Measurement, Units, Dimensions and Vectors

What you need to know

• There are seven fundamental (base) quantities in the SI system: length (metre), mass (kilogram), time (second), electric current (ampere), thermodynamic temperature (kelvin), amount of substance (mole) and luminous intensity (candela). Everything else you meet in Physics is derived from these.
• A derived unit is just base units multiplied or divided: the newton is kg m s^-2, the joule is kg m^2 s^-2, the watt is kg m^2 s^-3, the pascal is kg m^-1 s^-2. If you can rebuild these from F = ma, W = Fd and P = W/t you never need to memorise them.
• Dimensions are written with capital letters in square brackets: [M] for mass, [L] for length, [T] for time. Velocity is LT^-1, acceleration LT^-2, force MLT^-2, work ML^2T^-2, power ML^2T^-3, pressure ML^-1T^-2.
• A correct equation must be dimensionally homogeneous: every term on both sides carries the same dimensions. Check v^2 = u^2 + 2as this way, L^2T^-2 = L^2T^-2 + (L)(LT^-2) = L^2T^-2, so the equation passes.
• Dimensional analysis can find the form of a law but never the dimensionless constant. It will tell you that T is proportional to sqrt(l/g) for a simple pendulum, but it can never produce the 2 pi.
• Prefixes must be converted before substitution: 1 km = 10^3 m, 1 cm = 10^-2 m, 1 mm = 10^-3 m, 1 micrometre = 10^-6 m, 1 nm = 10^-9 m, 1 tonne = 10^3 kg, 1 g = 10^-3 kg.
• To convert km/h to m/s divide by 3.6; to convert m/s to km/h multiply by 3.6. So 72 km/h = 20 m/s and 25 m/s = 90 km/h.
• Area converts as the square of the length factor and volume as the cube: 1 m^2 = 10^4 cm^2 and 1 m^3 = 10^6 cm^3. This is the single commonest conversion slip in the whole paper.
• A scalar has magnitude only: distance, speed, mass, time, work, energy, power, temperature, density, pressure, potential difference, electric charge. A vector has magnitude and direction: displacement, velocity, acceleration, force, weight, momentum, impulse, torque, electric field intensity.
• Two perpendicular vectors are added by Pythagoras: R = sqrt(A^2 + B^2), with direction tan(theta) = B/A. For two vectors at any angle theta between them use the parallelogram law, R = sqrt(A^2 + B^2 + 2AB cos theta).
• Resolving is the reverse of adding: a vector F at angle theta to the horizontal has horizontal component F cos theta and vertical component F sin theta. Always measure theta from the axis you are taking the cosine along.
• An object is in equilibrium under coplanar forces when the sum of horizontal components is zero and the sum of vertical components is zero; for three non-parallel forces you may instead use the triangle of forces or Lami's theorem.
• Choose the instrument by the size of what you are measuring: metre rule reads to 0.1 cm, vernier calliper to 0.01 cm, micrometer screw gauge to 0.001 cm. A stopwatch, not a wall clock, times a pendulum, and you time 20 oscillations and divide to cut reaction-time error.
• Errors come in two kinds. Random error (a shaky hand, parallax, reaction time) scatters readings about the true value and is reduced by repeating and averaging. Systematic error (zero error on a balance, a stretched tape) shifts every reading the same way and averaging will not remove it.