CHECK YOUR EXISTING KNOWLEDGE

Introduction

If algebra and trigonometry owe their origins to the Greeks, then 17^th century Europe was the birthplace of modern science … and, quite frankly (in Europe), nothing much came in between – the ‘dark ages’ were named for their dearth of intellectual enlightenment rather than levels of ambient light. Historians generally regard this period as extending from the fall of the Roman Empire in the 5^th century AD to the onset of the mediaeval period in the late 9^th century. By contrast, anthropologists have extended it until the European Renaissance (literally ‘new birth’) in the 12^th century; some, indeed, mark it as late as the 15^th century – a thousand years of academic regression and stagnation.

Intellectually, the period can be regarded as starting with the destruction of the Royal Library of Alexandria, which, since the 3^rd century BC, had held an immense store of knowledge; contemporary accounts suggest hundreds of thousands of scrolls. Ironically, the date of destruction – indeed, whether the library was truly destroyed, broken up into smaller collections or dissipated by lack of patronage and more pressing political concerns – is unknown; one of the more salient features of the Dark Ages was the lack of documentary evidence in a period where perhaps only one person in a thousand was literate.

Outside of Europe, the flame of human development was kept flickering by the Chinese, Indians and Arabs who became sophisticated in astronomy (at that time indistinguishable from astrology) and mathematics (remember that oh-so-useful zero). Indeed, some historians regard the spark that ignited the renaissance as the opening up of the Near and Far East by travellers such as Marco Polo, bringing not just technological innovations including gunpowder, spectacles and pasta but scientific knowledge to augment that from the classical literature extant from the previous millennium.

The principal tributary to which the river of modern science can be traced is probably Roger Bacon (?1214–?1292), a Somerset-born Franciscan friar, philosopher, alchemist and medic; the ‘Father of Science’, ‘Dr Miribalis’ (Fig. 2.1). In his most famous work, Opus Maius, he outlined his unsurpassed knowledge of gunpowder, optics, mathematics, moral philosophy, theology, grammar…and experimental science, of which he was probably the inventor and certainly an ardent practitioner.

Figure 2.1 • ‘Dr Miribalis’, Roger Bacon, a Franciscan monk who lived and died in the 13th century, is regarded by many as the father of modern science for his introduction of the scientific method of hypothesis tested by experimental observation in contrast to the theological argumentation of the day.

It is Bacon who is credited with having invented the ‘Scientific Method’ – hypothesis tested by experimentation and observation. This may now seem basic in the extreme but, before Bacon, science was a matter of metaphysical debate and religious dogma; he sowed the seeds from which sprang the randomized controlled trials in today’s biomedical journals.

Bacon, though, would not have thought of himself as a scientist but as a philosopher – a lover, and seeker, of knowledge…as would those who followed him, culminating with Isaac Newton. They were natural philosophers, interested in the secrets of the world around them (‘scientists’ were not invented until the 18^th century) and it was these men who laid down the ground rules for the physics that we still use today.

However, for such men to flourish, two things were needed: freedom of expression and cross-fertilization of ideas (combined, perhaps, with a slight sense of competition). Bacon had little of either and, like other individual geniuses in the centuries that followed him, most notably Leonardo da Vinci (1452–1519) and Galileo Galilei (1564–1642), his work was often curtailed or even destroyed. The all-powerful authority of the Church regarded science as a direct challenge to the fundamentals of the bible and was ruthless in suppressing anything that contradicted that view.

Galileo suffered particularly badly in this respect, suffering imprisonment by the Inquisition, house arrest and finally (under pain of a rather horrible death) being made to recant his ‘heretical’ support for the Copernican heliocentric system. The Church had determined that it simply wasn’t possible for sunspots to rotate about the Sun or Jupiter’s moons around Jupiter: the Pope had decreed that the Earth was the centre of the Universe around which everything rotated. The Pope was appointed by God, and therefore infallible; the Inquisition took care of anything that might interfere with that, including Galileo.

The opportunity for the requisite level of intellectual freedom and learned discourse came with the Restoration of the British monarchy in 1660, following a protracted Civil War, regicide and Cromwell’s puritanical ‘Commonwealth’, which proscribed everything from Christmas to dancing.

