By the end of this lesson you will be able to
- explain why a number without a unit is not an answer, and name the units this course uses
- convert between units with the prefixes milli and kilo, and between minutes and hours or seconds
- rearrange a formula of the form a = b × c to find either of the other two quantities
- round an answer sensibly and check that it makes sense before you trust it
- read a value from a data-sheet table and type an answer the way the course's number exercises expect
Margaret Ellis hands you a stores order for Bramble's spare parts. One line says “driver standby current 350” and another says “cable length 40 cm”, while the data sheet uses amperes and metres. “Before you order anything,” she says, “make every figure use the same units, and check that each one makes sense.”
Why units matter
A number on its own says very little. If Henry tells you a bracket is 5 long, do you take it as 5 millimetres, 5 centimetres or 5 metres? The differences are enormous. A unit is the agreed size of one step of a measurement, such as one metre or one kilogram. A quantity is a number and a unit together. Mixing units, or dropping them, has caused costly failures in real engineering, and a wrong unit gives a wrong answer even when the arithmetic is perfect.
So the first habit of a technician is to write the unit with every number, and to check the unit before using it. This course uses the metric system throughout, the system in which units scale by tens, hundreds and thousands.
The units you will use
Here are the units this course uses. Do not try to memorise them now. You will meet each again, in its own lesson, when it is needed.
| Quantity | Unit | Symbol | What it tells you |
|---|---|---|---|
| Length | metre | m | How long. Bramble's thigh is 0.40 m. |
| Mass | kilogram | kg | How much matter there is. Bramble is 70 kg. (Lesson 2.1 separates mass from weight.) |
| Time | second | s | How long something lasts. |
| Force | newton | N | A push or a pull. A small apple resting on your hand presses down with about 1 N. |
| Torque | newton metre | N m | A twisting force, such as a motor turning a joint. |
| Electrical push | volt | V | How hard the battery pushes electricity round a circuit. |
| Electrical flow | ampere (amp) | A | How much electricity flows. |
| Electrical resistance | ohm | Ω | How much a part opposes the flow. |
| Power | watt | W | How fast energy is being used. |
| Energy | watt-hour | Wh | An amount of energy. Bramble's battery holds about 888 Wh. |
| Temperature | degree Celsius | °C | How hot. The workshop is at 22 °C. |
| Turning speed | revolutions per minute | rpm | Complete turns each minute. Bramble's fans turn at about 4000 rpm. |
| Frequency | hertz | Hz | Repeats each second. Bramble's cameras take 30 pictures a second, which is 30 Hz. |
The units for electricity (volt, ampere, ohm, watt) are explained properly in Module 3. For now it is enough to know that each is a different kind of quantity, and that they cannot be swapped for one another.
Prefixes: milli, kilo and mega
Real quantities are often much smaller or larger than one basic unit, so we attach a prefix to the unit. A prefix is a short word that scales a unit by a fixed number.
| Prefix | Symbol | Meaning | Example |
|---|---|---|---|
| milli | m | one thousandth (÷ 1000) | 1 mm = 0.001 m, and 1 mA = 0.001 A |
| kilo | k | one thousand (× 1000) | 1 kg = 1000 g, and 1 kW = 1000 W |
| mega | M | one million (× 1,000,000) | 1 MΩ = 1,000,000 Ω |
Two other conversions turn up constantly. There are 100 centimetres (cm) in a metre, and so 10 millimetres in a centimetre. And there are 60 minutes in an hour and 60 seconds in a minute.
Converting from one unit to another
To convert, decide first whether the new unit is bigger or smaller than the old one. If the new unit is bigger, you need fewer of them, so the number gets smaller. If the new unit is smaller, you need more of them, so the number gets larger. Then multiply or divide by the right amount, and check the direction.
1. Bramble's forearm is 0.26 m. A millimetre is smaller than a metre, so the number gets larger: 0.26 × 1000 = 260 mm.
2. A driver board draws 350 mA. An ampere is bigger than a milliampere, so the number gets smaller: 350 ÷ 1000 = 0.35 A.
3. A parts tray holds 2.5 kg. A gram is smaller than a kilogram: 2.5 × 1000 = 2500 g.
4. A test lasts 45 minutes. An hour is bigger than a minute: 45 ÷ 60 = 0.75 h.
Try a few conversions yourself below. The explorer changes the same quantity into other units as you move each slider.
Rearranging a formula
A formula is a rule that links quantities. Many formulas in this course have the same shape: one quantity equals two others multiplied together. Write this as a = b × c. If you know any two of the three, you can find the third, by rearranging the formula, which means changing it so that the unknown stands alone.
- To find a, multiply: a = b × c.
- To find b, divide a by c: b = a ÷ c.
- To find c, divide a by b: c = a ÷ b.
The number of turns equals the turning speed multiplied by the time: turns = speed × time. Here a = turns, b = speed in rpm (turns per minute) and c = time in minutes.
A joint turns at 60 rpm for 5 minutes: turns = 60 × 5 = 300 turns.
How long does it take to make 300 turns at 60 rpm? Time = turns ÷ speed = 300 ÷ 60 = 5 minutes.
At what speed does a joint make 300 turns in 5 minutes? Speed = turns ÷ time = 300 ÷ 5 = 60 rpm.
Notice that the units agree: rpm means turns per minute, so it goes with time in minutes. If the time had been in seconds, you would convert it first.
Rounding and significant figures
A calculator gives many digits, but most of them mean nothing. The significant figures of a number are its meaningful digits, counted from the first digit that is not zero. The number 0.26 has two significant figures, and 260 measured to the nearest 10 also has two. Three rules are enough for this course.
