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Chapter 2

Newtonian Physics

2.1Observation of the night sky

The earliest astronomers spent several hours night after night observing the objects in the sky. They were patient and diligent, and made careful notes of their observations. They were true scientists in the modern sense of the term. This means they relied only on their observation in order to draw conclusions regarding the reality of objects in the visible universe, rather than the authority of priests or sacred books.

They observed that the sky was filled with different luminous objects that rose in the east and traveled slowly towards the west before disappearing below the horizon. These celestial objects included the sun, the moon, the stars, and a few star-like spots of light of differing brightness. The stars seemed to be fixed to an invisible hollow sphere, and it was as though this sphere rotated around the earth. The axis of rotation appears to pass through a star in the north which we call the North Star or the Pole Star. The sun and the moon and the other star-like objects seem to revolve round the earth with different speeds. There were five such star-like lights visible at different times in the night sky: Mercury, Venus, Mars, Jupiter and Saturn. These star-like objects — together with the sun and the moon — were called Wanderers or Planets (from the Greek for wanderer).

The ancient Greek philosopher Aristarchus (died 230 BCE) suggested that instead of imagining all these heavenly objects to be orbiting the earth, it is easier to imagine the earth rotating about its axis. But this idea was so revolutionary that it was rejected. It was hard to understand how such a massive object as the earth could be set in motion, and how birds which flew above the earth were not left behind due to the rotation of the earth. Moreover, a stone thrown vertically upwards came back vertically downwards, which seemed to suggest a static earth. Many discoveries had to be made before all these objections could be satisfactorily answered.

2.2Measurement of time

The seven-day week is probably traceable to the seven Planets seen in the sky.¹ Each day of the week was dedicated to a particular one of these heavenly lights. We can see echoes of this in our English names: Saturday for Saturn’s day, Sunday for the sun’s day and Monday for the moon’s day. We also find this in the French names for the other days of the week, with the suffix di for day: Tuesday mardi (Mars), Wednesday mercredi (Mercury), Thursday jeudi (Jove = Jupiter) and Friday vendredi (Venus).

The pattern of the stars in the night sky changes gradually from night to night, but is repeated after a period of 365 days, or more accurately 365 and a quarter days. This was established by the ancient Egyptians who painstaking recorded the position of the bright star Sirius shortly after sunset night after night and found that Sirius returned to its original position after 365.25 days. This period of about 365 days also matched the climatic cycle of the seasons, which includes average temperatures, positions of the sun in the sky at noon, etc. So the idea that there is a natural period of time which we call a year was established in ancient Egypt, long before the year was associated with the earth’s motion round the sun.

The Egyptians also divided the day into 12 equal periods and the night into 12 equal periods. Each period was called an hour. Clearly, a day hour was in general different from a night hour. Moreover, the actual length of an hour also varied with the season, day hours being shorter in winter and longer in the summer, with the reverse holding for night hours.² Later the hour was defined as a 24th part of a day-night duration, making it a constant measure of length at any time of day or night and all through the year. Today the scientific unit of time is the second, which is defined by an atomic clock.

As the moon orbits the earth its phase as seen from earth changes from new moon to crescent moon to half moon to gibbous moon to full moon and all the way back to new moon. So the lunar cycle also defines a period of time. This period was called a month, the word month being related to the word moon. This is not an invariant period, because the length of a lunar month varies slightly from cycle to cycle. The average length of a lunar month is a little less than 30 days. So there are 12 lunar months in a year, with a few extra days left over. These days are added (unevenly) to the months to make the sum of the days of the 12 months equal to 365. Now, since the year is closer to 365.25 days, if we were to limit the year to 365 days, after 4 years we would be behind by 1 day. So 1 day is added to the year every four years (leap year). If the last two digits of the year is a multiple of 4 then it is a leap year. But even this arrangement is not perfect. Every now and then a small further adjustment is needed. So the rule is that if the last two digits are 0, then we look at the next two digits and if this number is divisible by 4, then it is a leap year. If not, it is not a leap year. So 1900 was not a leap year, and 2100 will not be a leap year. But 2000 was a leap year, and the next leap year ending in two zeros will be 2400.

Exercise 2.1.

(a) How many seconds are there in a month of 30 days?

(b) How many seconds are there in a year (365.25 days)?

