Forces of Nature

by Professor Brian Cox

Although seemingly calm and serene at low tide, the Bay of Fundy, in Canada, holds the record for the greatest tidal range – the vertical difference between the high tide and low tide – of 56 feet.

  ‘Puerilities’ is a word I intend to use more often. In an essay written in 1616 entitled ‘Discourse on the Tides’, Galileo likened the movement of the Earth’s oceans to the movement of water in a vase. He reasoned that because the water is distorted by changes in the orientation and acceleration of the vase, so the oceans are distorted in their movement by the orientation and acceleration of the Earth. He posited a mechanism of positive and negative acceleration to explain the back-and-forth motion of the tides, a theory that has often been labelled his ‘great mistake’. The irony is, both Galileo and Kepler were partly right. Here is the explanation for the origin of the tides.

  Johannes Kepler believed in the Copernican, heliocentric view of the Solar System and created this model to illustrate the six planets that he believed existed in our Solar System and how they related to one another.

  Kepler was correct in the sense that the tides are caused by the Moon’s gravitational effect on the Earth. He didn’t put it in those terms, of course, because Newton had yet to publish his theory of Universal Gravitation. Galileo was correct because the Earth is accelerating. He just didn’t appreciate towards what.

  Let’s accept that the tides have something to do with the Moon; its orbit can be described in precisely the same way that we described the Earth’s orbit around the Sun. The Moon is being pulled towards the Earth by the force of gravity but is continually missing it because it continues to try to move in a straight line, in accord with the principle of inertia.

  We now need to introduce Newton’s third and last law of motion. It states:

  TO EVERY ACTION, THERE IS AN EQUAL AND OPPOSITE REACTION.

  This means that forces always come in pairs. If the Earth exerts a gravitational pull on the Moon, the Moon exerts an equal and opposite gravitational pull on the Earth. This means that the Earth has to fall towards the Moon, accelerated by the force of gravity along a line connecting their centres. Why doesn’t the Earth career towards the Moon? For the same reason that the Moon doesn’t career towards the Earth – because it falls and misses. The Earth must also be in orbit! But around what? The answer is that we were a little lax in our language when we said that the Moon orbits around the Earth. It does to a good approximation, but in fact it orbits around a point slightly displaced from the centre of the Earth known as the centre of mass of the Earth-Moon system. To get an instinct for what’s happening, imagine two moons of equal mass orbiting around each other in circular orbits. Everything is perfectly symmetric, and they both orbit around a point that is equidistant between their centres. This is called the centre of mass of the system. If one of the moons is more massive than the other, the centre of mass will be closer to the massive moon, and they will both orbit around this offset point. The Earth is 81 times more massive than the Moon, so the centre of mass point about which they orbit is very close to the centre of the Earth, but not quite at the centre; it is displaced by 4,700 kilometres, which is about 1/81 of the distance between the Earth and the Moon. This is why it’s superficially reasonable, but not accurate, to say that the Moon orbits around the Earth. It’s only reasonable in a superficial sense because it dodges the problem of how the Earth can accelerate towards the Moon – as it must – and keep missing!

  From the point of view of the observer sitting at the centre of the Earth, the force of gravity is trying to accelerate the Earth towards the centre of the Moon in a straight line, in accord with Newton’s law of universal gravitation.

  The fact that the Earth is in orbit around the centre of mass of the Earth-Moon system is critical to an understanding of the tides. The key is to switch perspective, or frame of reference, just as we did when we explored the Coriolis Force and its effect on storm systems. We’re hopping between reference frames again, searching for fictitious forces – physicists are always doing this because it’s bloody useful, and we know how to do it! (See illustration.)

  Let’s picture what’s happening from the point of view of an observer sitting at the centre of the Earth. This is the point that is orbiting around the centre of mass of the Earth-Moon system. We can assume that the centre of the Earth is going round in a perfect circle, which it very nearly is. This reference frame is not inertial, because it’s rotating, and we will therefore expect fictitious forces to be present. But which? This time it’s not the Coreolis Force, which appears when things roll around in rotating reference frames, but the Centrifugal Force. What does this one do? From the point of view of the observer sitting at the centre of the Earth, the force of gravity is accelerating the Earth towards the centre of the Moon in a straight line, in accord with Newton’s law of universal gravitation. And yet, the Earth doesn’t approach the Moon – it stays a fixed distance away from the centre of mass of the Earth–Moon system if the orbit is circular. This must mean that the observer at the centre of the Earth experiences a force acting with the same strength as the Moon’s gravitational pull, but in the opposite direction, to precisely cancel it out. This force, equal and opposite to the gravitational force, is called the Centrifugal Force. It’s none other than the familiar force we experience if we sit on a fast-rotating fairground ride. We are thrown outwards, and if it’s the right sort of ride, the little cars we sit in will rise up and outwards as the speed increases. The force that does this is the Centrifugal Force.

  The centre of mass.

