Units, prefixes and standard form in physics

In nineteen eighty-three, a passenger jet over Canada ran out of fuel at forty-one thousand feet, and both of its engines stopped. The fuel had been worked out in pounds, but that aircraft was built to work in kilograms. It took off with less than half the fuel its crew believed was on board. The pilots glided it down onto an old airfield, and everyone on board survived. Every multiplication in the fuel sum was right. The unit was wrong.

Every measurement in physics comes in three parts. The quantity is what you measured. The unit is what you measured it in, and the symbol is the unit written short. Take mass. The quantity is mass, the unit is the kilogram, and the symbol is a small k and a small g. Scientists everywhere use one agreed set of units, called SI, short for the International System of Units. Length goes in metres, mass in kilograms, and time in seconds. Electric current goes in amperes, and temperature in kelvin. Those five are the base units you need, and every other unit in this course is built from them, and they serve combined science and triple physics students, Foundation and Higher, on every board. Force is measured in newtons, energy in joules, power in watts, and voltage in volts. Each of those is built out of the five base units. A built unit tells you what it means. One watt is one joule of energy every second. A two thousand four hundred watt kettle hands over two thousand four hundred joules each second it runs. That is why SI matters. The equations in this course are built so that SI numbers going in give an SI answer coming out, with no extra factor to remember. Sixty joules in two seconds: is that lamp thirty, sixty or a hundred and twenty watts? Thirty watts. Sixty joules shared over two seconds is thirty joules for each second, and a watt is one joule per second.

One SI base unit is not a plain word, and it is the kilogram. Kilo means a thousand, so a kilogram is a thousand grams, with the thousand built into its name. That makes the gram a unit that is not SI. Two hundred and fifty grams has to become nought point two five kilograms before it goes near an equation. Everyday life is full of units like that. Clocks run in minutes and hours, rulers in centimetres, and kitchen scales in grams. Physics wants seconds, metres and kilograms. Which one is already SI: two hundred and fifty grams, three minutes, or two kilograms? Two kilograms. The kilogram is the SI unit of mass, kilo and all. The grams and the minutes both need converting first. Time is the conversion you will make most often. A minute is sixty seconds, and an hour is sixty minutes. That makes an hour sixty times sixty, which is three thousand six hundred seconds. Three minutes is three times sixty, which is one hundred and eighty seconds.

A prefix is a word on the front of a unit that makes it bigger or smaller by a power of ten. Kilo is the one you already know. It means a thousand, ten to the three. Going up from kilo, mega means a million, ten to the six. Giga means a thousand million, ten to the nine. Tera means a million million, ten to the twelve. Going down, centi means a hundredth, ten to the minus two. Milli means a thousandth, ten to the minus three. Micro means a millionth, ten to the minus six. Nano means a thousand-millionth, ten to the minus nine. Those eight prefixes are all a GCSE paper uses. To take a prefix off, you multiply by its power of ten. Two point four kilowatts is two point four times a thousand, which is two thousand four hundred watts. Now use those prefixes on a length. What is fifteen centimetres in metres? Nought point one five metres. Centi means a hundredth, so this is fifteen hundredths of a metre. Here is the habit this video is built around. Before any sum, write a CONVERT line: the quantity, the number as the question gives it, an arrow, and the number in SI. Take a kettle labelled two point four kilowatts, switched on for three minutes. Power: two point four kilowatts, arrow, two thousand four hundred watts. Time: three minutes, arrow, one hundred and eighty seconds. Now the sum can happen. A watt is a joule every second, so the kettle gives two thousand four hundred joules for each of one hundred and eighty seconds. Two thousand four hundred times one hundred and eighty is four hundred and thirty-two thousand joules. Every number in that sum came off the CONVERT line, not off the question. Your handle for this video: the equation only gets to exist after the arrow has been drawn. A radio station broadcasts at ninety-five point eight megahertz. Hertz is the unit of frequency, and one hertz means one wave every second. What is ninety-five point eight megahertz in hertz? Write its CONVERT line. Mega is a million, so multiply by a million, and the decimal point moves six places right. Ninety-five point eight megahertz, arrow, ninety-five million eight hundred thousand hertz.

