Physical quantities and measurement

01

Physical quantities and units

A physical quantity describes something measurable, such as a length, a time or a mass. Its unit tells us what the number means.

25 cm and 0.25 m describe the same length. Changing the unit changes the numerical value, but does not change the object. Writing just "25" would leave the length unclear: 25 millimetres, centimetres and metres are very different.

A physical quantity is usually expressed as a numerical value and a unit. Some quantities are dimensionless: for example, a ratio of two lengths has no unit because the length units cancel.

Base quantities and SI units

SI is the international system of units. Use these six base quantities and their SI units in this course. Other quantities can be formed by combining base quantities.

Write the kelvin as K, without a degree sign. The SI base unit of mass is the kilogram, kg, even though the gram is also a familiar mass unit. One kilogram is 1000 g.

A quantity symbol is different from a unit symbol

In m = 2.0 kg, the first m is a symbol for the quantity mass; kg is its unit. In l = 2.0 m, lowercase l represents length and m is the unit metre. Read the meaning given in the question instead of deciding what a letter means on its own.

For more journeys, see distance, speed and velocity. To change the units before calculating, use prefixes and physical scales.

02

Prefixes and physical scales

A prefix changes the size of a unit. Milli means one thousandth, so a millimetre is one thousandth of a metre.

The numerical value can change while the physical quantity stays the same. See quantities and units if that distinction is unfamiliar.

Powers of ten make very large and small numbers easier to write. 10³ = 1000, while 10^-3 = 1 / 1000 = 0.001. A negative exponent means a fraction here; it does not mean a negative length or current.

Capital letters matter: mA means milliamperes, but MA means megaamperes. The prefix kilo is lower-case k; the upper-case K is the unit kelvin.

Area and volume need more than one conversion

A square measuring 1 cm along each side has an area of 1 cm². Each side is 0.01 m, so convert both dimensions when calculating its area.

The same reasoning applies to a cube: 1 cm³ = (0.01 m)³ = 0.000001 m³. Apply the length conversion to all three dimensions.

Does the size of the answer make sense?

An order of magnitude gives the approximate scale of a quantity as a power of ten. For example, 10⁰ m is 1 m. The objects below vary in size; these values are useful reference scales, not exact measurements of every example.

If a calculation gives 0.02 m for the width of a human hair, pause: 0.02 m is 2 cm, about the width of a small coin. A hair is much thinner, with a width of roughly 10^-4 m. Check the original reading and conversion before accepting the answer.

03

Choosing measuring instruments

Choose an instrument that can cover the measurement and distinguish the detail you need.

A metre rule is useful for a book, but its divisions are too coarse to measure the thickness of one sheet of paper well. A micrometer can resolve a small thickness, but cannot directly span a room.

Range

The span of values the instrument can measure, such as 0 to 150 mm for a pair of calipers.

Resolution

The smallest change you can read from the scale or display, such as a 0.01 mm display step.

Precision

How closely repeated measurements agree. Readings with little spread are precise, but may all share the same error.

Accuracy

How close a measurement is to the true or accepted value. Checking the zero and using the instrument correctly help avoid a consistent offset.

A smaller display step does not guarantee an accurate measurement. A digital balance could show the same value repeatedly while its zero is wrong. Its readings would agree closely but remain displaced from the true value.

Match the instrument to the object

These are example ranges and resolutions. Check the actual instrument's markings and specification before choosing it.

Metre rule

0 to 1 m; 1 mm divisions.
A book or a straight rod. Align the rule with the length and read both ends.

Measuring tape

0 to 5 m; 1 mm divisions.
A room or a longer object. Keep the tape straight and taut; a sagging tape follows a longer path.

Digital calipers

0 to 150 mm; 0.01 mm display step.
A small external diameter or thickness. Keep the object square to the jaws and close them gently.

Digital micrometer

0 to 25 mm; 0.001 mm display step.
A thin wire or sheet. Use the ratchet as instructed so that excessive pressure does not squash the object.

Digital calipers may also have inside jaws for an internal diameter and a depth rod for a hole. Choose the correct contact surfaces. For a diameter, align the measurement across the widest part through the centre; tilting the instrument can give the wrong length.

Ask three questions before measuring

1. Will it fit? Check the object's expected size against the instrument's range.
2. Can I distinguish enough detail? Check the divisions or display step against the required measurement.
3. Can I use it properly? Consider alignment, access, the object's shape and whether contact could deform it.

For a very thin sheet, another useful approach is to measure the thickness of a closely packed stack and divide by the number of sheets. This gives an average thickness; gaps or compressing the stack would affect the result.

Once an instrument is chosen, use sensible measurement methods to read it, check its zero and record the result.

04

Making useful measurements

A useful measurement depends on how you take the reading, not just which instrument you choose.

Choose a suitable range and resolution first. Then look for a specific source of error and a method that addresses it.

Read the difference between the ends

Place a rule parallel and close to the length being measured. If the object does not start at zero, read both ends and subtract. This is also useful when a rule's zero end is worn.