  Great. But what’s that got to do with the tides? We’re about halfway through, so perhaps you should have a break for a cup of tea and come back refreshed. As an aside, I find something amusing about this explanation for the tides, which is quite a wonderful explanation if you have the patience to follow it. Let me tell you what I find amusing. Imagine, as I tell you, that there is a hint of Joe Pesci in Goodfellas in my voice. I have a love–hate relationship with television. I love most of it, to be honest, but I sometimes find it a superficial medium. The trick is to find a way of exploring ideas in sufficient depth within a television programme that is the length of a single undergraduate lecture, in a visual and entertaining way. I get into a lot of ‘creative debates’ about the definition of ‘sufficient depth’, as you might imagine. Usually we are exploring grand ideas about the origin of the Universe or the beginning of life on Earth, and because the answers to these ideas are speculative, there is room for a bit of hand-waving. In a programme about tides, however, there can be no hand-waving because the reason for the tides is known. I think the tides are a good thing to explain. But I offer a wry smile. ‘Sufficient depth’ is a well-defined concept in this instance. It is defined as being the explanation. Such is the trap, lying in wait for the unwary television executive, in wanting to make a television series about simple questions that actually have answers, rather than complicated questions that don’t.

  Here is the rest of the explanation for the tides. Recall that the Earth is in a little orbit around the centre of mass of the Earth-Moon system. At the centre of the Earth, the Moon’s gravitational pull is perfectly balanced by a fictitious force called the Centrifugal Force. Now consider a point on the surface of the Earth directly beneath the Moon. That too will be in a little orbit, and it will have to go around in a circle of precisely the same radius as the point at the centre of the Earth, because the Earth is a solid ball of rock and a point on the surface can’t move in a different way to the centre. This means that the Centrifugal Force experienced at a point on the surface beneath the Moon must be exactly the same as that experienced by the centre of the Earth. But – and this is the crucial point – the Moon’s gravitational pull at the Earth’s surface directly beneath it is stronger than it is at the centre of the Earth, because the surface of the Earth is closer to the Moon than the centre. The two forces won’t precisely balance! There will be a little too much gravitational pull at the surface directly beneath the Moon, and it is this little
extra pull that deforms the oceans and raises a tide beneath the Moon.

  Usually we are exploring grand ideas about the origin of the Universe or the beginning of life on Earth, and because the answers to these ideas are speculative, there is room for a bit of hand-waving.

  The effect of the variance of gravitational pull and the Centrifugal Force of the Moon can be witnessed all over the globe in our tides. These tides change the appearance of our coastal landscapes every day, and we now have the technology to monitor them and predict them to be able to use that information for maritime purposes – as well as leisure pursuits such as surfing!

  Now consider the situation on the other side of the Earth. Again, the Centrifugal Force must be the same, because every point on the Earth’s surface has to orbit in a circle of precisely the same radius as every other point, but now we are further away from the Moon than the centre of the Earth is, so we’ll experience a weaker gravitational pull from the Moon. This means that the Centrifugal Force, which always points away from the Moon, will be slightly too large, and this will also result in oceans being deformed away from the surface, raising a tide. This is why there are two tides on Earth every day – one beneath the Moon and one on the opposite side of the planet.

  It is the Moon that dictates the tidal forces on Earth, caused as a result of the imbalance between the Moon’s gravitational pull and the Centrifugal Force.

  The tidal forces are the result of the imbalance between the Moon’s gravitational pull and the Centrifugal Force, which is present because the Earth is orbiting around the centre of mass of the Earth-Moon system. Although we usually perceive them because of the large deformation of the surface of the oceans, they are sufficiently large that the Earth’s crust is deformed by a measurable amount, shifting the rocks every day by as much as half a metre. This is not a great shift, but GPS systems are adjusted to take account of the changes in the Earth’s gravitational field caused by the rock tides, and geologists monitor the impact of these tides on the Earth’s fault lines and the potential they have to trigger earthquakes and volcanic eruptions.

  The far side of the Moon, in an image taken by the Luna 3 space probe, 28 October 1959.

  The tides.

  Our explanations of rotating storm systems and tides are quite beautiful in my view, because they embody one of the central themes of this book – that apparently complex and disconnected naturally occurring phenomena can be explained using a simple, underlying framework – in this case Newton’s laws of gravitation and motion. I don’t believe we need a reason to seek an explain for these things beyond the fact that it’s interesting and fun. But the explanations of the tides and the rotation of storm systems both benefitted from us jumping between different frames of reference, which is to say looking at physical phenomena from different points of view. This idea is the launch-pad for something deeper. As we discussed at the beginning of this chapter, Albert Einstein elevated the idea that the laws of Nature must take the same form in all frames of reference to a fundamental principle. Our Universe is built this way. Implementing this requirement forced him to discard Newton’s laws and redraw our intuitive picture of space and time, the grand arena that is so very tempting to take for granted.

  Calculate your own speed of rotation

  At the Equator, the circumference of the Earth is 40,070 kilometres and the day is 24 hours long, so the speed is 1670 kilometres/hour (1037 miles/hour). This decreases by the cosine of your latitude so that at a latitude of 45 degrees, cos(45) = .707 and the speed is .707 x 1670 = 1180 kilometres/hour. You can use this formula to find the speed of rotation at any latitude.

