# Measurements

We often take for granted the availability of measuring instruments. We know that there are slight differences between them: for example, two different bathroom scales may vary by as much as half a kilogram or more. When weighing ourselves, that is probably accurate enough, but if it were measuring out the amount of sugar for a cake, the variation of 0.5 kg would be disastrous. We need greater accuracy for certain applications.

And who says what a kilogram is, anyway? Or a second, or a metre? When we buy a watch, how do we know the manufacturer is using exactly the same second as another manufacturer? What if one watch is made in China and another in Switzerland? How can we be sure the two countries are using the same standard time interval for the second?

## Units and Measures

The S.I. kilogram and metre. The kilogram is still measured as an artifact, kept at the BIPM, in Paris, but the other six base units are measured against natural phenomena.

The history of units and measures follows the path of changing needs and technological capabilities. As trade increased and spread further afield, the need for a universal system for measuring weight and length became essential for economic growth. If traders could not rely on their goods matching orders, they would lose customers. With the Industrial Revolution came mass production, and a global trade system, and the need for a single system of standardised measures was paramount.

The S.I. International System of Units and Measures has been accepted by the UN and all scientific organisations as the world's standard for measurements. This system specifies seven base units, as well as many other units derived from them. The base unit of length is the metre, for time the second, and for weight the kilogram. These units are now used everywhere in the world, except three countries: Liberia, Myanmar, and the USA.

The S.I. International System of Units and Measures also uses a series of S.I. Prefixes, which make it easy to do mathematics with measurements.

# Scientific Notation

All data in science should use the system of scientific notation which makes the number of significant figures clear. e.g. 300 is ambiguous: does it mean exactly 300, or between 299 and 301, or even something between 290 and 310? In scientific notation, the degree of known precision is always given by the number of significant figures. For example:

• Roughly 300 (between 250-349) is \$3 x 10^2\$,
• Close to 300 (between 290-310) is \$3.0 x 10^2\$,
• Precisely 300 (between 299-301) is \$3.00 x 10^2\$,
• Very precisely 300 (between 299.9-300.1) is \$3.000 x 10^2\$.

A rule of thumb is 3 significant figures, wherever possible.

# Errors in Measurements

There has never been a measurement that had no error! Errors can arise by mistakes made by the measurer or the instrument is not used properly, or is not set up well.

In science, it is very important that errors are treated carefully. If they are not reported with the measurement, the answer to a calculation can be wrong, and there is no way of knowing by how much.

Errors fall into two general categories: random and systematic:

## Random Errors

These are errors which occur with no particular pattern. They do not repeat in a regular way, and may be identified readily from a graph of results, since they stand out from the general trend of a curve.

Random errors are typically both imprecise and inaccurate. Repeating the measurement should allow the random error to be identified and eliminated from the dataset. If it repeats, it is more likely to be a systematic error.

One statistical method to eliminate random errors which may distort a dataset is to remove the top and bottom ten-percentile. This means only the data from 10 - 90% of results are used, on the assumption that the random errors which will distort the dataset will occur outside this general range.

## Systematic Errors

A systematic error occurs when there may be high precision in measurement but nevertheless low accuracy.

Precision is how well a measurement is taken, and how close to that measurement every subsequent measurement with the same instruments and conditions is.

Even though the precision may be good, the accuracy to the actual value may be poor. For example, an instrument that has been poorly calibrated will always return a value that is incorrect by the same amount, and no amount of care in the measurement technique will eliminate that error in accuracy. This is a systematic error.

An illustration is an archer who is very precise. On a still (no wind) day, he can hit the bullseye every time. However, if there is a wind, and he does not adjust for it, his precision will cause a tight cluster of arrows all a certain distance off target.

Random error is what it sounds like - each measurement that is significantly far from most other measurements is subject to some erratic error, and will not repeat. On a graph of the results, random errors are easy to identify as they stand out on their own. Systematic errors tend to shift the entire results curve a certain regular distance from the true results.

### Measuring position, speed and velocity with a motion sensor

A motion sensor is a useful and precise instrument for measuring position and motion of objects moving between a few centimetres to about 2m away. An acoustic sensor, like the one illustrated here, uses the echo of an acoustic signal to calculate the position of any object in the direction of its transmitter/receiver.

Some brands may be fitted to a special reader, or directly to a computer via a USN connector. Software provides an interface, and may detect the sensor type automatically.

Parameters, such as the length of recording, may be set. To start recording press the launch or start button. The data may be displayed as a table or a graph.

This motion graph was produced when the sensor was allowed to swing freely towards and away from a wall.

The default setting for duration is 5 seconds, but you may change this to whatever suits the experiment you are conducting.

On the left of this graph you will see the actual data points. You may export these data to an Excel file for further analysis and graphing.

Motion sensor readout: a falling object produces a data table and graph of poistion and velocity against time

## Density

Density (ρ) = mass/volume

Density is a measure of how much volume a substance requires for a given amount of it (mass). Its units can be g/cm3 (or mL), kg/L, or t/m3.

Water has a density normally of very close to 1.0 kg/L. This is because when the kilogram was invented, it was given the value of the mass of 1L of water. An object will float if its density is less than that of water. Since salt increases the density of water, objects float more easily in salt water than pure water.

Filling cavities in a heavy material with air is an easy way to reduce its density, and permit it to float. That is why it is possible for ships weighing thousands of tonnes to float. Iron has a density 8 times that of water, so there must be at least 8 times as much air as steel in the ship for it to float.

Volume can be measured by displacement of water. This can be done more accurately using a discharge tube, which allows the displaced water to flow out to be measured.

The volumes of regular geometric shapes can be calculated from the following formulae:

Cube: \$l^3\$

Prism: \$l ⋅ w ⋅ h\$

Sphere: \$4/3 πr^3\$

Cylinder: \$πr^2⋅h\$

Cone and pyramid: \$1/3 πr^2⋅h\$

where l = side length, w = width, h = height, r = radius

Materials vary greatly in their densities.

The average density of the planet is 5.515 tonnes per cubic metre, more than five times that of water, explaining why water rises to the surface to form oceans.

## Other densities

These are some densities of metals:

 Lead (Pb) 11.3 g/cm3 Copper (Cu) 8.96 g/cm3 Aluminium (Al) 2.70 g/cm3 Iron (Fe) 7.9 g/cm3 Magnesium (Mg) 1.74 g/cm3 Uranium (U) 19.1 g/cm3 Tungsten (W) 19.25 g/cm3 Gold (Au) 19.3 g/cm3

### Archimedes and the Eureka Moment

Archimedes, the Greek scientist and inventor, is best known for the anecdote known as his Eureka moment in the bath, in which he famously ran down the street naked crying 'Eureka!' (I have found it!). This was in reference to his realisation that volume could be determined by displacement of water, thus allowing him to solve a puzzle set to him by the king of Syracuse.

Content © Andrew Bone. All rights reserved. Created : July 1, 2013 Last updated :March 5, 2016

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