Calculations, units and the data booklet

Almost every physics question ends in a number with a unit. Getting that number right depends on a small set of habits that apply in every topic: working in SI units, converting prefixes correctly, checking that an equation makes sense by its units, rounding to a sensible precision, and knowing roughly how big the answer should be. This set of skills is tested directly in some question parts, and indirectly in nearly all of them.

SI base units

The International System of Units (SI) is built on a small number of base units, defined independently of one another. All other units are derived units, products and quotients of base units. Physics in this course uses six of the seven SI base units; the seventh, the candela for luminous intensity, does not appear.

Derived units

A derived unit is found from the defining equation of its quantity. The newton comes from $F=ma$, so $1\text{ N}=1{\text{kg m s}}^{-2}$; the joule comes from work done, $W=Fs$, so $1\text{ J}=1\text{ N m}=1{\text{kg m}}^2{\text{s}}^{-2}$. Writing a unit in base units means following these definitions back until only base units remain.

Checking equations by their units

A physically correct equation is homogeneous: every term on each side has the same units. Checking this is a quick way to catch an error in an equation that has been recalled or rearranged, and it can identify the unit of an unknown constant.

The last point is the limitation of the method. A units check cannot detect a wrong dimensionless factor such as ½, $2\pi$ or $\sqrt{2}$. A homogeneous equation can still be wrong; an equation that is not homogeneous is certainly wrong.

Prefixes and conversions

Prefixes multiply a unit by a power of ten. The data booklet lists them from peta to femto, so they need not be memorised; converting them correctly takes practice.

Squared and cubed units

When a unit is squared or cubed, so is its prefix. Since $1\text{ cm}={10}^{-2}\text{ m}$, $1{\text{cm}}^2=({10}^{-2}\text{ m}{)}^2={10}^{-4}{\text{m}}^2$ and $1{\text{cm}}^3={10}^{-6}{\text{m}}^3$. Similarly $1{\text{mm}}^2={10}^{-6}{\text{m}}^2$, which matters for the cross-sectional area of a wire.

Non-SI units

Some fields use units that are more convenient than SI ones. The data booklet gives their conversions; each must be converted to SI before being used in an equation that contains SI constants.

A temperature difference has the same size in kelvin and in degrees Celsius, so a change of 20 °C is a change of 20 K. Only an absolute temperature, in gas laws or in $L=\sigma A{T}^4$, must be converted.

Significant figures and standard form

Counting significant figures

The significant figures of a number are the digits that carry meaning about its precision. Non-zero digits are always significant. Zeros between non-zero digits are significant (2.05 has three). Leading zeros are not (0.0042 has two). Trailing zeros after a decimal point are significant (3.40 has three), because the writer chose to include them.

Standard form (scientific notation) writes a number as $a\times {10}^n$ with $1\le a<10$. It removes any doubt about trailing zeros: 4500 could have two, three or four significant figures, but $4.5\times {10}^3$ has two and $4.50\times {10}^3$ has three.

How precise should an answer be?

• For multiplication and division, give the answer to the same number of significant figures as the least precise piece of data. $2.0\times 3.14=6.3$, not 6.28.

• For addition and subtraction, the number of decimal places is limited by the least precise value: 12.6 m + 0.85 m = 13.5 m.

• Keep extra figures in intermediate steps and round only the final answer.

• In practice, 2 or 3 significant figures, matching the data, are almost always accepted.

Orders of magnitude and estimation

The order of magnitude of a quantity is the power of ten closest to it. A person’s mass of 70 kg is of order ${10}^2$ kg; the Earth’s radius, $6.4\times {10}^6$ m, is of order ${10}^7$ m (on a logarithmic scale, 6.4 is nearer 10 than 1). Two quantities "differ by three orders of magnitude" when their ratio is about ${10}^3$.

The length scale spans about 42 orders of magnitude from a proton to the observable universe. A few reference values in each quantity make it possible to judge whether any answer is reasonable:

Making an estimate

An estimate replaces exact data with sensible rounded values and works to one or two significant figures. State each assumed value, keep the arithmetic in powers of ten, and check the answer against experience.

Rearranging equations and using a calculator

Rearranging an equation in symbols before substituting numbers is quicker and safer than substituting first. The algebra is easier to check, the units can be checked on the rearranged form, and the same expression can be reused if the data change.

On a calculator, enter powers of ten with the exponent key (EXP, EE or ×10ˣ), not by typing "× 10 ^". Typing $6.67\times 10\wedge -11$ without brackets can be evaluated in the wrong order inside a fraction. Put the whole numerator and the whole denominator in brackets, or evaluate them separately.

Ratios and proportionality

Many calculations become simpler when written as ratios. If $y\propto x^n$, then ${\displaystyle \frac{y_2}{y_1}}={\left({\displaystyle \frac{x_2}{x_1}}\right)}^n$, and every constant cancels. For example, the intensity from a point source obeys $I\propto {\displaystyle \frac{1}{d^2}}$, so moving three times as far away reduces the intensity by a factor of $3^2=9$, whatever the power of the source.

Scalars, vectors and their components

A scalar has a magnitude only: mass, time, energy, speed, temperature. A vector has both a magnitude and a direction: displacement, velocity, acceleration, force, momentum, field strength. Scalars add by ordinary arithmetic. Vectors add by taking their directions into account, either with a scale diagram or with components.

