Modelling, quantities and units

Modelling assumptions and what each one removes

A model replaces the real object by a simpler one that can be analysed. Calling a crate a particle means its size and shape are ignored, so all its mass is concentrated at one point, every force acts at that point, and rotation is excluded from the problem.

Each of the remaining modelling words removes one complication. A light string has no mass, so the tension is the same all along it, provided any pulley it passes over is smooth, since a rough pulley would let the tension differ across it. A smooth surface or pulley has no friction. An inextensible string cannot stretch, so while it is taut the objects it joins move together along it and their accelerations have the same magnitude along the string. For a taut string in a straight line, or over a single smooth pulley, that gives one shared acceleration in every equation. A uniform beam carries its weight at its midpoint and a rigid one does not bend.

A few more turn up in question stems. A rod is rigid and has length but no thickness. A lamina is flat with no thickness at all. A bead is threaded on a wire and can slide along it. A peg is a fixed point that a string can pass over. A further assumption is usually left implicit: air resistance and wind are ignored. Name it when a question asks for a criticism of the model.

Units and quantities

Mechanics runs on SI units: metres, seconds and kilograms, plus everything built from them, so newtons for force and m s⁻¹ and m s⁻² for motion. Convert before calculating: a speed left in km h⁻¹ and substituted into a suvat equation makes every subsequent value incorrect. Weight is a force, W = mg, with g = 9.8 m s⁻² unless a question says otherwise.

Quantities split into scalars, which have size only, and vectors, which have size and direction. Distance and speed are scalars; displacement and velocity are vectors. In one dimension the direction is carried entirely by the sign, so choose a positive direction at the start of a question, state it, and interpret the sign of every result against that choice.

Extension, dimensions: M, L and T

A unit is a chosen size for a quantity. A dimension is the kind of quantity it is, which is what is left once the choice of unit has been taken away. Mechanics needs three: mass M, length L and time T. Square brackets name one, so a mass has [m] = M, a distance [d] = L and a time [t] = T. This section and the next are an extension, not required for Edexcel 9MA0; AQA and OCR assess dimensional analysis in Further Mathematics.

Reading dimensions off units is the method, and it covers any quantity whose units you know. A speed in m s⁻¹ is a length over a time, so [v] = L T⁻¹, and an acceleration in m s⁻² gives [a] = L T⁻². A newton is a kilogram metre per second squared, so [F] = M L T⁻². Work is a force times a distance, [W] = M L² T⁻², and power is work per unit time, [P] = M L² T⁻³. Momentum mv and impulse Ft agree at M L T⁻¹, as they should, since the impulse-momentum principle sets one equal to the other.

Describe the drawing

Two stacked bands. The upper one turns each base unit into its dimension: kilogram into M, metre into L, second into T. The lower one lists seven mechanical quantities with their units on the left and the dimensions those units force on them on the right, from speed at L T to the power of minus one down to power at M L squared T to the power of minus three. Momentum and impulse arrive in the same right-hand cell, M L T to the power of minus one, which is the impulse-momentum principle showing up in the dimensions.

The reading runs both ways. Read M L⁻¹ T⁻² back into units and it is kg m⁻¹ s⁻², a newton per square metre, which is a pressure. That direction is worth practising, because it is how you name the unit of a quantity a question has handed you as a dimension.

Some quantities have no dimensions. A ratio of two quantities of the same kind cancels, so the coefficient of friction is a force over a force, the coefficient of restitution a speed over a speed, and an angle in radians an arc length over a radius. Each is dimensionless, written [μ] = 1. Pure numbers are dimensionless too, ½ and 2 and π alike, which is why the method cannot determine a numerical constant. The argument of a sine, an exponential or a logarithm must also be dimensionless.

Extension, checking and predicting with dimensions

Terms that are added, subtracted or equated carry the same dimensions, so the two sides of a correct equation agree. That is a test you can run on anything. Take P = Fv: [F][v] = M L T⁻² × L T⁻¹ = M L² T⁻³, which is [P], so the relation survives. Take v² = u² + 2as: the left is L² T⁻², and [a][s] = L T⁻² × L = L² T⁻² matches it, with the 2 contributing nothing. Disagreement is decisive, since an inconsistent equation is wrong. Agreement is weaker: an incorrect dimensionless factor passes the test.

Describe the drawing

The equation s equals ut plus a half a t cubed, written once with the exponents of M, L and T under each term. The left side and the term ut both read nought, one, nought and are ticked. The last term reads nought, one, one, one power of time too many, and is crossed out. Beside it, greyed, sits the repair: a half a t squared, whose triple matches the other two.

Run the test backwards and it predicts. Propose that a quantity is a product of powers of the quantities it could depend on, with a dimensionless constant k in front, write the dimensions of both sides, and match the index of M, then of L, then of T. Three equations come out, one per base dimension, and they solve like any simultaneous set. The proposal is a modelling assumption rather than a derivation, and k is left undetermined, so what the argument delivers is the shape of a formula.