Physical Quantities and Units

A physical quantity is a magnitude and a unit

Every physical quantity is written as a numerical magnitude together with a unit. A length is not 2.4; it is 2.4 m, where 2.4 tells you how many and m tells you how much of one. A number on its own means nothing in physics: 2.4 could be a length, a mass or a current, and only the unit distinguishes them. Changing the unit changes the number, never the quantity itself: the same length is 2.4 m, 240 cm or 2.4 x 10^3 mm.

Carrying units through a calculation is the cheapest check you have. If you want a speed from a distance of 1500 m and a time of 300 s, then 1500 m / 300 s = 5.0 m s-1, and the unit of the answer confirms that you divided rather than multiplied. If the unit of your answer is not the unit of the quantity you were asked for, the calculation is wrong whatever the arithmetic says.

You are also expected to estimate quantities sensibly, to the nearest power of ten or so. An estimate is never quoted to three significant figures: the mass of an adult is about 70 kg, not 69.3 kg, and the height of Mount Everest is about 1 x 10^4 m. Learn the anchor values in the table below; they are the ones a question is most likely to ask you to judge.

SI base quantities, derived units and prefixes

The syllabus uses five SI base quantities, each with its own SI base unit: mass (kg), length (m), time (s), electric current (A) and thermodynamic temperature (K). Base units are the ones that are defined directly, so they are not written in terms of anything else. Note that the base unit of mass is the kilogram, not the gram: the kilo is already built into the name, so there is no such thing as a base unit of 'gram' and you must not treat the kg as 10^3 of some larger unit.

Every other unit is a derived unit, written as a product or quotient of base units. You get there by taking the defining equation of the quantity and replacing each symbol with its unit. Force is defined by F = m a, so 1 N = 1 kg x 1 m s-2 = 1 kg m s-2. Work is defined by W = F s, so 1 J = 1 N m = 1 kg m2 s-2. Learn to do this from the equation rather than memorising the list, because a question can ask you for any quantity in the syllabus.

Prefixes save you from writing long strings of powers of ten. Each prefix is a fixed factor, and it works with base units and derived units alike: 2.4 kN is 2.4 x 10^3 N, 470 microA is 470 x 10-6 A = 4.7 x 10-4 A, and 3.2 GJ is 3.2 x 10^9 J.

The safe routine is always to convert to base units before substituting into an equation. Then every intermediate value is in base units, the answer comes out in base units, and the only decision left is which prefix to use when you write the final value. Watch the kilo in kg: a density is in kg m-3, so convert grams to kilograms and centimetres to metres before you divide.

• pico (p): x 10-12, nano (n): x 10-9, micro (mu): x 10-6
• milli (m): x 10-3, centi (c): x 10-2, deci (d): x 10-1
• kilo (k): x 10^3, mega (M): x 10^6, giga (G): x 10^9, tera (T): x 10^12

Checking an equation with base units: homogeneity

An equation is homogeneous when both sides reduce to the same base units. This must be true of every correct physical equation, because you cannot add metres to seconds in the same way that you cannot add apples to volts. So the check is quick and decisive: write the base units of the quantity on the left, do the same for the expression on the right, and compare.

Take the constant-acceleration equation v2 = u2 + 2 a s. On the left, v is a velocity, so v2 has units (m s-1)2 = m2 s-2. On the right, u2 also has units m2 s-2. The term 2 a s has units (m s-2) x m = m2 s-2, and the '2' is a pure number with no unit, so it does not affect the comparison. Both sides are m2 s-2, so the equation is homogeneous.

Now try a wrong equation: a student writes s = u t + a t. The term u t has units (m s-1) x s = m, which matches the metre on the left. But a t has units (m s-2) x s = m s-1, so the second term is a velocity and cannot be added to a displacement. The equation is not homogeneous, and that alone proves it is wrong; the correct version is s = u t + 0.5 a t2.

Homogeneity is necessary but not sufficient. The equations s = a t2 and s = 0.5 a t2 are both homogeneous, and only one of them is physically correct, so a units check can never confirm that a dimensionless factor such as a half or a 2 pi is right. It can only rule an equation out. Units also cannot check the argument of a trigonometric or exponential function, since those arguments are dimensionless numbers.

Random errors and systematic errors

No reading is exact. The error of a measurement is the difference between the measured value and the true value, and the uncertainty is the range within which the true value is expected to lie. The starting point is the instrument's resolution, the smallest change it can detect: 1 mm on a metre rule, 0.1 mm on vernier calipers, 0.01 mm on a micrometer screw gauge, 0.01 s on a digital stopwatch. Choosing a coarser instrument immediately makes every percentage uncertainty larger.

Random errors are unpredictable. They vary in size and in sign from one reading to the next, so repeated readings are scattered about a mean. They come from reading a scale by eye, from parallax when the eye is not level with the pointer, from reaction time when a stopwatch is started and stopped, and from small changes in temperature, vibration or supply voltage. You reduce them by repeating and averaging, and by improving technique: read scales at eye level, time many oscillations instead of one, and take each reading to the finest division the instrument allows.

Systematic errors act in one direction. Every reading is shifted by the same amount, or scaled by the same factor, so the readings are consistently too large or consistently too small. A zero error is the commonest example: an instrument that does not read zero when it should. A micrometer screw gauge that reads +0.03 mm when it is closed adds 0.03 mm to every diameter; a top-pan balance that reads 0.3 g with an empty pan adds 0.3 g to every mass; a stopwatch that runs slow makes every time too small. A wrongly calibrated scale, a thermometer reading 2 degrees C high at every temperature, and a metre rule read from a worn end all do the same thing.

