Notes · MathematicsUK · A-Levels

Quantities and units in mechanics

This foundational mechanics topic establishes the quantities, units and modelling language used throughout the mechanics content. It covers SI base and derived units for length, time, mass, velocity, acceleration and force, the standard modelling assumptions (particle, light, smooth, inextensible), and the distinction between scalars and vectors.

3 sections~8 min reading time3 competenciesLevel Foundation 1 · Standard 2

T·171717 / 20

Exam profile

AO1 · Use correct SI units for mechanical quantitiesAO2 · State and criticise modelling assumptionsAO3 · Set up mechanics models with appropriate simplifications

Operators:stateexplaindescribeshowjustify

basic level

AS-Level covers SI units and the standard modelling assumptions in mechanics.

higher level

The full A-Level applies the same language throughout kinematics, forces and moments, expecting fluent use of assumptions.

Depth

Text

Contents · 3 sections

Quantities and units in mechanics

SI units and derived quantities#

FoundationDfE GCE Mathematics — mechanics quantitiesAQA 7357 (Mechanics)

Quantities and their SI units

Fig. 1Base units (m, kg, s) combine into the derived units for velocity, acceleration and force.

Mechanics is built on the SI system of units. The base quantities you need are length, measured in metres (m), time in seconds (s), and mass in kilograms (kg). Every other mechanical quantity is derived from these, and using consistent SI units throughout a calculation — converting any centimetres, grams or minutes at the outset — is the single most reliable way to avoid errors.

Velocity is a rate of change of displacement, so its unit is metres per second, written

\text{m s}^{-1}

; acceleration is a rate of change of velocity, giving

\text{m s}^{-2}

. These derived units follow directly from the definitions, and writing them with negative-index notation (

\text{m s}^{-1}

rather than m/s) is the convention expected in the A-Level.

Force is defined through Newton's second law and has the unit newton (N). One newton is the force that gives a mass of one kilogram an acceleration of one metre per second squared, so

1\ \text{N} = 1\ \text{kg m s}^{-2}

. Weight, being a force, is also measured in newtons, and is quite distinct from mass (in kilograms) — a distinction that must be kept clear.

The acceleration due to gravity is a standard constant used throughout mechanics, taken as

g = 9.8\ \text{m s}^{-2}

(some questions use

9.81

). The weight of a mass

m

is the force

W = mg

, so a 2 kg mass has weight

2 \times 9.8 = 19.6\ \text{N}

. Knowing this relationship and the value of

g

is assumed throughout the mechanics papers.

1\ \text{N} = 1\ \text{kg}\,\text{m}\,\text{s}^{-2}, \qquad W = mg

The newton and weight

Force is mass times acceleration; weight is mass times $g$ , and is measured in newtons.

Worked example

Weight from mass

Find the weight of a 4 kg object, taking $g = 9.8\ \text{m s}^{-2}$ .

Use W = mg
$W = mg = 4 \times 9.8$ .
Evaluate
$W = 39.2\ \text{N}$ .
$W = 4 \times 9.8 = 39.2\ \text{N}$

Result: The weight is $39.2\ \text{N}$ (directed vertically downwards).

Typical mistakes

Confusing mass and weight, or quoting a weight in kilograms.
Mixing units within a calculation (e.g. leaving a distance in cm).

Active revision

A particle has mass 500 g. Find its weight in newtons, taking $g = 9.8\ \text{m s}^{-2}$ , and state the units of its acceleration.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Modelling assumptions in mechanics#

StandardDfE GCE Mathematics — mechanics quantitiesAQA 7357 (Mechanics)

A particle in equilibrium on a surface

Fig. 2Modelled as a particle on a smooth surface, only the weight and the normal reaction act.

Mechanics problems are made tractable by modelling assumptions, each of which removes a complication so the mathematics stays manageable. Modelling an object as a particle treats it as a point mass with no size, so its rotation and dimensions can be ignored and all forces act at a single point. This is reasonable when the object's size is irrelevant to the motion, as for a ball in flight.

Strings and rods carry their own standard assumptions. A light string or rod is assumed to have negligible mass, so it does not add to the forces; an inextensible string keeps a constant length, so connected particles share the same speed and acceleration; and a rigid rod does not bend. A rod may also be uniform, meaning its mass is evenly distributed so its weight acts at its midpoint (its centre of mass).

Surfaces and the medium are modelled too. A smooth surface exerts no friction, so only a normal reaction acts; a rough surface does exert friction. The phrase 'air resistance is negligible' removes the drag force, and modelling a pulley as smooth means the tension is the same throughout the string passing over it. Each assumption should be recognised and its effect understood.

