Forces of Nature. Andrew Cohen

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СКАЧАТЬ due to the attractive force of gravity around 4.6 billion years ago. The Sun formed first, followed by the planets. Let’s fast-forward a few million years to a time when the infant Sun is shining in the centre of a planet-less Solar System. Circling the young Sun are the remains of the cloud of dust and gas out of which the Sun formed, containing all the ingredients to make a planet. This is known as a protoplanetary disc. The fine details of the formation of planets are still a matter of active research, and the mechanisms may be different for rocky planets such as the Earth and gas giants such as Jupiter. For Earth-like planets, random collisions between dust particles can result in the formation of objects of around 1 kilometre in diameter known as planetesimals. These grow larger as they attract smaller lumps of rock and dust by their gravitational pull, increasing their mass, which increases their gravitational pull, attracting more objects, and so on. This is known as runaway accretion, and computer simulations using Newton’s laws suggest that through a series of collisions between these ever-growing planetesimals, a small number of rocky planets emerge from the protoplanetary disc orbiting the young star.

      Models of planetary formation can be checked using the telescopic observation of young star systems. In 2014 the ALMA (Atacama Large Millimeter/submillimeter Array) observatory in Chile captured a beautiful image of a planetary system forming inside a protoplanetary disc around HL Tauri, a system less than 100,000 years old and only 450 light years from Earth. A series of bright concentric rings is clearly visible, separated by darker areas. It is thought that these dark gaps are being cleared by embryonic planets orbiting around the star and sweeping up material – they are the shadow of the planetary orbits. It is interesting to note that planetary formation appears to be well advanced in this very young system. This image is perhaps a glimpse of what our Solar System looked like 4.5 billion years ago.

      Rocky planets begin life as small, irregular planetesimals and evolve over time into spheres. To make progress in understanding why, we might make an observation; all objects in the Solar System are not spheres. The Martian moon Phobos has a radius of approximately 11 kilometres. It is a misshapen lump. Smaller still are the asteroids, comets and grains of dust that formed at the same time as the planets. The Comet 67P/Churyumov–Gerasimenko is less than 5 kilometres across and is an intriguing dumbbell shape. Analysis of data from the Rosetta spacecraft, in orbit around the comet at the time of writing, has shown that 67P was formed by a low-velocity collision of two larger objects. Perhaps this is a snapshot of the processes that previously resulted in the formation of much larger objects such as planets and moons. Smaller lumps of rock merge together under the influence of gravity, and if there is enough material in the vicinity, as there would have been early in the life of the Solar System, the objects will undergo many such collisions and grow. Why isn’t comet 67P spherical?

      ‘FORÇA, EQUILIBRI, VALOR I SENY’

      (STRENGTH, BALANCE, COURAGE AND COMMON SENSE)

      Let’s return to the human towers. What sets the maximum height of a tower? Consider an artificial situation in which the tower is a vertical stack of humans, one on top of the other. If there are only two people in the stack, then the force on the person at the base is the weight of the person above. Let’s understand that sentence. What is weight? Your weight at the Earth’s surface is given by Newton’s equation; it is defined to be the force exerted on you by the Earth. What numbers should we put into the equation to calculate it? Your mass: 75kg. The mass of the Earth: 5.972 x 10²⁴ kg. Newton’s gravitational constant, G: 6.6738 x 10^-11 m³ kg^-1s^-2. What should we use for r? This is the distance from the centre of the Earth to the centre of you. That sounds a bit vague. More precisely, r is the distance between the centre of mass of the Earth and your centre of mass, but it’s a very good approximation to simply insert the radius of the Earth into Newton’s equation. This is because you are only around a couple of metres tall, and the average radius of the Earth is 6,371,000 metres, so moving your centre of mass around by a few tens of centimetres isn’t going to change the calculation much.

      Plugging in the numbers, Newton’s equation tells us that the force on you at the Earth’s surface – your weight – is approximately 736 Newtons (a force of 1 Newton produces an acceleration of 1 m/sˆ2 on a 1kg mass).

      We now need to introduce another of Newton’s laws – his third law of motion, also published in the Principia: To every action, there is an equal and opposite reaction. This says that the Earth exerts a force on you and you exert an equal and opposite force on the Earth. We can now understand what happens when the human towers get higher and higher. If one person stands on another’s shoulders, there is a downward force on the lower person of around 730 Newtons. If another person of the same mass climbs up, the force on the person at the base doubles to 1460 Newtons. If another two people climb up to form a tower five people high, the force on the base person is 2920 Newtons, and so on. Clearly, at some point, the person at the base isn’t strong enough to hold the tower up, and the whole thing will collapse. This is where the skill of the castellers comes in. By having a base, made up of many individuals, the forces can be distributed across the human structure, and this allows the towers to get higher before catastrophe strikes. There is clearly a trade-off; a larger base can support a larger layer above, which in turn can support a larger layer above, and so on. But a larger layer weighs more, and exerts a larger force on the layer below. The ingenious geometrical solutions to this gravitational conundrum emerge through a combination of trial and error, instinct and skill, and this is what makes the Tarragona Castells competition so compelling. For our purposes, it is the principle that matters. As the tower gets higher, the forces on the base increase, and ultimately a limit will be reached.

      Perhaps you can see where this is leading. High human towers are more difficult to sustain because the force on the base becomes increasingly large as the mass of the tower increases. This suggests that the size of structures that rise above the surface of a planet is limited by the structural strength of the rock out of which the planet is made, and the mass of the planet, which sets the gravitational pull and therefore the weight of the structure. On Earth, the tallest mountain as measured from its base on the sea floor is Mauna Kea, on the island of Hawaii. This dormant volcano is 10 kilometres high, over a kilometre higher than Mount Everest. Mauna Kea is sinking because its weight is so great that the rock beneath cannot support it. Mars, by contrast, is a less massive planet. At a mere 6.39 x 10²³ kg, it is around 10 per cent of Earth’s mass and has a radius about half that of Earth. A quick calculation using the equation here will tell you that an object on the surface of Mars weighs around 40 per cent of its weight at the Earth’s surface. Since Mars has a similar composition to Earth, its surface rock has a similar strength, and this implies that more massive mountains can exist on Mars because they weigh less – and this is indeed the case. The Martian mountain Olympus Mons is the highest mountain in the Solar System; at over 24 kilometres in altitude, it is close to the height of three Everests stacked on top of each other. Such a monstrous structure is impossible on Earth because of the immense weight – a result of the Earth’s greater mass and therefore stronger gravitational pull at the surface.

      We see that there must be a limit to the height to which a structure can rise above the surface of a planet. The more massive the planet, the stronger the gravitational pull at its surface, and the lower the height of structures that the surface can support. As the planets get more and more massive, their surfaces will get smoother and smoother because of the stronger gravity. Less-massive planets can be more uneven. We are approaching an answer to our question; we have a mechanism for smoothing out the surface of a planet, but why should this mean that planets get smoothed into a sphere?

      Imagine a mountain on the surface of a planet. Let’s say it is at the North Pole. Now, in your mind’s eye, imagine rotating the planet through, say, 90 degrees, so the mountain sits on the Equator. Has anything changed? All the arguments about the maximum height of the mountain still apply, because the gravitational force at the surface depends only upon the radius and mass of the planet СКАЧАТЬ