Edexcel A-Level Physics

Base and derived quantities

Every physical quantity is a number multiplied by a unit, and the whole of physics can be written in terms of a small set of base quantities, each measured in an SI base unit. A base quantity is not defined in terms of any other quantity. The ones used in A-level physics are in the table.

A derived quantity is defined by an equation that links it to base quantities, and its unit is the matching combination of base units. Speed is length divided by time, so its unit is $\mathrm{m\,s^{-1}}$. Many derived units have their own names, and it is worth being able to unpack each into base units because that is how equations are checked.

Each entry comes from a defining equation. The newton follows from $F = ma$, the joule from $W = Fs$, the watt from $P = W/t$, the volt from $V = W/Q$ (energy per unit charge, so $\mathrm{J\,C^{-1}}$) and the ohm from $R = V/I$. The tesla comes from $F = BIl$, so $\mathrm{T} = \mathrm{N\,A^{-1}\,m^{-1}} = \mathrm{kg\,s^{-2}\,A^{-1}}$.

Some units are not SI but appear throughout the course. The electronvolt (eV) is the energy gained by an electron moving through a potential difference of one volt, so $1\,\mathrm{eV} = 1.60\times10^{-19}\,\mathrm{J}$. The atomic mass unit (u) is used for nuclear masses, and $1\,\mathrm{u} = 1.66\times10^{-27}\,\mathrm{kg}$.

Prefixes and standard form

Prefixes rescale a unit by a power of ten. The ones needed are:

Convert to base units before substituting into an equation, because an equation such as $F = ma$ only gives newtons when the mass is in kilograms. Areas and volumes need the factor squared or cubed: $1\,\mathrm{mm^2} = 10^{-6}\,\mathrm{m^2}$, not $10^{-3}$, and $1\,\mathrm{cm^3} = 10^{-6}\,\mathrm{m^3}$.

Checking equations with units

An equation is homogeneous when the base units on both sides are the same, and every correct equation is. The check does not prove an equation is right, since it cannot detect a dimensionless factor such as $\tfrac{1}{2}$ or $2\pi$, but it does show up a wrong one immediately.

Suppose a student proposes that the period of a mass on a spring is $T = 2\pi\sqrt{k/m}$, where $k$ is the spring constant in $\mathrm{N\,m^{-1}}$. The unit of $k/m$ is $\frac{\mathrm{N\,m^{-1}}}{\mathrm{kg}} = \frac{\mathrm{kg\,m\,s^{-2}\,m^{-1}}}{\mathrm{kg}} = \mathrm{s^{-2}}$ so the square root has unit $\mathrm{s^{-1}}$, which is a frequency and not a period. The correct equation, $T = 2\pi\sqrt{m/k}$, has unit $\sqrt{\mathrm{s^2}} = \mathrm{s}$.

The same method finds the unit of a constant. The gravitational constant $G$ appears in $F = Gm_1m_2/r^2$, so $G = Fr^2/(m_1m_2)$ and its unit is $\frac{\mathrm{kg\,m\,s^{-2}\times m^2}}{\mathrm{kg^2}} = \mathrm{m^3\,kg^{-1}\,s^{-2}}$

Estimating

An estimate is a value good to within about a factor of ten, reached by choosing sensible round numbers for each quantity and combining them. It is used to judge whether a calculated answer is reasonable and to solve problems where precise data are not given. The method is to break the quantity into parts that can each be guessed, to keep one significant figure, and to state the assumptions.

To estimate the useful power of a person running up a flight of stairs, take a mass of $60\,\mathrm{kg}$, a height gained of $3\,\mathrm{m}$ and a time of $10\,\mathrm{s}$. The energy gained is $mg\Delta h = 60\times9.81\times3 \approx 1.8\times10^{3}\,\mathrm{J}$, so the power is about $1.8\times10^{3}/10 \approx 2\times10^{2}\,\mathrm{W}$. The person's body transfers more than this, because muscles are not perfectly efficient, but the order of magnitude of $10^2\,\mathrm{W}$ is right.

A second example is the mass of air in a classroom measuring $8\,\mathrm{m}$ by $6\,\mathrm{m}$ by $3\,\mathrm{m}$. The volume is about $1.4\times10^{2}\,\mathrm{m^3}$, and air has a density of about $1.2\,\mathrm{kg\,m^{-3}}$, so the mass is about $1.7\times10^{2}\,\mathrm{kg}$, or roughly $2\times10^{2}\,\mathrm{kg}$.

Useful reference sizes for estimating are an atom at about $10^{-10}\,\mathrm{m}$, a nucleus at about $10^{-15}\,\mathrm{m}$, a person's mass of about $60$ to $80\,\mathrm{kg}$, a walking speed of about $1.5\,\mathrm{m\,s^{-1}}$ and atmospheric pressure of about $10^5\,\mathrm{Pa}$.

Scalars and vectors as quantities

A scalar has magnitude only (mass, time, energy, speed, distance, temperature), and a vector has magnitude and direction (force, velocity, displacement, acceleration, momentum, field strength). A vector is written in bold, $\mathbf{F}$, or with an arrow, $\vec{F}$, and its magnitude as $F$ or $\lvert\vec{F}\rvert$. Adding scalars is ordinary arithmetic, but adding vectors has to allow for direction, which is covered where vectors are used in mechanics.