2.3Ptolemy’s model

Since the sun presents a circular face it seemed natural to assume that the sun is actually a sphere. Likewise, although the moon goes through phases, it made sense to assume that the moon is a sphere and the dark part of the moon represents the night regions. The shapes of the other planets could not be discerned with the human eye, but it seemed logical to assume they too are spheres. What about the earth? The earth looks more or less flat, but the early scientists figured out that the earth is actually a sphere.

The Greek philosophers came to this conclusion through careful observation and logical deduction. Chief among their observations were the following:

1. A ship dips below the horizon when it travels far from the shore, showing that the ocean surface is curved.

2. The noon day sun is seen further south in the sky in more northern climates.

3. The shadow of the earth on the moon’s surface during a lunar eclipse is circular.

So the earth and all the planets were known to be spherical. The sphere was therefore a perfect three-dimensional shape. It made sense to say that the circle was the perfect two-dimensional shape. This had major consequences for understanding the movement of the solar system.

Ptolemy (died 178 CE) was an Egyptian astronomer who studied the celestial objects with great precision. He concluded that the heavenly bodies all travel around the earth in circles — with each circle being called a deferent — with somewhat differing speeds. With the exception of the sun, the moon, and the stars, the celestial objects (i.e. the five planets) also move in a cyclic path — called an epicycle — about the main circular orbit or deferent. This dual motion was necessary to explain why these planets seem to backtrack their path in the sky every now and then instead of progressively moving along in a single direction.

Ptolemy’s model prevailed for over a thousand years and was the dominant model of the solar system used by scientists until it was seriously challenged by Copernicus.

2.4The Copernican revolution

Copernicus (died 1543) showed that it is easier to explain all the astronomical data of the movements of the heavenly bodies by assuming that the earth rotates on its axis and that it goes round the sun and that all the planets also orbit the sun, except that the moon orbits the earth and travels along with the earth in its orbit of the sun. Copernicus’ model is called the heliocentric — sun at the center — model, as opposed to the geocentric — earth at the center — model of Ptolemy. Because Copernicus’ theory brought about a major shift in our most basic understanding of the visible universe, any major change of generally accepted ideas has come to be called a Copernican shift.

But the Copernican revolution did not happen overnight. Copernicus was fully aware of the controversy that his theory would create, and so he published his thesis only on his deathbed. He was condemned by the Christian leaders of Europe, both Catholic and Protestant, on the charge that his model contradicted the Bible. Though the Bible has not changed, today no Christian leader claims any contradiction between the Bible and the Copernican model of the solar system. But the religious objection was not the most serious obstacle to the universal acceptance of Copernicus’ theory. Let us recall that the philosopher Aristarchus (died 230 BCE) had put forward a heliocentric model of the solar system several centuries earlier. The opposition to Aristarchus certainly did not come from religion but from observation. The clouds, the air in general, and creatures that flew through the air all acted as though the earth were stationary. It required Newton to explain how these phenomena could be consistent with a rotating earth.

After Copernicus, astronomers abandoned Ptolemy’s model and made further observations that both confirmed and refined Copernicus’ model.

Tycho Brahe (died 1601) constructed an extremely accurate observatory. He recorded the position (i.e. the angular position of a celestial object when viewed from a point on the ground) of the heavenly bodies at different times over a period of several years.

Johannes Kepler (died 1630) made use of Brahe’s data and found that they conflicted slightly with the data used in support of Ptolemy’s and Copernicus’ models. Kepler found that Brahe’s data could be explained by suggesting that the earth and the planets did not travel round the sun in perfect circles, but rather in ellipses (oval paths), with the sun at one focus of the ellipse.

With the invention of the telescope, Galileo Galilei (died 1642) was able to show that the planet Venus showed all the phases — full, gibbous, half, crescent — as the moon. This was perfectly explainable on Copernicus’ heliocentric model, but not on Ptolemy’s geocentric model. Ptolemy’s model did not allow for a full phase of Venus to be visible from earth. Thus there was unquestionable evidence — in addition to Brahe’s observations and the calculations of Kepler — that the planets do indeed orbit the sun.

A year after Galileo’s death saw the birth of Isaac Newton who became the father of what we today call Physics in the modern sense of the term. Newton used the theories of Kepler and the observations of Galileo to establish a mathematical description of physics which still endures today, though it has been greatly modified by Albert Einstein’s theory of Relativity and the Quantum theory developed by several physicists of the early twentieth century.

