units of length, area, volume, mass and weight

7/30/2019 Units of length, area, volume, mass and weight

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Measurement

We seem to live in an eleven-dimensional space, most of whose dimensions are folded up into

something called the Calabi-Yau Manifold. Here is a section of a quintic Calabi-Yau threefoldprojected into three-dimensional space (though obviously then projected onto the virtuallytwo-dimensional surface of this monitor or piece of paper):

According to Doctor Who, various beings live in Calabi-Yau space, including the Guardians ofTime, Chronovores, the Great Old Ones (for example Nyarlathotep and Cthulhu) and I reckon

also the reapers and so on. I also sometimes wonder if an ex-friend of mine belongs there.

However, all of this can be safely ignored if you consider yourselves to be entities consisting ofa single world-line existing in space and time and having finite mass, as I expect you do. As far

as we're concerned for the purposes of this document, there are three dimensions of space,one of time and one of mass, and these are the things I'm going to talk about here.

Everything in a small region of space can be pretty accurately located at a particular moment

using three numbers to describe its position. For instance, my head is currently about a metrefrom the wall to my left, a metre and a half from the floor and three metres from the French

windows behind me. The fact that I only need three numbers to describe where my head is.


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Consider this humble toilet roll:

This has a location within this room which can be described using those three numbers, usingthe X, Y and Z axes:


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If you wanted to tell someone where that toilet roll was, you would only need three numbersto do it, and those numbers would represent measurements along those three axes.

The simplest measurement to describe is probably length. The metric system uses a unit

called the metre (often written as m), to measure length. This was originally defined asfollows. Here's Earth:


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If you imagine a line like this:

drawn from the North Pole to the Equator through Calais in Artois, France, it will be exactly 10

000 kilometres long. This is because, just after the French Revolution, a metre was defined asa ten millionth of the distance between the North Pole and the Equator along a line which

passes through Calais. Nowadays, this isn't considered accurate enough so they use aparticular colour of light and count the number of waves in it instead. This older

measurement varies quite a bit anyway because of things like rocks expanding in summer andcontracting in winter.

If this document is on a piece of paper, that paper will be 0.211 metres wide (211 millimetres

or 21.1 centimetres) and 0.297 metres high (297 millimetres or 29.7 centimetres). Therefore,a metre is about three and a third A4 pieces of paper long. That means that if you started at

the equator with a large stack of A4 sheets of waterproof paper and put them end to end fromthere to the North Pole, you would need 33670033 and two-thirds of them. That would make

a pile 3367 metres high, which is not actually that much if you think about it in terms of shelf


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space in libraries, bookshops and so on.

Many other units in the metric system are defined using the metre. Area is length times depth,

or breadth, or height, and so on. It can be measured in square metres. If this is a piece ofpaper, it has an area of 62667 square millimetres, which can be written as 62667 mm 2. This is

the same as 626.67 cm2 or 0.62667 m2, (square metres). I will come back to the issue of how

many pieces of A4 paper would be needed to cover this planet entirely, because it's not simple.

The metric system is also known as the Systme International, SI for short.

The official SI unit of area is the are, which is a hundred square metres. That would be the

area of a square ten metres on a side, or nearly sixteen hundred sheets of A4. However, the areitself is rarely used as a unit of area and it's much more common to use the hectare, which is a

hundred times bigger. This is the unit of area used to measure things like fields and floorspacein large buildings. A hectare is ten thousand square metres, so a square a hundred metres on a

side would have an area of one hectare. This is very close to the area of Trafalgar Square:

Volume is how big something is. A cube has a volume of the length of one of its edges

multiplied by itself, then multiplied by itself again. A cubic metre can be written as m3, i.e.with a 3.

Once again, the SI unit of volume is not the cubic metre but the litre, although science often

uses the term cubic decimetre for this. A litre, or cubic decimetre (dm3), is the volume of acube with an edge measuring ten centimetres or one decimetre (dm), a tenth of a metre.

Then there's mass. Mass is the quantity of a substance, which is different than its size. Forinstance, a litre of outer space is quite likely not to contain anything at all but is still a litre involume. The SI unit of mass is the kilogramme, which is the mass of a litre of distilled water at

4C, the maximum density of water. The base unit, however, is a thousandth of that thegramme, or gram. A million grammes, rather than being called a megagramme, is referred

to as a tonne.

Weight is not the same as mass, although for masses at rest on the surface of the ocean thedifference between the two concepts would be absolutely minute. Weight is the force on an

object due to gravity. It is related to mass and when people say weight they are usuallyreferring to mass by the wrong word. To illustrate the difference, here is a picture of Neil

Armstrong:


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In this picture, Mr Armstrong has a mass of 77 kilogrammes. Here is a picture of Neil

Armstrong on the Moon:


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In this picture, minus his spacesuit, Neil Armstrong has the same mass as he had in the first

picture, but he weighs much less.

Weight is measured in newtons. Actually, weight is not measured in newtons very muchbecause nearly everyone usually uses units of mass to describe weight. A newton is how much

force it takes to accelerate a kilogramme by one metre per second per second. In the first

picture, the entire mass of our planet is pulling Mr Armstrong towards it with a force of about754 newtons, but in the second, he is being pulled towards the centre of the Moon with a forceof 126 newtons because the gravity of the Moon is only 1/6 of ours. However, his mass, which

is expressed in kilogrammes, is the same 77 kilogrammes.

Formulae for volume

The simplest shape to work out the volume of is the cube:

A B

This is easy because its volume is simply the cube of the length of one edge. If you call one

edge AB and assume it's 3 cm long, the volume of this cube is (AB)x(AB)x(AB)=AB3, or in thiscase 3x3x3=27 cm3.

Just slightly more complicated is the cuboid. Cuboids have the same number of faces, edges,

corners and angles as cubes but at least two of those faces are proper rectangles rather thansquares. This is a cuboid:

and this is another one:


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Since cuboids have up to three different lengths of edge, their volume is xyz, with each letter

representing the length of an edge. For instance, this box:

has dimensions of 6x9x10.5 cm, and therefore x=6 cm, y=9 cm and z=10.5 cm, giving it avolume of 6x9x10.5 = 567 cm3, or just over half a litre.

While we're at the simple stage, I just want to mention surface area. Going back to the cube

example, which was 3x3x3 cm, it consists of six square faces with edges three centimetreslong. Since the area of a square is the same as the length of its edge multiplied by itself, in this

case 9 cm2, and it has six faces, it has a total surface area of 9x6, or 54 cm 2.

There are also formulae for other shapes with only straight edges, but these are not so widelyused.

PART 2 TO FOLLOW SHORTLY

units of length, area, volume, mass and weight

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