Wonders of the Universe - Brian Cox [63]
Towering over every other mountain in the Solar System is the extinct volcano, Olympus Mons. Rising to an altitude of around 24 kilometres (15 miles), it is almost the height of three Mount Everests stacked on top of each other. The fact that a smaller planet has higher mountains is not a coincidence; it is partly down to environmental factors such as the rate of erosion and the details of the planet’s geological past, but there is also a fundamental limit to the height of mountains on any given planet: the strength of its surface gravity. Mars has a radius approximately half that of Earth’s, and since it is only 10 per cent as massive, a little calculation using Newton’s equation will tell you that the strength of the gravitational pull at its surface is approximately 40 per cent of that on our planet. This changes everything’s weight.
Here on Earth we don’t often think about the difference between mass and weight, but the distinction is very real. The mass of something is an intrinsic property of that thing – it is a measure of how much stuff the thing is made of. This doesn’t change, no matter where in the Universe the thing is placed. In Einstein’s Theory of Special Relativity, the rest mass of an object is an invariant quantity, which means that everyone in the Universe, no matter where they are or how they are moving, would measure the same value for the rest mass.
Weight is different. For one thing, it is not measured in kilogrammes, it is measured in the units of force – newtons. This is easy to understand if you think about how you would measure your weight. When you stand on bathroom scales, they measure the force being exerted on them by you; you can see this by pressing down on them – the harder you push, the greater the weight reading. The force you are exerting on the scales is in turn dependent on the strength of Earth’s gravity. This should be obvious; if I had taken the scales up in the Vomit Comet and tried to stand on them, they wouldn’t have read anything because I would have been floating above them – hence the word ‘weightless’. In symbols, the weight of something on Earth is defined as:
The immense Olympus Mons can exist on Mars because the planet has 40 per cent of Earth’s gravitational pull. However, move this extinct volcano to our planet and it would sink into the ground because of its enormous weight.
W is weight, m is the thing’s mass, and g is the familiar measure of Earth’s gravitational field strength – 9.81 m/s2 – with a couple of caveats that we’ll get to below! (For absolute accuracy, the correct definition of weight is; the force that is applied on you by the scales to give you an acceleration equal to the local acceleration due to gravity – i.e. the force the scales exert on you to stop you falling through them.) So, here on Earth a human being with a mass of 80kg weighs 785 newtons; on Mars, the same 80-kg person would weigh approximately 295 newtons.
So your weight depends on a few things; one is your mass, another is the mass of the planet you are on. Your weight would also change if you were accelerating when you measured it, which is another manifestation of the equivalence principle. So, if you took Olympus Mons and stuck it on Earth, then as well as dwarfing every other mountain on the planet, it would also weigh around two and a half times as much as it does on Mars. This enormous force would put its base rock under such intense pressure that it would be unable to support the mountain, so it would sink into the ground. A planet the size of ours cannot sustain a mountain the size of Olympus Mons – it would weigh too much. The highest mountain on Earth, as measured from its base, is Mauna Kea, the vast dormant volcano on Hawaii. It is over one kilometre (half a mile) higher than Everest, and it is gradually sinking. So Mauna Kea is as high as a mountain can be on our planet, and this absolute limit is set by the strength of our gravity.
The definition of weight can get a bit convoluted,