The two behave completely differently in the laboratory. Averaging a large number of readings cancels random errors, because the scatter falls on both sides of the mean, but it does nothing at all to a systematic error: the mean of many readings is simply the same wrong value. The only cure for a systematic error is to find it and correct it, by measuring the zero reading and subtracting it, or by calibrating the instrument against a known value. A precise set of readings that is consistently off the true value is the signature of a zero error.

Precision and accuracy are not the same thing

Precision describes how close repeated measurements are to one another, so it is a statement about the spread of your readings and about the instrument you used. A set of readings that all lie within 0.02 mm of each other is precise; a set spread over 0.12 mm is not. Accuracy describes how close a measurement is to the true value, so accuracy only has meaning when the true value is known, for example when you measure a standard mass or check a value given in the question.

The two can occur in any combination, and the combination tells you what is wrong. Precise but not accurate readings are all close together but all shifted the same way, which is the signature of a systematic error or zero error: the technique is good and the instrument is faulty. Accurate but not precise readings are scattered, but their mean is close to the true value, which is the signature of random error that has been averaged out. Neither precise nor accurate points to poor technique and a miscalibrated instrument together.

Resolution and sensitivity are separate again. The resolution of an instrument is the smallest change it can detect, and the sensitivity is the change in output per unit change in input. An instrument can be very sensitive but coarse, or coarse but perfectly calibrated. When you choose an instrument, ask two questions: is its resolution fine enough for the size of the quantity, and can I read it to that resolution? Measuring 20 mm with a metre rule gives a percentage uncertainty of about 5 per cent, whereas measuring 200 mm with the same rule gives about 0.5 per cent, so measuring a larger length or using a finer instrument is the way to improve a result.

Assessing the uncertainty in a derived quantity

Every measurement is quoted as value +/- uncertainty, for example 60.0 +/- 0.6 g or 8.0 +/- 0.4 cm3. The absolute uncertainty has the same unit as the value; the percentage uncertainty is the absolute uncertainty divided by the value and multiplied by 100. From repeated readings the uncertainty is taken as half the range between the largest and the smallest reading, and a single reading cannot have an uncertainty smaller than half the smallest scale division.

Two rules carry you through almost every question. When quantities are added or subtracted, add their absolute uncertainties. When quantities are multiplied or divided, add their percentage uncertainties. The word is add, never subtract: the uncertainties always make the answer less certain, whatever the arithmetic does. If a quantity is raised to a power n, its percentage uncertainty is multiplied by n, so the percentage uncertainty in the volume of a sphere is three times the percentage uncertainty in the radius.

Here is the rule used on a real measurement. A student finds the mass of a metal block with a top-pan balance and its volume by displacement in a measuring cylinder. The density is the mass divided by the volume, so the percentage uncertainties add. The absolute uncertainty in the density is then that percentage of the density, and the density is quoted with its uncertainty to the same number of decimal places.

Quote the uncertainty to one or two significant figures and give the value to the same decimal place: 7.5 +/- 0.5 g cm-3, not 7.5 +/- 0.45 g cm-3. Finally, look at which measurement dominates. If the percentage uncertainty in the volume is five times that in the mass, there is no point in buying a better balance; you improve the result by measuring the volume more precisely.

Scalars, vectors and resolving vectors into components

A scalar has magnitude only: distance, speed, mass, time, energy, work, power, density, pressure, temperature, charge and potential difference are all scalars, and they add like ordinary numbers. A vector has both magnitude and direction: displacement, velocity, acceleration, force, weight, friction, tension, momentum, impulse, electric field strength and magnetic flux density are all vectors. Direction is not decoration: two 5 N forces on the same object may add to 10 N or to zero depending on the angle between them, so a vector answer that gives no direction is incomplete.

Adding coplanar vectors. Draw them head to tail and the resultant is the single vector from the tail of the first to the head of the last; the parallelogram construction gives the same resultant. Subtracting a vector means adding it reversed, which is why a change in velocity is the final velocity plus the reversed initial velocity. If the two vectors act along the same line, add or subtract their magnitudes. If they are perpendicular, find the magnitude with Pythagoras and the direction with a tangent: for 3.0 km north followed by 4.0 km east the resultant displacement is the square root of (3.0^2 + 4.0^2) = 5.0 km, at an angle whose tangent is 4.0 / 3.0, that is 53 degrees east of north.

Resolving does the opposite: it replaces one vector by two perpendicular components that together have exactly the same effect. For a force F at an angle theta to the x-direction, the x-component is F cos theta and the y-component is F sin theta. Choose the axes so that as many forces as possible lie along them, and remember that the components are independent: the x-component affects only motion in x, which is why projectile motion splits so neatly into a constant-velocity horizontal part and an accelerating vertical part.

A standard use is a body on a slope of angle theta. The weight mg resolves into mg sin theta down the slope and mg cos theta perpendicular to the surface, while the normal contact force balances the perpendicular component. For a 2.0 kg block on a slope of 20 degrees: mg sin 20 = 2.0 x 9.81 x 0.342 = 6.7 N down the slope, and mg cos 20 = 2.0 x 9.81 x 0.940 = 18.4 N into the surface. Take care with which angle theta is measured from, because using the wrong one swaps sine and cosine.