2.5Newton’s laws

Several Greek philosophers made significant contributions to our understanding of nature, but it was Aristotle who drew up a system for understanding fundamental physical phenomena. Aristotle believed — as did most Greek philosophers — that all matter was composed of the elements: earth, air, water, fire and ether. Vertical motion was natural. Solid objects fell to the ground because they sought to be close to the earth, since they were also made of earth. Liquids fell or flowed downwards because they sought to be united to the seas, the domain of the element called water. Flames shot upwards because the domain of fire was above. The purpose of all natural motion was goal oriented. Horizontal motion, on the other hand, was violent, and needed an agent to sustain it. The motion of the stars was circular, because this was the property of ether.

But Aristotelian physics was not based on carefully controlled observation. Laboratory experiments showed that bodies had this property called inertia which would cause a moving body to continue its motion in the absence of friction or air resistance.

Newton raised the concept of inertia to the status of a law of physics. He was thus able to show that all motion could be explained on the basis of his three laws of motion:

First Law:

Every object remains at rest, or moves with a constant speed in a straight line, unless compelled to do otherwise by an external force.

As the earth rotates, the air that is close to the ground also moves with the same speed as the ground. Every object that floats or flies through the air is also carried along with the motion of the earth. Inertia keeps everything going together. So we cannot tell from an observation of the atmosphere that the earth is in motion, any more than we can tell that an airplane is moving by observing an object that is dropped inside the plane. Of course, because the motion of the earth is a rotation and not a simple translation along a straight line, there are other factors that give rise to atmospheric phenomena such as hurricanes, and it is these that reveal the rotation of the earth.

Second Law:

An external force F applied to an object of mass m would impart an acceleration a to the object in the direction of the force such that F = ma.

Acceleration is not just increase of speed, but any change of velocity in magnitude or direction. So an object moving on a circular path is accelerating because its direction of motion is changing. It can be shown with some calculation that an object moving along a circle of radius r with a constant speed v experiences an acceleration of magnitude v²/r directed towards the center of the circle.

An important quantity in the study of motion is the momentum of an object, written as p and defined as the product of mass and velocity: p = mv. Acceleration is the rate of change of velocity. So the product ma is the rate of change of momentum. So Newton’s Second Law can be stated as: Force = Rate of change of momentum.

Exercise 2.2.

(a) A car of mass 1500 kg accelerates from rest to a speed of 60 kmph in 10 seconds. Find the average force applied on the car by the engine. (Convert kmph into meters per second. Remember, average acceleration = change of velocity divided by time.)

(b) Taking the distance of the earth to the sun as 150,000,000 km find the circumference of the earth’s orbit in meters. Hence find the speed of the earth as it travels through space. Use the result of Exercise 2.1 (b).

(c) Find the acceleration of the earth as it orbits the sun.

Third Law:

When an object A applies a force on an object B, the object B simultaneously applies an equal and opposite force on object A.

The Third Law explains why we do not go through the floor. Our weight applies a force on the floor, and the floor applies an equal and opposite force — called the normal force — on our feet.

These laws have successfully explained horizontal motion. What about vertical motion? According to the Second Law an accelerating body must be driven by a force. So if an apple accelerates to the ground it is because there is a force acting upon it. Moreover, this law also states that the force is proportional to the acceleration. Not all bodies are pulled to the earth by the same force. Some objects are heavier than others. But heavier and lighter objects fall at the same acceleration in the absence of air resistance. The Second Law can also be written as a = F/m. The acceleration equals the force divided by the mass. Since all falling objects accelerate at the same rate in a vacuum where there is no air resistance, Newton’s Second Law indicates that there must be a force — the force of gravity — acting on an object that is proportional to the mass of the object. The greater the mass, the greater the force, and so the ratio of force to mass stays constant.

So the force exerted by the earth on an apple is proportional to the mass of the apple. Newton’s Third Law requires that if the earth exerts a force on the apple, the apple must exert an equal and opposite force on the earth. Through an argument from symmetry it follows that the force exerted by the apple on the earth should be proportional to the mass of the earth. So this suggests that the mutual force of attraction between two objects should be proportional to the product of the masses of the two objects.