Resolving a vector

Any vector can be replaced by two perpendicular components that together have the same effect. For a vector of magnitude $A$ at an angle $\theta$ to the horizontal, the data booklet gives

The cosine always goes with the component adjacent to the angle. If the angle is measured from the vertical instead, the roles swap: the vertical component is $A\cos \theta$ and the horizontal one $A\sin \theta$. Resolution is always into two perpendicular directions, usually horizontal and vertical, or along and perpendicular to a slope.

Adding vectors

To add vectors, draw them tip to tail: the resultant runs from the tail of the first to the tip of the last. Two perpendicular vectors give a right-angled triangle, so Pythagoras and the tangent give the magnitude and direction. For vectors at other angles, resolve each one, add the horizontal components and the vertical components separately, then combine the two totals. Subtracting a vector is adding its negative, a vector of the same size pointing the opposite way, which is how a change in velocity $\mathrm{\Delta }v=v-u$ is found. Multiplying a vector by a scalar changes its magnitude (and reverses it if the scalar is negative) but not its line of action.

Variables and relationships

A continuous variable can take any value within a range (length, time, temperature, current). A discrete variable takes only separate values, often whole numbers (the number of coils, the number of slits, the number of layers of insulation). The distinction matters when choosing a graph: continuous data are plotted as points joined by a line or curve of best fit, while discrete or categorical data are often shown as a bar chart.

Every proportional relationship is linear, but a linear relationship is proportional only when its intercept is zero. A correlation describes a trend in data and does not by itself show that one variable causes the other.

Percentage change and percentage difference

A percentage change compares a new value with an original one: ${\displaystyle \frac{\text{new}-\text{original}}{\text{original}}}\times 100\mathrm{\%}$. A percentage difference between an experimental value and an accepted value is ${\displaystyle \frac{\mathrm{\mid }\text{experimental}-\text{accepted}\mathrm{\mid }}{\text{accepted}}}\times 100\mathrm{\%}$; it is sometimes called the percentage error. A student who measures the speed of sound as 352 m s⁻¹ against an accepted 343 m s⁻¹ has a percentage difference of ${\displaystyle \frac{9}{343}}\times 100\mathrm{\%}=2.6\mathrm{\%}$.

Approximations and neglecting small effects

Physics models routinely leave out effects that are too small to matter: fluid resistance on a dense ball falling a short distance, the internal resistance of a fresh cell compared with a large external resistance, the mass of a light string. A useful test is whether the neglected effect would change the answer by more than the uncertainty in the data. If a 1 % correction is applied to data known only to 5 %, it makes no difference to the result.

What the data booklet gives

A clean copy of the physics data booklet is available for every paper. Knowing its layout saves time and errors. It contains:

• Mathematical equations: areas and volumes of common shapes (but not the surface area of a sphere), and the resolution of vectors.

• The rules for propagating uncertainties through sums, products, quotients and powers.

• Fundamental constants, with the values to use: $g=9.8{\text{m s}}^{-2}$, $G=6.67\times {10}^{-11}{\text{N m}}^2{\text{kg}}^{-2}$, $c=3.00\times {10}^8{\text{m s}}^{-1}$, $e=1.60\times {10}^{-19}\text{ C}$, $h=6.63\times {10}^{-34}\text{ J s}$, and others.

• SI prefixes and unit conversions (eV, u, kWh, ly, pc, AU, radians).

• Electrical circuit symbols and a chart of the electromagnetic spectrum with typical wavelengths.

• Equations for each theme, A to E, with the additional higher level equations listed separately.

What must be recalled

The booklet lists equations, not definitions, and it leaves out some relationships that follow from others. These must be remembered or derived:

• The definitions of quantities and units, such as pressure, specific heat capacity, emf, the volt and the ampere.

• The surface area of a sphere, $4\pi r^2$, needed for luminosity and intensity.

• The wavelengths of harmonics on strings and in pipes, from the pattern of nodes and antinodes.

• The refractive index as $n={\displaystyle \frac{c}{v}}$, and the critical angle from Snell’s law with a refraction angle of 90°.

• The radius of a charged particle’s path in a magnetic field, from $qvB={\displaystyle \frac{m{v}^2}{r}}$.

• Kepler’s third law, from equating gravitational and centripetal force.

• The number of nuclei remaining after $n$ half-lives, $N=N_0{\left(\frac{1}{2}\right)}^n$.

• The mean intensity of sunlight over the whole Earth's surface, $S\mathrm{/}4$.

Data that are not constants, such as the mass and radius of the Earth, specific heat capacities, latent heats, refractive indices or the speed of sound in air, are given in the question when they are needed.

Self-check

• To do:

  I can name the six SI base units used in physics and the quantities they measure.

• To do:

  I can express any derived unit in SI base units by following its defining equation.

• To do:

  I can check an equation for homogeneity and find the unit of an unknown constant.

• To do:

  I can convert prefixes, including squared and cubed units and speeds in km h⁻¹.

• To do:

  I can convert between eV and J, u and kg, kWh and J, and °C and K.

• To do:

  I can count significant figures, write numbers in standard form and round answers sensibly.

• To do:

  I can state the order of magnitude of a quantity and make an estimate with stated assumptions.

• To do:

  I can find equations and constants quickly in the data booklet and know which relationships must be recalled.