P: Quantities and units in mechanics — AQA A-Level Mathematics
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P: Quantities and units in mechanics explained
Mechanics begins with measurement. The SI system defines seven base units, but this section focuses on three fundamental quantities: length (metre, m), time (second, s) and mass (kilogram, kg).
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Every other mechanical quantity is derived from these. Velocity is displacement per unit time, so its unit is m s⁻¹. Acceleration is velocity change per unit time, so m s⁻². Force is mass times acceleration, giving the newton (N = kg m s⁻²). Weight is the force of gravity on a mass, W = mg, also measured in newtons. A moment is the turning effect of a force about a point, M = Fd, so its unit is the newton metre (N m). When solving problems, always convert to SI base units before substituting into formulae, and check that your final unit matches the quantity you claim to have found.
Your focus
- Identify length, time and mass as fundamental quantities and give their SI units.
- Derive the units of velocity, acceleration, force, weight and moment from defining equations.
- Convert between units and apply the correct unit to each mechanical quantity in calculations.
P: Quantities and units in mechanics exam tips
Marking Points
- States the three fundamental quantities as length, time and mass, with SI units metre (m), second (s) and kilogram (kg).
- Defines velocity as rate of change of displacement and gives its unit as m s⁻¹.
- Defines acceleration as rate of change of velocity and gives its unit as m s⁻².
- Derives the newton from F = ma as kg m s⁻² and uses it for both force and weight.
- Distinguishes weight as a force (W = mg, in newtons) from mass (in kilograms).
- Defines moment as force times perpendicular distance from a pivot and gives its unit as N m.
Examiner Tips
- 💡Write the unit alongside every numerical answer; a correct value with a wrong or missing unit can lose marks.
- 💡When a question asks you to 'show that' a unit is equivalent, expand the defining equation in base units, e.g. N = kg m s⁻².
- 💡Check whether a quantity is a vector or scalar before giving its unit; velocity and acceleration are vectors, speed and mass are scalars.
Common Mistakes
- Treating mass and weight as the same: mass is a scalar in kg, weight is a force in N; correct by always writing W = mg and labelling units.
- Using grams or centimetres in formulae: convert to kg and m first, otherwise derived units will be wrong.
- Writing the unit of a moment as N/m or J: a moment is force times distance, so the unit is N m, not N m⁻¹ or joules.