new knowledge.

#### Newton's Three, plus One, Laws

We presented Newton's laws by explaining his thinking and his accomplishments, while avoiding attempts to analyze the words with which he tried to explain his discoveries. Newton wrote an equation for calculating the motion of an planet, inventing the mathematics for solving it and showed that the solution was an ellipse, with the sun in one focus. Wikipedia gives us a glimpse at what Newton himself wrote. You will find there the literal text of Newton's three laws in the original Latin and in an English translation. How does what Newton wrote and what textbooks paraphrase relate to what we have explained? We summarize this here, with the laws in reverse order, for ease in understanding them.
• His third law, usually stated as "action equals reaction," means this:

"The sum of all forces (static and dynamic) acting on a body is zero" This is the prescription for setting up an equation of motion. You may call any subset of those forces an "action" and the remainder of them a "reaction." That does not affect the calculation. In the case two forces are gravity and inertia, the case of planets, one can consider gravity and "action" and inertia a "reaction."

• Newton's second law describes the force of inertia in an inertial frame of reference. In the terminology we have used in this knol, the force of inertia is a dynamic force. Some textbooks use the term "balanced forces" in the case of static equilibrium. When such texts then talk about an object accelerating, they say the forces are "unbalanced."

We prefer to consider dynamic forces, the force of inertia and the force of friction, as forces acting together with static forces. Thus the equation of motion still describes a "balance of forces." The force of inertia is in balance with (or is a reaction to) the net sum of all static forces.

• The Newton's first law is a corollary of his second law. When all other forces are made small, as when we are so far away from the stars that inertia is the only force to consider, then the equation of motion is simply m * y'' = 0 . The solution of this equation is uniform motion, i.e. motion along a straight line, with constant velocity.

Some textbooks say that Newton's first law explains that Newton's laws are only valid in inertial frames. That limitation in his laws does exist, but it deserves a more detailed, explicit explanation, which we attempt in the next section.

• Newton's law of gravity is necessary to explain the observed orbits of the planets. This is the "inverse square" law, explained in the previous section and applet. The law of universal gravity is relevant for motion of planets, which were the experimental basis of his discovery. Note that his third law is more universal and applies to all forces, which may include gravity but do not have to do so.

#### Inertial Systems and the Limitations of Newton's Mechanics

The ideal which Einstein emphasized is to formulate the laws of physics so that they are valid in all frames of reference, moving or not. When laws are the same in all frames, they are said to be "invariant" (unchanging). We can then do our calculations in any frame of reference. We will get different curves in different frames of reference, different orbits, but when we transform them by the rules of geometry, they will all agree. It is still important to choose the frame of reference carefully, since the calculations and the results are simpler in some frames.

Newton's equations, as he formulated them, are not invariant. They are valid only in special frames of reference called inertial frames. In an inertial system, inertia obeys the second of Newton's laws. In non-inertial systems it does not. Newton and his contemporaries considered a frame fixed to the stars as representing absolute space. As we see things now, such a frame is just one inertial system. All frames which are moving uniformly with respect to the stars are "inertial systems." We can use Newton's laws in any of them. For example, we know that in the frame of reference attached to the stars, the orbits of the planets are simple; they are ellipses. Those simple orbits, ellipses, are obtained by Newton's laws in any inertial system. A simple example of a non-inertial frame is your own experience on a bus which suddenly changes direction. You are pulled to one side. It is inertia of course. However, in the frame of reference attached to the bus, this force does not follow the second law. Instead of leaving you at rest, inertia pushes you around. For many practical problems we can treat the Earth as an inertial system. In the experiment described by the spring-pendulum applet, we used a frame of reference attached to the floor, that is to the Earth. The fact that Earth rotates with respect to stars does not change the results of most Earthly experiments appreciably. It is observable in some experiments which last a long time and involve rotating masses. For example, hurricanes, large rotating masses persisting for days, are affected by the rotation of the Earth.

A second example which shows that Earth is not exactly an inertial system is a modern device invented a century after Newton, the gyroscope. It is used for navigation, in the space shuttle as well as on planes and ships. In an inertial system, the axis of its rotating disc always points in the same direction. However, in a rotating frame, that axis rotates. It obviously cannot remain stationary in both. It rotates in a frame attached to the Earth. Similar effect due to rotation of Earth is Coriolis effect.

The challenging question, "With which stars does the the axis of a gyroscope remain aligned?" has been asked and answer is complex and not quite final. For our purposes, we consider them to be the stars that we can see, that is, the stars of our galaxy. In situations of high velocities or of gravitational fields of high intensity, Newton's theory does not work. Einstein's Theory of General Relativity must be used in those situations. Newton's mechanics, today called Classical Mechanics, contains both dynamics and statics. His concepts ruled physics up until the 20th century. They were generalized further and are part of the disciplines of  "modern physics," which now includes Quantum Mechanics and Einstein's Theory of Relativity. Fundamental aspects of these theories are still based on principles first formulated by Newton in his Principia. The Theory of relativity is explained in the follow up knols called Relativity triptych.

## References

<=== Principia full text
Halley demonstrated univeral validity of Newton's new theory by applying it to a comet...
• Text of the laws (Latin and English) in wikipedia. It follows most textbooks in saying that first law defines inertial frames. Wikipedia also assumes knowledge differential calculus. It thus combines both the complications which we intentionally avoided. In texts put together by committees such accumulations of complications are not unusual. Wikipedia article nevertheless is a useful complement to this exposition.

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