Measurement

Dimensions and Units

But, before we can do mechanics, we need to discuss measurement. In order to make measurements, we need three fundamental quantities: Length, Time and Mass. These are known as dimensions – if you measure how far it is from here to there, you have measured a length (often denoted as L). If you measure an area, ignoring any constants due to the exact shape of the area, that is the product of two lengths – area has the dimensions of L². Similarly volume is L³.

It is also possible to construct quantities that have more complicated dimensions. For example, speed can be thought as “how far” divided by “how long” or length/time = LT^-1.

Length as a dimension has no quantitative element. We can try to measure a length as much as we like, but without a definition of how much length is “1”, we cannot really get any useful information. So, we need to define units for length, time and mass.

It is important to have “good” units. The units should be precisely defined, unvarying in time, available to everyone. The length of a specific piece of chalk, for example, is a very poor measurement, because:

a) it is not precisely defined – what length are we talking about? The chalk might have rounded edges so it is not unambiguous.

b) the definition of length will change with time – as the chalk gets handed around and measured with, more and more of it will naturally wear off the end, making the unit of length shorter!

c) The chalk is not easily accessible to everyone – there is only of this piece of chalk, so it would need to be handed from person to person so that people could make measurements –no one would get anything done.

Now, we can play with these problems, making some better at the expense of the others – for example, we could define the unit of length as the length of any piece of chalk, whatever piece you had with you. This would alleviate problems b and c at the expense of worsening the precision of the unit. This situation held for the vast majority of human history, where units of measurement were based on things like body parts – a foot really was the length of your foot, a cubit was the distance from your elbow to the tip of your longest finger. Although these rulers were ubiquitous, they were far from precise – most tall men would measure a cubit larger than a short man.

In physics we tend to use the SI units as out standard (also known as mks units), although the cgs system is also common.

Length: the meter (m) defined as the distance traversed by a wave of light in a vacuum over 1/299 792 458 s (This definition fixes the speed of light at exactly 299 792 458 m/s)

Time: the second (s) defined as 9 192 631 770 periods of the radiation emitted by Cesium.

Mass: 1 kg defined as the mass of a specific block of platinum kept in a vault in Paris.

The unit of mass is not a great definition as the block is kept locked away (for its protection).

More can be read on the establishment of the SI units in Peter Galison’s Book "Einstein's Clocks, Poincaré's Maps: Empires of Time".

Other quantities we might compute are based upon these fundamental units: speed, as stated above is a distance divided time – in the SI units, we measure speed in m/s. All quantities can be constructed this way – with dimensions L^aM^bT^c.

Eg: force ~ MLT^-2 (as we’ll see), which in SI units is kgm/s², but this is a cumbersome unit to carry around in the many places that forces are used in physics, so as a form of shorthand (and homage to a great mind) we refer to the unit of force as the newton (N).

Units named after people are, by convention, not capitalized when written out in full (to differentiate it from the person’s name), but when written as an initial, are capitalized.

Dimensions themselves are powerful tools to discover formula through dimensional analysis and to check the validity of equations.