Within months of Charles II regaining the English throne, the finest minds of the day were meeting on a regular basis; two years later, a Society for ‘The promoting of Physico-mathematicall Experimentall Learning’ was formed and, in 1663, Charles II granted it a royal charter. The Royal Society still exists today and there are few higher accolades in the field of science than the honorific ‘FRS’ (Fellow of the Royal Society) after one’s name. The membership of the time reads like a ‘Who’s Who’ of physics – Robert Hooke and Christopher Wren (who are probably better remembered as the men behind the rebuilding of London after the Great Fire of 1666, in particular St Paul’s cathedral), Robert Boyle and, of course, Isaac Newton. It was these men and their peers who provided much of the groundwork that still forms the basis of physics today and which forms the content of this chapter.

Velocity, angular velocity and acceleration

In everyday speech, ‘velocity’ and ‘speed’ are interchangeable; in physics, they are (usually) not. Both are measured in units of distance (displacement) per unit time (metres per second; miles per hour etc.); however, velocity is a vector quantity (having magnitude and direction) whilst speed is scalar (having magnitude alone).

Acceleration is change in velocity (deceleration is expressed as negative acceleration; both are measured in metres per second per second = ms⁻²). This can, therefore, be caused by a change in the magnitude of the velocity or a change in its direction (or both). Put in terms of driving a car, using the brake or accelerator causes a change in both speed and velocity; however, using the steering wheel causes a change in velocity even if the speed remains constant.

In terms of graphical representation (Fig. 2.2), if time (t) is plotted on the x-axis and displacement (s) on the y-axis, then the slope of the graph represents the velocity: the change in displacement, Δs, divided by the change in time Δt. Whereas this is straightforward if the graph is a straight line (i.e. constant velocity). If it is not, then the mean velocity needs to be obtained using a method of calculation known as calculus. If you are unfamiliar with this, read the Fact File below before proceeding; if calculus is old hat, then you will easily appreciate that velocity can be written as:

Figure 2.2 • A graph of displacement against time. The slope (tangent) of the graph represents the velocity. When the value of this tangent is plotted against time, then tangent of the resultant graph represents acceleration. In this case, as the change of velocity (i.e., the acceleration) was constant, the graph is a straight line. The final example demonstrates the case where the velocity is variable. In order to calculate the velocity from the displacement against time graph, we must repeatedly take the tangent and then work out the average. This is done using differential calculus.

Similarly, if we plot the results of this graph with velocity (v) on the y-axis and time (t) on the x-axis, we get a gradient representing acceleration (a). If this is a straight line, then acceleration (the rate of change of velocity) is constant; if not, then calculus can calculate the average gradient:

This, for example, is why the Earth is accelerating around the sun, although its speed is constant (bar the loss due to friction of a millisecond per century), its velocity is constantly changing as it describes an ellipse around the sun. This can be inconvenient on occasions; however, if we measure displacement in terms of the number of degrees (θ) traversed per unit time we can measure the angular velocity (ω):

Because the rotation is about a specific axis, which can be changed, angular velocity is also a vector quantity.

Mass and momentum

In the same way that physicists differentiate speed and velocity, ‘mass’ and ‘weight’ are also separate entities. Mass (m) is a measure of the quantity of matter in an object and, within the SI system, is measured in kilograms. It does not matter where the mass is – outer space, the centre of the Earth, the surface of the sun – the mass remains constant. By contrast, weight (W) is the force by which a mass is attracted to a gravitationally massive object (such as the Earth):

where g is the acceleration of free fall, which can vary depending on location (Table 2.1). Although we measure our weight in kilograms, it should in fact be measured in newtons; we can get away with it because we are in a (reasonably) constant gravitational field. If we were in free-fall or outer space, our weight would be 0 N, although our mass would be unchanged.

Table 2.1 Gravitational forces on selected astronomical bodies

This brings us to another property, momentum (I), which is dependent on mass rather than weight. If we were to run into a brick wall, it would hurt: how much it hurt would depend on how heavy we are and how fast we are running, so it is no surprise to learn the momentum is the product of these two factors:

However, even if we were weightless – say, in outer space – it would hurt just as much when we hit the wall; our mass is still the same.