Crucially, you must be able both to state the assumptions a model uses and to criticise them — to say what effect an assumption has on the answer and when it might break down. Ignoring air resistance, for example, over-estimates a projectile's range; assuming a string is light is reasonable for thread but not for a heavy chain. This evaluation is exactly the AO3 skill the mechanics questions reward.

Worked example

Interpreting assumptions

A ball is thrown and modelled as a particle with air resistance neglected. Explain the effect of each assumption.

Particle
Treating the ball as a particle ignores its size and spin, so all its weight acts at a single point and rotation is not modelled.
No air resistance
Neglecting air resistance removes the drag force, simplifying the motion to constant acceleration under gravity.
Effect
Real air resistance would slow the ball, so the model over-estimates the range and the maximum height reached.

Result: The assumptions give a constant-acceleration model; neglecting air resistance over-estimates range and height.

Typical mistakes

Forgetting that a smooth surface means no friction acts (only the normal reaction).
Assuming a rod's weight acts at one end rather than at its centre when it is uniform.

Active revision

A child on a sledge slides down a snowy slope. State three modelling assumptions you would make, and comment on the effect of one of them.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Scalars, vectors and dimensional reasoning#

StandardDfE GCE Mathematics — mechanics quantitiesAQA 7357 (Mechanics)

Scalar and vector quantities

Fig. 3Vectors carry direction and must be combined by vector methods; scalars have magnitude only.

A scalar quantity has magnitude only, while a vector has both magnitude and direction — a distinction that is central in mechanics. Distance and speed are scalars; displacement and velocity are the corresponding vectors. Mass is a scalar, but weight and force are vectors. Keeping track of which quantities carry direction is essential, because vectors must be combined by vector methods, not simple arithmetic.

The distinction has real consequences. A particle can travel a large distance yet have zero displacement if it returns to its start, and its average speed (total distance over time) can differ from the magnitude of its average velocity (displacement over time). Recognising when a problem requires the vector quantity — often signalled by a change of direction — prevents a common class of errors.

Forces are vectors and combine by the triangle or parallelogram law, exactly as in the vectors topic. When several forces act, their vector sum is the resultant; when they balance, the resultant is zero and the particle is in equilibrium. Resolving a force into perpendicular components, usually horizontal and vertical, is the standard technique for handling forces at angles.

Dimensional consistency is a useful check: both sides of a physical equation must have the same units. In

v = u + at

, for instance, each term has units of

\text{m s}^{-1}

(

at

being

\text{m s}^{-2} \times \text{s}

), confirming the equation is dimensionally sound. A formula whose terms do not match in units is certainly wrong, so this check catches errors quickly.

\text{average speed} = \frac{\text{distance}}{\text{time}}, \qquad \text{average velocity} = \frac{\text{displacement}}{\text{time}}

Scalar and vector averages

Speed uses total distance; velocity uses displacement, which has direction.

Worked example

Distance versus displacement

A particle moves 8 m east, then 3 m west, in a total of 5 s. Find the average speed and the magnitude of the average velocity.

Total distance
Distance travelled $= 8 + 3 = 11$ m; average speed $= \dfrac{11}{5} = 2.2\ \text{m s}^{-1}$ .
Net displacement
Displacement $= 8 - 3 = 5$ m east.
Average velocity
Magnitude $= \dfrac{5}{5} = 1\ \text{m s}^{-1}$ east.
$\frac{5}{5} = 1\ \text{m s}^{-1}$

Result: Average speed $2.2\ \text{m s}^{-1}$ ; average velocity $1\ \text{m s}^{-1}$ east — they differ because the particle reversed direction.

Typical mistakes

Treating displacement as if it were distance when the direction of motion changes.
Adding forces as if they were scalars instead of combining them as vectors.

Active revision

A runner completes one lap of a 400 m track in 50 s, returning to the start. Find the average speed and the magnitude of the average velocity, and explain the difference.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

References & sources

Sources

Department for Education

Mathematics: AS and A level content (GCE subject content)

AQA

AQA A-level Mathematics 7357 specification

Quantities and units in mechanics

3 sections~8 min reading time3 competenciesLevel Foundation 1 · Standard 2

Quantities and units in mechanics

Key points

Weight from mass

Key points

Interpreting assumptions

Key points

Distance versus displacement

Sources

Quantities and units in mechanics

Key points

Weight from mass

Key points

Interpreting assumptions

Key points

Distance versus displacement

Sources