This note covers how physical quantities are measured and written down: SI units and prefixes, the instruments used for length, time and volume, and the difference between scalars and vectors. Every later topic depends on measuring accurately and using units correctly, so it is worth getting these habits right from the start.
Key ideas
A physical quantity is something you can measure, such as length, mass, time or temperature. A measurement always has a number and a unit. "12" means nothing in physics, but "12 m" does.
The base quantities you need are length (metre, m), mass (kilogram, kg), time (second, s) and temperature (kelvin, K, or degrees Celsius, °C). Other units, such as the newton or the joule, are built from these.
Prefixes
Prefixes make very large or very small numbers easier to write.
- kilo (k) = 103, mega (M) = 106, giga (G) = 109
- centi (c) = 10⁻2, milli (m) = 10⁻3, micro (μ) = 10⁻6, nano (n) = 10⁻9
To convert, replace the prefix with its power of ten. For example, 2.5 km = 2.5 × 103 m = 2500 m, and 350 mm = 350 × 10⁻3 m = 0.350 m.
Measuring length and volume
- A ruler measures to the nearest millimetre. A tape measure is used for longer distances.
- A micrometer screw gauge measures small thicknesses, such as a wire, to 0.01 mm. You read whole millimetres on the sleeve and hundredths of a millimetre on the rotating thimble, then add them.
- Calipers measure the outside or inside width of an object more precisely than a ruler.
- A measuring cylinder measures the volume of a liquid. Read it with your eye level with the bottom of the curved surface (the meniscus).
- The volume of an irregular solid is found by displacement: the rise in the water level when the object is fully submerged equals its volume.
To reduce the effect of a ruler's width on a small measurement, measure many identical items together and divide. For example, measure the thickness of a stack of 10 identical sheets, then divide by 10.
Measuring time
Clocks and stopwatches (analogue or digital) measure time. Human reaction time is about 0.2 s, so timing a short event once gives a large percentage error. For a repeating motion such as a pendulum swing, time many oscillations and divide by the number. This spreads the reaction-time error over the whole count. Repeating the whole measurement and taking an average also improves reliability.
One period T is the time for one complete oscillation: T = (total time) ÷ (number of oscillations).
Scalars and vectors
- A scalar has size (magnitude) only. Examples: distance, speed, mass, time, energy, temperature.
- A vector has size and direction. Examples: displacement, velocity, acceleration, force, weight, momentum.
Vectors are added by combining their directions as well as their sizes. If two vectors act in the same line, add them when they point the same way and subtract them when they point opposite ways. (Extended) For two vectors at right angles, draw them head to tail on a scale drawing. The resultant is the straight line from the start of the first to the end of the second. You can also use Pythagoras for the size and trigonometry for the angle.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Scalar | A quantity with magnitude only. |
| Vector | A quantity with magnitude and direction. |
| Resultant | The single vector that has the same effect as two or more vectors acting together. |
| Period, T | Time for one complete oscillation, in seconds. |
| T = t ÷ n | Period from the time t for n oscillations. |
| Frequency, f = 1 ÷ T | Number of oscillations per second, in hertz (Hz). |
| R2 = A2 + B2 (Extended) | Size of the resultant R of two perpendicular vectors A and B. |
| Micrometer reading | Sleeve reading + thimble reading (each division on the thimble is 0.01 mm). |
Worked examples
Example 1 A student times 20 complete swings of a pendulum and records 31.4 s. Find the period and the frequency.
Step 1: period = total time ÷ number of oscillations = 31.4 ÷ 20 = 1.57 s.
Step 2: frequency = 1 ÷ T = 1 ÷ 1.57 = 0.637 Hz.
Answer: T = 1.57 s, f = 0.637 Hz.
Example 2 A micrometer shows 5.5 mm on the sleeve and 28 divisions on the thimble. Each thimble division is 0.01 mm. What is the diameter of the wire? Another student measures a stack of 10 identical paper sheets as 1.20 mm thick. How thick is one sheet?
Step 1: thimble reading = 28 × 0.01 mm = 0.28 mm.
Step 2: diameter = 5.5 + 0.28 = 5.78 mm.
Step 3: thickness of one sheet = 1.20 mm ÷ 10 = 0.120 mm.
Answer: wire diameter 5.78 mm; one sheet 0.120 mm thick.
Example 3 (Extended) A boat steers east at 4.0 m/s through water. A river current flows north at 3.0 m/s. Find the size and direction of the boat's resultant velocity.
Step 1: the two velocities are at right angles, so draw them head to tail (4.0 units east, then 3.0 units north).
Step 2: size: R2 = 4.02 + 3.02 = 16 + 9 = 25, so R = 5.0 m/s.
Step 3: angle north of east: tan θ = 3.0 ÷ 4.0 = 0.75, so θ = 36.9°.
Answer: 5.0 m/s at 36.9° north of east.
Common errors
- Writing a number with no unit, or using the wrong one (for example cm where m is needed in a formula).
- Reading a measuring cylinder from above or below instead of level with the meniscus.
- Timing one swing of a pendulum instead of many, then quoting the result as accurate.
- Adding two perpendicular vectors by simply adding their sizes (4.0 + 3.0 = 7.0). The resultant is 5.0, not 7.0.
- Calling speed a vector. Speed is a scalar; velocity is the vector.
Exam tip: Before substituting into any equation, convert every quantity to base units (m, kg, s). Write the unit on every line of your working so you can spot a slip quickly.
Check yourself
- Convert 4500 μm to metres and to millimetres.
Answer
4500 μm = 4500 × 10⁻6 m = 4.5 × 10⁻3 m = 4.5 mm. - A stopwatch shows 24.6 s for 30 oscillations of a mass on a spring. Find the period.
Answer
T = 24.6 ÷ 30 = 0.82 s. - Sort these into scalars and vectors: mass, force, energy, velocity, distance, acceleration.
Answer
Scalars: mass, energy, distance. Vectors: force, velocity, acceleration. - (Extended) A walker goes 6.0 km east then 8.0 km north. What is the size of the resultant displacement?
Answer
R2 = 6.02 + 8.02 = 36 + 64 = 100, so R = 10 km. (The distance walked is 14 km.)