Newton extended this law to include all the heavenly bodies such as the moon, the sun and the planets. What about circular motion? It is not ethereal motion, but a form of acceleration, as we have seen. And acceleration requires force. If we tie a stone to a string and twirl it around, the stone will fly in a circular path because the tension in the string provides the force that generates the acceleration (equal to v²/r where v is the speed of the stone and r the length of the string). In the case of a planet circling the sun, the force comes from gravity. Using Kepler’s data of planetary motion Newton inferred that the further a planet is from the sun, the smaller is its acceleration, and that this acceleration is inversely proportional to the square of the distance of the planet from the sun. So Newton was able to conclude that the force between two objects of masses m₁ and m₂ at a distance of r from each other is proportional to the product m₁m₂ and inversely proportional to r². So this proportionality can be written as

A proportionality relation can be written as an equation by introducing a constant factor. So Newton’s Law of Gravitation is fully written as

where G is called the Universal Gravitational Constant. Scientists use the Syst`eme Internationale or SI system for defining the units of measurement. This system is also called the MKS system because its units are meter, kilogram and second.³ G = 6.67 × 10⁻¹¹ in SI units.

What is the gravitational force acting on an object of mass m near the surface of the earth? Let us call the mass of the earth M. Let the radius of the earth be R. Consider an object a short distance above the ground, where by short distance we mean not more than about 10 kilometers. This is small compared to the radius of the earth which is about 6370 km. The force exerted by the earth on this object of mass m is therefore

where the distance between the mass and the earth is taken to be the distance between the mass and the center of the earth, which is R.

If this object were dropped from a small height above the ground, it would fall with an acceleration which we shall call g that is given by Newton’s Second Law as F = mg. Therefore

Cancelling m from both sides, we obtain

The mass of the earth M = 5.97 × 10²⁴ kg, the radius of the earth R = 6.37 × 10⁶ m and G = 6.67 × 10⁻¹¹ in SI units. Putting these numbers into the equation, we obtain the acceleration g of a falling object to be 9.8 m/s². So the force of gravity acting on an object of mass m is given by mg. We call this force the weight of the object in a technical sense. So a mass of 10 kg would have a weight of 98 newtons (written 98 N) where the newton is the SI unit of force.

Exercise 2.3.

(a) Earlier you found the acceleration of the earth as it orbits the sun. Knowing the distance of the earth to the sun and the value of G find the mass of the sun. (Hint: See Eq. (2.2). Here g is the acceleration of an object in the gravitational field of the earth. Replace this by the acceleration of the earth in the gravitational field of the sun.)

(b) The mass of the moon is 7.35×10²² kg and the radius of the moon is 1726 km. Find the acceleration with which an object would fall due to the force of gravity close to the surface of the moon.

(c) If an object is fired horizontally close to the ground it could go into orbit round the earth if the speed of the object is sufficiently high. Using the fact that the centripetal acceleration of an object moving along a circle is v2/r find the minimum horizontal velocity v with which a rocket must be fired for it to go into stable orbit round the earth. For simplicity assume the earth is a smooth sphere with no hills, trees, buildings or birds, and ignore air resistance.

2.6Work and energy

When a force acts on an object and displaces the object through a distance, we say that the force performs work on the object. If the displacement is exactly along the direction of the force, the work done is simply the product of the force and the displacement. If the force is at an angle to the displacement, the work done is less than the product of force and displacement. If the force is at right angles to the displacement, the work done is zero. Mathematically, we say work W = FS cos θ where θ is the angle between the force of magnitude F and the displacement of magnitude S.⁴

When an apple falls from a tree, work is done by the force of gravity on the apple. Because the force of gravity is in the same direction as the displacement the work done is positive. If the apple drops through a vertical height h, the work done by gravity on the apple is mgh.

What is the effect of this work done by the force of gravity? The immediate effect is that the apple speeds up. A moving object is said to have kinetic energy by virtue of its motion. If the mass is m and the speed is v the kinetic energy of the object is given by

Prior to the apple’s fall it was at some height above the ground. If this height is h meters, then we say that it had a potential energy of mgh. As it falls, its potential energy changes into kinetic energy and work is being done on the apple by the force of gravity.

The Work-Kinetic Energy Theorem states that the work done by a force on a body is equal to the change of kinetic energy of the body.

Potential and kinetic energy are forms of mechanical energy. We see that potential energy can be converted into kinetic energy. The reverse happens when we throw an object upwards. In the next chapter we shall see that there is another kind of energy called heat energy, and that it is possible to convert heat into mechanical energy and vice versa. So there is a universal law — called the Law of Conservation of Energy — which states that energy cannot be created or destroyed, but may be converted from one form to another. In SI units energy is measured in joules (J).

Exercise 2.4.