  Einstein’s Theory of Special Relativity

  The subject of motion is unexpectedly rich. Subtleties are evident even in Newton’s Principia. After dealing with the motion of objects in general, and developing many of the tools that modern-day physicists take for granted, Newton’s focus turned to the motion of the planets around the Sun in order to address age-old questions about the tides and the passing of the days, months and years. He was also keenly aware that an understanding of space and time is necessary, and he carefully stated his assumptions about the existence of absolute space and absolute time. That Newton felt it necessary to state that absolute time exists as an assumption is, to my mind, a clear example of his brilliance as a physicist. Newton treated this assumption as we now treat the law of inertia; as an axiom, in agreement with observations at the time, but not provable from first principles. It is a remarkable thing that he identified such an assumption and considered it worthy of note, even though in the seventeenth century, and surely today in most people’s minds, it must ‘go without saying’ that there is not much to say about time other than that it is absolute and that it ticks. And so we return to our musings at the beginning of the chapter about Monet’s field of poppies, vanished forever – perhaps – with the passing of the years. We are now in a position to explore the tantalising ‘perhaps’.

  That Newton felt it necessary to state that absolute time exists as an assumption is, to my mind, a clear example of his brilliance as a physicist. Newton treated this assumption as we now treat the law of inertia; as an axiom, in agreement with observations at the time, but not provable from first principles.

  Albert Einstein’s theories of relativity are among the greatest of human achievements; over a century later, they are still part of the essential foundations of physics.

  Why did Einstein replace Newton’s Laws of Motion?

  Central to our exploration of motion has been the idea of an inertial frame of reference. If you’ve grown weary of the term, if you recall I suggested a drinking game. If you go down this route, you are about to discover a link between vintage wine and memories.

  To recap, the idea is that it isn’t possible to work out which inertial reference frame you are in; they are all absolutely equivalent to each other and the notion of ‘at rest’ is always a relative one. In simpler language, this means that you can’t tell whether or not you are moving. If you accelerate, the story is different, and fictitious forces appear. Albert Einstein thought very deeply about these ideas – more deeply, in my opinion, than anyone else. Einstein is the archetypal wild-haired, sockless genius. In later life he looked otherworldly, appearing to inhabit an abstract space beyond Earthly trivia alongside his theories. This is, of course, a cliché; Einstein was a great physicist, but he discovered no Royal Road to understanding because no such road exists. He worked hard, thought deeply and learnt how to do sums. That said, his theories of relativity are certainly amongst the greatest of human achievements. Over a century after their publication, they are still part of the essential foundations of modern physics.

  Einstein discovered his Theory of Special Relativity by elevating the idea that all inertial reference frames are equivalent to a great principle; an axiom; a fundamental property of our universe. It was his guiding light. To understand why this was so important to Einstein, we need to revisit a concept we explored in Chapter One, symmetry.

  The statement that all inertial frames are equivalent is a statement of symmetry. If you recall, symmetry in mathematics and physics means doing something with the result that nothing changes. A square has a particular symmetry in the sense that we can change our point of view by rotating around the square by 90 degrees and everything will look the same. We can ask a similar question about physical laws such as Newton’s laws of motion. Do the laws remain the same if we change our perspective? One such symmetry relates to the question: do the laws of Nature look the same in all inertial frames?

  Here’s another way of looking at it. The laws of physics describe real things and how they behave. Newton’s laws, as we’ve seen, say that a rolling ball will continue to roll in a straight line unless acted upon by a force. If there is a force, the equation that describes what will happen to the ball is F = ma. Let’s imagine that we are watching a rolling ball, and we decide to change our perspective by hopping into a different inertial reference frame. We wil
l choose a frame of reference that is flying towards the rolling ball and see how our description of what’s happening changes. Note well that ‘our description of what’s happening’ is another way of saying ‘the laws of Nature’, so what we’re really saying is that we want to know how the laws of Nature change. The ball will still appear to move in a straight line because there are no forces acting, but its speed will look different. If we fly towards the ball at 20 m/s, and the ball was rolling towards us at 10 m/s, then common sense informs us that we’ll see the ball rolling towards us at 30 m/s. As long as we account for the change in speed by adding the speed up in this way, we can use Newton’s laws and we’ll get all of our predictions correct. Our description of the physics of the situation is left unchanged by our shift in perspective. This is a symmetry of Newton’s laws; they remain the same if we jump between inertial frames of reference and keep track of the change in the speeds of all the objects in a simple and intuitive way.

  A strike at goal illustrates what these clever physicists were trying to impart – a rolling ball will continue to roll in a straight line unless acted upon by a force.

  A little piece of jargon: accounting for the change in speed in this way is known as a Galilean Transformation, in honour of Galileo. In full physics mode, we can say that Newton’s laws are invariant under Galilean Transformations – this is a symmetry of Newton’s laws.

  Albert Einstein was awarded the Nobel Prize in Physics in 1921. As for other Nobel winners, the idea of symmetry in the laws of Nature was also of great importance.