AQA A-Level Physics

SI units

Every physical quantity is a number multiplied by a unit, and the whole of physics is built on a small set of base units from which every other unit is derived. The base quantities you need are mass (kilogram, kg), length (metre, m), time (second, s), amount of substance (mole, mol), temperature (kelvin, K) and electric current (ampere, A).

A derived unit is a combination of base units. The newton is the force that gives a mass of 1 kg an acceleration of $1\,\mathrm{m\,s^{-2}}$, so $1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}$. The joule is a newton metre, $1\,\mathrm{J} = 1\,\mathrm{kg\,m^2\,s^{-2}}$, and the watt is a joule per second, $1\,\mathrm{W} = 1\,\mathrm{kg\,m^2\,s^{-3}}$. Writing an equation's units in base units is a quick check that the equation can be right: both sides must reduce to the same combination.

Prefixes and standard form

Prefixes scale a unit by a power of ten, so that very large and very small values can be written without long strings of zeros.

Standard form writes a number as $a \times 10^n$ with $1 \le a < 10$, so $0.000\,047\,\mathrm{m}$ is $4.7\times10^{-5}\,\mathrm{m}$ or $47\,\mu\mathrm{m}$.

Converting between units

Some units of the same quantity are not related by a simple power of ten, and you convert using their definitions. The electronvolt is the energy gained by an electron accelerated through a potential difference of 1 V, so $1\,\mathrm{eV} = 1.60\times10^{-19}\,\mathrm{J}$. The kilowatt hour is the energy transferred by a 1 kW device in one hour, so $1\,\mathrm{kW\,h} = 1000\,\mathrm{W}\times3600\,\mathrm{s} = 3.6\times10^{6}\,\mathrm{J}$.

Random and systematic errors

An error is the difference between a measured value and the true value. Two different kinds behave differently, and the remedy for each is different too.

A random error makes repeated readings scatter unpredictably either side of the true value. It comes from things such as reaction time, small fluctuations in the reading or the difficulty of judging a scale. Taking many readings and averaging them reduces its effect, and a graph drawn through many points averages it out.

A systematic error shifts every reading by the same amount or the same proportion in one direction. A zero error on a micrometer, a stopwatch that runs slow or a meter that is wrongly calibrated all do this. Repeating the measurement does nothing to it. It is reduced by calibrating the instrument against a known standard, checking the zero, or changing the method. On a graph a systematic error shows as a non-zero intercept where the line should pass through the origin.

Vocabulary of measurement

A set of readings can be precise but inaccurate, when a systematic error moves them all to the wrong place, and a set can be centred on the true value but imprecise, when the random scatter is large.

Uncertainty

Every measurement is quoted with an uncertainty, a range within which the true value is expected to lie. The absolute uncertainty has the units of the quantity, the fractional uncertainty is the absolute uncertainty divided by the value, and the percentage uncertainty is the fractional uncertainty multiplied by 100.

A single reading from a scale is usually given an uncertainty of half the smallest division, or the smallest division if the readings are hard to judge. A digital meter's uncertainty is normally taken as one in the last digit. For repeated readings the uncertainty can be estimated as half the range between the largest and smallest readings (the spread), when the number of readings is small. A reading of a length from a ruler starts at one end and finishes at the other, so if both ends have to be read the uncertainties add.

The number of significant figures in a result should match its uncertainty. Quoting $9.812\,345\,\mathrm{m\,s^{-2}}$ when the uncertainty is $0.1\,\mathrm{m\,s^{-2}}$ claims a precision that the measurements never had, and $9.8\pm0.1\,\mathrm{m\,s^{-2}}$ is the honest statement.

Combining uncertainties

• When quantities are added or subtracted, add their absolute uncertainties.
• When quantities are multiplied or divided, add their percentage (or fractional) uncertainties.
• When a quantity is raised to a power $n$, multiply its percentage uncertainty by $n$.

Subtracting two nearly equal readings gives a small difference with the full sum of the absolute uncertainties, so its percentage uncertainty can become large. That is why a length found as the difference between two close readings is best measured directly.

Worked example

A cylinder has a mass $m = 0.250\pm0.002\,\mathrm{kg}$, a diameter $d = 4.00\pm0.05\,\mathrm{cm}$ and a length $L = 10.0\pm0.1\,\mathrm{cm}$. Find its density with its uncertainty.

The volume is $V = \pi (d/2)^2 L$, and the density is $\rho = m/V$, so $\rho \propto m\,d^{-2}L^{-1}$. The percentage uncertainties are

The total is $0.8 + 2.5 + 1.0 = 4.3\%$.

The value of the density, with lengths in metres, is

and $4.3\%$ of this is about $0.09\times10^{3}$, so $\rho = (1.99\pm0.09)\times10^{3}\,\mathrm{kg\,m^{-3}}$.

Graphs, error bars and gradients

A point on a graph can carry error bars, short lines above and below (and, if needed, to the left and right) showing its uncertainty. The bar length is the absolute uncertainty in that reading. If the uncertainty in one axis is too small to draw it can be left off, and the uncertainty in the other axis is then the one that matters.

A straight-line graph is analysed through its gradient and intercept. The best-fit line is the one that passes through all the error bars with points balanced on either side. To find the uncertainty in the gradient, draw the worst acceptable line, which is the steepest or shallowest line that still passes through all the error bars, and compare its gradient with the best line.

The percentage uncertainty in the gradient is that value as a percentage of the best gradient. The uncertainty in the intercept is found the same way, by comparing the intercept of the best line with that of the worst line. A gradient should be measured with a large triangle, spanning most of the line, because a small triangle magnifies the reading uncertainty.

When a relationship is not linear it can often be made so. If $y = ax^n$ then plotting $\lg y$ against $\lg x$ gives a straight line of gradient $n$ and intercept $\lg a$. If $y = ae^{kx}$ then plotting $\ln y$ against $x$ gives gradient $k$ and intercept $\ln a$.

Estimation and orders of magnitude

An order of magnitude is the nearest power of ten. Estimating to the nearest order of magnitude is a skill of its own: choose sensible values for the quantities involved, combine them with the physics, and accept an answer that may be out by a factor of a few.

Some useful sizes to hold in mind: a person's mass is about $10^2\,\mathrm{kg}$, a car's about $10^3\,\mathrm{kg}$, a walking speed is about $1\,\mathrm{m\,s^{-1}}$, an atom's diameter is about $10^{-10}\,\mathrm{m}$ and a nucleus's about $10^{-15}\,\mathrm{m}$, and the Earth's radius is about $6\times10^{6}\,\mathrm{m}$.

Worked example

Estimate the gravitational potential energy gained by a person climbing a flight of stairs. Take a mass of $70\,\mathrm{kg}$ and a height gain of about $3\,\mathrm{m}$.

which is of order $10^{3}\,\mathrm{J}$. Using $g = 10\,\mathrm{m\,s^{-2}}$ instead of $9.81$ makes a difference of 2%, which is far smaller than the uncertainty in the other estimates and so does not matter here.