Converting is not tidying up at the end, and two appliances show why. A phone charger is labelled five watts and runs for thirty minutes. A hairdryer is labelled two kilowatts and runs for three minutes. Someone multiplies the numbers exactly as they are printed. Five times thirty gives one hundred and fifty for the charger. Two times three gives six for the hairdryer. Their verdict is that the charger transferred twenty-five times more energy. That verdict is wrong. What is wrong with the comparison? The comparison fails because the numbers are in different units, watts with minutes against kilowatts with minutes. Converting afterwards cannot rescue it, because nobody wrote the units down. Convert first instead. Charger: thirty minutes, arrow, one thousand eight hundred seconds. Hairdryer: two kilowatts, arrow, two thousand watts, and three minutes, arrow, one hundred and eighty seconds. The charger gives five times one thousand eight hundred, which is nine thousand joules. The hairdryer gives two thousand times one hundred and eighty. That comes to three hundred and sixty thousand joules. The hairdryer transferred forty times more energy, not twenty-five times less. Same numbers, same arithmetic, and the only change was converting before the sum. One examiner's report on a Higher paper says this. The errors that students made in calculations were common errors, usually involving incorrect unit conversions or failing to convert units. In plain words, the method was right and the units were not. The fix is the CONVERT line, written before the sum, so every number is in SI before it is used.

Physics numbers get very big and very small, so they are written in standard form. That means a number from one up to ten, multiplied by a power of ten. Take the radio frequency, ninety-five million eight hundred thousand hertz. Put the decimal point straight after the nine, and count the places it moved. It moved seven. In standard form, that is nine point five eight times ten to the seven hertz. The unit rides along on the end, every time. Significant figures are the digits that carry meaning, counted from the first digit that is not zero. You round to them once, at the very end of a calculation. Suppose a calculator's display reads four point six eight seven five, times ten to the minus three, for a current in amperes. The question wants two significant figures. Which is right: four point seven amperes, or four point seven times ten to the minus three? Four point seven times ten to the minus three amperes. Only the front number, called the mantissa, gets rounded. The power of ten stays exactly where it was. Write four point seven amperes on its own, and the answer is a thousand times too big. A report on a different Higher paper notes this. Many students struggled to give the answer to two significant figures, perhaps caused by students not understanding how their calculators were giving the answer in standard form. The display had already put the answer in standard form, and the power of ten got left behind. The habit that fixes it is to say the whole number aloud, power of ten included, before you round.

Some questions want the answer out of SI, in a prefixed unit or in hours. That rounded current is a good place to start. Milli means ten to the minus three. So four point seven times ten to the minus three amperes is four point seven milliamperes, and the power of ten has turned into the prefix. Here's a different kind of return trip. A calculation's answer comes out in seconds, but the question asks for it in hours. Five thousand four hundred seconds into hours: multiply or divide, and why? Divide, by the three thousand six hundred seconds in one hour. An hour is a far bigger unit than a second, so the same time needs far fewer of them. That gives one and a half hours.

Three quick questions to finish, each one on a number this video has not used yet. A bag of flour holds five hundred grams. What is that in SI units? Nought point five kilograms. Grams are not SI, and a kilogram is a thousand grams, so five hundred divided by a thousand. Next one. A twenty-five minute bus journey: how many seconds is that? One thousand five hundred seconds. That is twenty-five minutes, times sixty seconds in each minute. This last one stretches further. Green light has a wavelength, the length of one wave, of five hundred and fifty nanometres. What is five hundred and fifty nanometres in metres, in standard form? Five point five times ten to the minus seven metres. Nano is ten to the minus nine, so that is five hundred and fifty times ten to the minus nine. Moving the point two places left makes the front number five point five, and the power goes up by two, to minus seven. Every one of those started the same way, with an arrow. The equation only gets to exist after the arrow has been drawn. That jet over Canada had its sum done right and its unit wrong. One CONVERT line, pounds to kilograms, and it would have left the ground with the fuel it needed.

This is the first video in Units and describing motion, so there is nothing behind it yet. Give it a thumbs up once you are comfortable with it, and any video without one is your list to come back to. Not sure yet? Save it for later; conversions start to feel normal about three questions in.

Next in the chapter: Scalars and vectors, where a number on its own stops being enough.

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