- Do not round in the middle of a calculation. Keep several extra digits, and round once at the end.
- Give the answer with about the same number of significant figures as the least exact number you started with. Two or three is usually right.
- When an exercise says how many decimal places to give, follow it. If it does not say, give a sensible answer, and do not worry about small rounding differences: the checker accepts a small margin.
An example: Bramble's battery is 6 kg of its 70 kg. The fraction is 6 ÷ 70 = 0.085714..., which is 8.5714... per cent. Writing 8.5714285 per cent would suggest a precision that nobody measured. “About 8.6 per cent”, or even “about 9 per cent”, is the honest answer.
Estimating and checking that an answer makes sense
A calculator will give you a wrong answer as confidently as a right one, if you type the wrong thing. So learn to estimate: work out roughly what the answer should be, using rounder numbers, before or after the exact sum. Then compare.
Take the fan speed of about 4000 rpm. How many turns is that each second? The exact sum is 4000 ÷ 60 = 66.7. A quick estimate: 60 lies between 50 and 80. Since 4000 ÷ 50 = 80 and 4000 ÷ 80 = 50, the answer must lie between 50 and 80. If your answer came out as 240,000 or as 0.015, the size alone tells you something has gone wrong: you multiplied by 60 instead of dividing, or turned the division upside down.
Ask yourself four quick questions each time.
- Does the unit belong to the kind of quantity I want? (A torque cannot be in volts.)
- Is the size believable? Bramble's forearm is 0.26 m, so a result of 26 m or 0.026 m is wrong. A torque of 0.002 N m for a knee that supports a 70 kg robot is nonsense too: even the knee motor alone is rated 1.6 N m, which is far more.
- Did the number move the right way when I converted? Going to a bigger unit must make it smaller.
- Are my units consistent? For example, rpm goes with a time in minutes, not seconds.
Common mistake: converting the number but not the unit
Writing “260 m” when you meant 260 mm is the same mistake as forgetting to convert at all. When you change a number, change the unit label with it, and read the whole thing back: “260 millimetres”.
Reading a data-sheet table
Bramble's data sheet lists its joints in a table. The way to read a table is always the same: find the row you want, find the column you want, read the value where they cross, and check the unit in the column heading. Here is part of it.
| Joint (each) | Motor rated torque (N m) | Motor rated speed (rpm) |
|---|---|---|
| shoulder | 1.0 | 3000 |
| elbow | 0.6 | 3000 |
| knee | 1.6 | 3000 |
| hand | 0.10 | 3000 |
The knee's motor rated torque is 1.6 N m: knee row, torque column, and the unit N m is in the heading. The rated speed of the shoulder's motor is 3000 rpm. (The words “rated torque” and their meaning come in later lessons. For now you are practising reading.)
How the number exercises work
In a number exercise you type only the number, without the unit. If the answer is 0.26 m, type 0.26. The exercise tells you the unit to work in, and how many decimal places to give when it matters. The checker accepts a small margin for rounding. If you fall into a common trap, such as forgetting a conversion, a hint explains the slip without giving you the number. You can also ask for the worked answer at any time.
Exercises
Give each number answer without its unit, using the unit the task names. Each exercise is checked when you press Check my answer.
Exercise 1 · Metres to millimetres
Bramble's forearm is 0.26 m long. What is that in millimetres?
Exercise 2 · Which unit fits?
Each line below is a quantity from Bramble's data sheet. Choose the unit that fits it.
Exercise 3 · Do these log entries make sense?
Thomas Hale's old log has several entries. Using Bramble's data sheet, choose every entry that is believable.
Exercise 4 · How long for the turns?
A joint turns at 60 rpm. Turns = speed × time, so time = turns ÷ speed. How many minutes does it take to make 450 turns?
Exercise 5 · Read the table, then convert
Use the data-sheet table in this lesson. The knee's motor turns at its rated speed for 90 seconds. How many turns does the motor make? Change the time to minutes first, and give the answer as a whole number.
Quick check
Four questions to confirm the main ideas. Choose an answer to see the explanation.
A stores list says a driver board draws “350” in its standby column. What is the real problem with that entry?
Without a unit the number cannot be used: 350 mA is 0.35 A, and 350 A would be a thousand times more current. Rounding and fractions do not fix a missing unit.
You convert 0.26 m to millimetres. Which statement about the result is correct?
A millimetre is one thousandth of a metre, so it takes 1000 of them to make a metre, and the number grows to 260. If the number had shrunk, you would know to look for a slip.
You know a = b × c, and you know a and c. Which is the correct way to find b?
Dividing both sides of a = b × c by c leaves b = a ÷ c. Multiplying, or dividing the wrong way round, gives a number of the wrong size.
A colleague's calculation gives Bramble's forearm length as 26 m. What should you do?
A forearm of 0.26 m cannot be 26 m, so the size test shows a slip, most likely a wrong conversion. Only changing the label to millimetres would give 26 mm, which is also wrong.
Summary
- A number without a unit is not an answer. Write the unit with every quantity and check it before you use it.
- Prefixes scale units: milli is one thousandth, kilo is one thousand, mega is one million. A bigger unit makes the number smaller, and a smaller unit makes it larger.
- For a = b × c, find b with a ÷ c and find c with a ÷ b. Keep units consistent, for example rpm goes with minutes.
- Round once at the end, to a sensible number of significant figures.
- Estimate first and check that each answer is the right size and moves the right way. A knee torque of 0.002 N m is nonsense.
Found a mistake on this page, or something unclear? Report a problem and mention “Hardware Lesson 1.2”.