Look perpendicularly at the scale. If the object and markings are at slightly different heights, looking from one side makes them appear to line up differently. This is parallax error. Keep the object close to the scale as well as placing your eye correctly.

Check the zero before trusting the display

Clean and gently close the jaws of digital calipers, then check the reading. If they correctly close with no object between them but the display is not zero, there is a zero offset. Use the instrument's zero control when appropriate, or correct a stated stable offset.

Repeating a reading does not remove a stable zero error. Repeats help you judge and reduce random variation through averaging. A consistent offset needs a correction or a properly adjusted instrument.

Time several complete oscillations

The period of a pendulum is the time for one complete oscillation. The bob must return to the same reference point moving in the same direction. Going from one side to the other is only half an oscillation.

1. Place a fixed reference mark beside the path, for example at the centre of the swing.
2. Start timing as the bob passes the mark in a chosen direction. Count this as zero.
3. Count one each time it next passes the mark in that same direction. Stop at 20.
4. Repeat the timing under the same conditions, find the mean elapsed time, then divide by 20.

The delay in reacting at the start and finish is a much smaller fraction of 40 s than of 2 s. Timing many oscillations reduces its fractional effect on the period. Repeating and averaging helps with variation in those delays, but cannot guarantee that all timing errors disappear.

The calculated 2.02 s does not mean a person directly timed one swing to the nearest 0.01 s. It comes from dividing a longer measured interval by an exact count of oscillations.

Use displaced water to measure an irregular solid

Choose a measuring cylinder large enough for the object and final water level, with divisions fine enough to distinguish the rise. Put it upright on a level surface. For water, read the bottom of the curved meniscus at eye level.

This works when the solid is fully submerged, does not dissolve or absorb water, and no water is lost. Remove trapped air bubbles: they displace extra water and would make the calculated solid volume too large. Avoid splashing when lowering the object.

Other common measurements

Record what was measured

Put the quantity and unit in each table heading, such as Time for 20 oscillations / s. The entries then contain the numerical readings. Keep readings consistent with the scale or display used; do not invent extra decimal places to make a result look more precise.

Keep enough digits during a calculation to avoid unnecessary rounding, then report the result with precision justified by the measurements. Show a mean or a corrected reading separately so the original observations remain clear.

Some quantities are calculated from several measurements. For example, measuring distance and elapsed time lets you calculate average speed. The examples below apply the same measurement principles to particular quantities.

05

Scalars and vectors

A scalar has magnitude only. A vector has both magnitude and direction.

Magnitude means size, given by a numerical value and its unit where applicable. A speed of 4 m/s tells us how fast an object moves. A velocity of 4 m/s east also tells us its direction.

Time, temperature and energy are also scalars. Force and acceleration are vectors. Acceleration describes a change in velocity per unit time, so the direction of that change matters.

A minus sign alone does not make a quantity a vector. A temperature of -5 °C is still a scalar. The value is below zero on the Celsius scale; it does not point in a spatial direction.

In a vector answer, state the direction as well as the magnitude, such as 6 N upwards. For signed values, say which direction you have taken as positive.

When two vectors act together, use a scale drawing to find their resultant.

06 / Pure

Adding vectors with a scale drawing

The resultant is the single vector equal to the sum of the vectors being added. A scale drawing lets you find its magnitude and direction.

A vector arrow represents both magnitude and direction. Keep both unchanged when moving an arrow for the construction.

First check the directions

Suppose two forces act at the same time on one object. If they are 3 N east and 2 N east, their resultant is 5 N east. If they are 3 N east and 2 N west, their resultant is 1 N east. Adding magnitudes alone would ignore the opposing direction.

For forces that are not along the same line, construct their vector sum head to tail. Draw the second arrow starting at the head of the first, without rotating it or changing its length. The resultant points from the first arrow's tail to the second arrow's head.

Moving the arrow is a drawing method for adding the vectors. In these examples, both forces act simultaneously on the same object; the arrows do not show forces applied one after the other.

Construct the scale drawing on paper to measure it. The diagram on a screen changes size with the display, so its lengths need not be physical centimetres. Use the supplied lengths and scale. The method is construction and measurement, not remembering one particular triangle.

The method also works without a right angle

Now two simultaneous forces are 4 N east and 3 N at 60° north of east. The second force is diagonal, rather than directly north.

1. Use the same paper scale, 1 cm per N. Draw AB 4 cm east.
2. At B, draw a light reference line continuing east. Use a protractor to measure 60° towards the north from that line.
3. Along that direction draw BC 3 cm long. Join A to C, with the arrowhead at C.
4. Measure AC: about 6.1 cm. Its direction is about 25° north of east.
5. The resultant is therefore about 6.1 N at 25° north of east.

The answer is less than 4 + 3 = 7 N because the forces are not in the same direction. An arrow from C to A would have the correct length but the wrong direction for the resultant.