Force

Once again, this is a term, the specificity of which, in physics, is in sharp contrast to the generality of meaning in the vernacular. To the scientist, force (F) is something that changes an object’s velocity, remembering that velocity is a vector quantity with both magnitude and direction. As we saw when discussing weight, force is measured in newtons (N) and can be calculated using the formula:

Newton’s laws and equations of motion

Reading this far, you will already have gathered that Sir Isaac Newton (Fig. 2.3) was an important man in the history of physics; arguably, he is the most important. Born in 1642, the year that Galileo died, Newton was something of a prodigy. Although he seems to have been a solitary, reclusive child, preferring introspection to classes (which probably bored him), he flourished whilst at Cambridge University; by the age of just 27, he was Lucasian Professor of Mathematics (the same chair is now held by Stephen Hawking). In 1665, when bubonic plague hit London and Cambridge, he withdrew to the family home in Lincolnshire and devoted himself to developing the ideas that would make him famous.

Figure 2.3 • Newton as he appeared at the time of publication of Principia in 1687.

Typically for Newton, he did not choose to publish them until 1687, and then only because he was afraid someone else might claim credit for them. Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural History), commonly referred to as the Principia, is probably the most important scientific book of all time, although Darwin’s On the Origin of Species might give it a close run.

Although he achieved excellence in many other fields: optics (he discovered diffraction), fluid mechanics, history, theology and alchemy and was an able administrator (for 28 years he was master of the Royal Mint) and politician (twice a member of parliament), it is for the work on mechanics and gravitation that he owes his principal fame.

His three laws of motion govern and quantify most of the everyday movement that we observe. Let us consider them in order.

Newton’s first law of motion

Every body continues in its state of rest or in uniform motion in a straight line unless acted on by an external force.

You may recognize this as being the definition of force, stated the other way around. Put in simpler terms, things keep on doing what they are already doing (standing still or moving) unless something happens to alter this.

The ‘standing still’ bit is obvious and completely in accordance with our everyday observations – objects don’t spontaneously start moving unless engendered by something or someone (the external force). The ‘moving’ bit requires a bit more consideration – at first glance it might seem at odds with everyday observation: if we throw a ball into the air, it doesn’t carry on travelling in a straight line, it describes a curve (called a parabola) and returns to Earth – we even have the saying ‘what goes up must come down’. Similarly, we all know that a freewheeling bicycle will not keep travelling forever; unless we pedal, it will slow and eventually stop.

Newton’s flash of genius was to realize that rather than being examples of deviation from this law, such objects are behaving in the observed manner because they are being acted on by external forces: the ball’s velocity is reduced and reversed by the force of gravity, pulling it towards the centre of the Earth; the bicycle (although post-dating Newton by 200 years) is nevertheless slowed by friction in the wheel bearings, between road and tyre and against the air molecules that must be pushed aside in order to progress. In a world where it is impossible to escape these effects, Newton managed to appreciate these forces for what they were and envisage what would happen if they were removed – indeed, if we threw our tennis ball in the vacuum of outer space, away from gravitational influences, it would keep going indefinitely. The nearest we can come to appreciating this is by reducing friction to a minimum and looking at the behaviour of objects moving on ice. Intuitively, we know that an ice-skater will slow far less than a bicyclist over a similar distance (the sport of curling relies on this) and, over a short distance, an object sliding across a smooth sheet of ice may not appear to slow at all. This tendency for an object to keep on doing what it is already doing (conservation of momentum) is termed inertia.

Fact File

ISAAC NEWTON 2

The story of Newton sitting in his mother’s orchard and asking himself why an apple had fallen downwards rather than upwards is probably apocryphal; however, his intuitive leap regarding his first law may have been due to the fact that his was living during a period, which began in Bacon’s time, known as the ‘Little Ice Age’. During the time of his confinement to the country, England experienced long, bitterly cold winters – indeed, during this period England recorded its lowest ever recorded atmospheric pressure (931 millibars) and, reputedly, its coldest ever temperature.

The River Thames was frozen sufficiently to bear the weight of not just skaters but of a coach and horses and, in the Fenlands of Lincolnshire, (which, lying below sea level, flooded every winter), there would have been skating aplenty. Had Newton not had access to those now climatically rare natural ice-rinks, perhaps he would never have received the spark that ignited his unique insights into motion.

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