Singapore-Cambridge A-Level Physics (H2)

Base quantities and SI units

A physical quantity is a number multiplied by a unit, so a statement such as a length of $2.5\,\mathrm{m}$ means 2.5 times the metre. Every unit used in physics is built from a small set of SI base units, each attached to a base quantity.

All other quantities are derived, and their units are products or quotients of the base units. A unit that has been given its own name, such as the newton, is shorthand for such a combination.

Each entry follows from a defining equation. Force is mass times acceleration, so $\mathrm{N} = \mathrm{kg\,m\,s^{-2}}$. Energy is force times distance, so $\mathrm{J} = \mathrm{N\,m}$. Power is energy per unit time, so $\mathrm{W} = \mathrm{J\,s^{-1}}$. The volt is a joule per coulomb, which is $\mathrm{kg\,m^2\,s^{-2}}$ divided by $\mathrm{A\,s}$, and the ohm is a volt per ampere.

Prefixes

Prefixes scale a unit by a power of ten and are written directly before the unit symbol.

A prefix belongs to the whole unit, so it is raised to the same power as the unit. Thus $1\,\mathrm{cm^3} = (10^{-2}\,\mathrm{m})^3 = 10^{-6}\,\mathrm{m^3}$ and not $10^{-2}\,\mathrm{m^3}$. In the same way $1\,\mathrm{mm^2} = 10^{-6}\,\mathrm{m^2}$, which matters when a wire's cross-sectional area is found from a diameter in millimetres. The kilogram is the one base unit that already carries a prefix, and masses in grams are converted to kilograms before they are used in equations.

Non-SI units in common use

Four units outside the SI are used freely, each with a fixed conversion.

The kilowatt hour is the energy transferred by a 1 kW device running for one hour, so $1\,\mathrm{kWh} = 1000\,\mathrm{W}\times3600\,\mathrm{s}$. The electron volt is the energy gained by an electron moving through a potential difference of one volt, which is $e\times1\,\mathrm{V}$. The unified atomic mass unit is one twelfth of the mass of a carbon-12 atom.

Checking an equation by homogeneity

An equation is homogeneous when every term has the same base units. A physical equation must be homogeneous, because only quantities of the same kind can be added or equated. A check on the units of each side is therefore a quick test for a mistake in an equation, and a way of finding the units of a constant.

Take the period of a simple pendulum, $T = 2\pi\sqrt{l/g}$. The left side has unit s. On the right, $l/g$ has unit $\mathrm{m}/\mathrm{m\,s^{-2}} = \mathrm{s^2}$, so its square root is s, and the $2\pi$ has no unit. Both sides are in seconds, so the equation passes.

A second example is the equation $F = mv^2/r$ for the force on a body moving in a circle. The right side has unit $\mathrm{kg\,m^2\,s^{-2}\,m^{-1}} = \mathrm{kg\,m\,s^{-2}}$, which is the newton, so it is homogeneous.

The test has limits. A homogeneous equation can still be wrong by a pure number, since numbers such as $\tfrac12$ and $2\pi$ have no unit. The equation $s = ut + at^2$ is homogeneous and yet the correct one has a factor $\tfrac12$ on the last term. Homogeneity can show that an equation is wrong, but it cannot show that it is right.

Homogeneity can also give the unit of an unknown constant. If $F = kx$ then $k = F/x$ has unit $\mathrm{N\,m^{-1}}$. In the law of gravitation $F = Gm_1m_2/r^2$ the constant $G$ has unit $\mathrm{N\,m^2\,kg^{-2}}$, which is $\mathrm{m^3\,kg^{-1}\,s^{-2}}$ in base units.

Making estimates

An estimate is a reasoned value that is correct to within about a factor of ten. It is found by using typical sizes that are known, writing them in standard form and keeping one significant figure. Useful values include the mass of an adult (about $70\,\mathrm{kg}$), the height of a door (about $2\,\mathrm{m}$), the speed of a car on a motorway (about $30\,\mathrm{m\,s^{-1}}$), the mass of a typical apple (about $0.1\,\mathrm{kg}$), the density of water ($1000\,\mathrm{kg\,m^{-3}}$) and the mains voltage in Singapore ($230\,\mathrm{V}$).

Worked example: estimate the kinetic energy of a car travelling at motorway speed. A car has a mass of about $1.2\times10^{3}\,\mathrm{kg}$ and a speed of about $30\,\mathrm{m\,s^{-1}}$.

A second example: estimate the number of atoms in a glass of water. A glass holds about $250\,\mathrm{cm^3}$, which is a mass of about $0.25\,\mathrm{kg}$. The molar mass of water is about $0.018\,\mathrm{kg\,mol^{-1}}$, so there are about $0.25/0.018 \approx 14\,\mathrm{mol}$ of molecules, or $14\times6\times10^{23} \approx 8\times10^{24}$ molecules. Each has three atoms, giving about $2.5\times10^{25}$ atoms. The answer is only an order of magnitude, and it is better to say so than to quote more figures than the inputs allow.

When a quantity is estimated from a formula, a check on the order of magnitude of the answer also catches slips with powers of ten. A result of $10^{9}\,\mathrm{m\,s^{-1}}$ for the speed of a car signals at once that a prefix or a power has been lost.