(a) What is the kinetic energy of the earth as it orbits the sun, ignoring the kinetic energy due to the rotation of the earth about its axis?

(b) 50,000 kg of water drop every second down a waterfall through a height of 30 m. If 80 per cent of this gravitational energy could be converted to electricity, how much electrical energy can be produced at the bottom of the fall per second? Energy produced per second is called power and expressed in units called watts (W).

2.7Determinism

A major consequence of Newtonian physics was that it created a sense of absolute universalism in the minds of many people. According to the laws of Newton every object follows a deterministic behavior. For example, consider two spheres moving along a straight line towards each other and moving away from each other along the same straight line after collision. If we know the masses and velocites of the spheres before they collide, we can predict the velocities of the spheres after they collide, assuming the total kinetic energy to be unchanged in the collision. In the following chapter we shall see how we came to know that all matter is made of atoms and molecules. If it were possible to measure the masses, positions and velocities of every single atom and molecule in the universe at a given moment of time, then it is theoretically possible to predict the exact configuration of all the atoms and molecules an hour later, a day later, a year later, a billion years later. This means that the exact state of the universe a billion years from now has already been determined. Nothing can change the flow of the history of the universe. Whether you will remain alive and if so what color clothing you will be wearing on a day exactly 10 years from today has already been determined. There is no such thing as free will because the circuits of the brain follow the laws of physics and whatever they do is dictated by necessity and they cannot do anything other than what is determined by the laws. So free will is an illusion. When we think we are choosing something over another we are simply moving in the direction determined by the laws of physics. Ironically, this sort of scientific determinism leads to conclusions very similar to nonscientific beliefs in astrology and fate.

With the advent of the quantum theory and relativity in the early years of the 20th century, Newtonian physics was replaced by modern physics which basically states that Newtonian principles need to be revised when we consider very small objects such as atoms and very fast moving objects such as light. Modern physics does not posit a deterministic universe. It is intrinsically impossible to make an exact measurement of all the quantities such as position, velocity, etc. of even a single atom, leave alone billions of atoms. Not only is precise measurement impossible, but prediction is also ruled out by the laws of quantum mechanics. Thus determinism has collapsed. One can no longer use Newtonian physics to argue against the possibility of free will. The philosophical debates regarding free will became far more sophisticated as a result of quantum theory.

2.8Summary

The heavenly bodies appear to be moving in a circular motion round the earth. But the cumulative results of painstaking observations made by astronomers through the centuries enabled Copernicus to conclude that the sun and not the earth is the center of the solar system. But if the earth also moved in a circular orbit, then one could no longer accept the ancient belief that circular motion was a special property of the heavenly bodies.

The Copernican worldview required a scientific explanation for this perpetual motion of the planets round the sun. Newton was the first scientist to explain the kinematics of both celestial and terrestrial objects. He provided a simple mathematical system that explained all sorts of motion. The force of gravity attracts all objects to each other. The force between two objects is proportional to the product of the masses and inversely proportional to the square of the distance between them.

In addition to the law of gravity Newton also enunciated the three laws of motion. All objects have this property called inertia, whereby they tend to remain at rest or move in a straight line with constant speed. In order to change this state, an external force has to be applied. This force is numerically equal to the product of the mass of the object and its acceleration under this force. Moreover, when one object applies a force on another, the second object simultaneously applies an equal and opposite force on the first.

Newton’s laws of motion and gravity form the basis of all classical mechanics, the branch of physics dealing with motion and forces. These laws successfully explained the motion of the planets as well as all motion on earth. They seemed to be absolutely infallible rules followed by all matter in the universe. This led many people to believe that the universe is deterministic. Everything happens the way it does because it cannot happen any other way. All things have been predetermined by the laws of motion.

¹I use the word Planet with a capital P to denote the term as it was used in antiquity — meaning the five visible planets plus the sun and the moon.

²The day/night difference and the seasonal variations would be greater in regions further removed from the equator than Egypt.

³The United States continues to use the outdated British FPS system. This is an example of systemic inertia.

The ratio sin θ is defined as shown.

⁴Consider a triangle with one right angle = 90⁰. The side opposite to this right angle is called the hypotenuse. If θ is one of the other two angles, then the side between the 90⁰ angle and the angle θ is called the adjacent side of the angle θ and the third side is called the opposite side. We define cos θ as the number obtained by dividing the length of the adjacent side by the length of the hypotenuse.