OCR A Level Physics A: Units, measurement uncertainty and vectors (H556)

This guide teaches sections 2.1, 2.2 and 2.3 of Module 2 in the OCR Advanced GCE in Physics A specification (H556), Version 3.0 (March 2026), for first assessment in 2017. It covers units, errors and uncertainties, and scalars and vectors, with worked examples. For a shorter overview, see the Foundations of Physics study guide. Condensed recall is in the revision notes and exam-style work is in the practice questions. Course pages: OCR A Level Physics hub, checklist and free diagnostics.

What this unit covers

Section	What you must be able to do
2.1.1(a)-(b)	Give a quantity as number and unit; estimate quantities
2.1.2(a)-(f)	Base units, derived units, homogeneity, prefixes, axis and column labels
2.2.1(a)-(b)	Systematic (including zero) and random errors; precision and accuracy
2.2.1(c)	Absolute and percentage uncertainties when data are combined
2.2.1(d)	Error bars, best-fit and worst lines, percentage difference
2.3.1(a)-(b)	Scalars and vectors with examples; adding and subtracting vectors
2.3.1(c)-(d)	Vector triangle for two coplanar vectors; perpendicular components

2.1.1 Physical quantities and estimates

A physical quantity is a number multiplied by a unit. “The force is 12” is incomplete; “12 N” is a measurement. The unit is part of the answer.

Make estimates by rounding to one significant figure and stating each assumption.

Worked example. Estimate the power of an adult running up a flight of 20 steps in 9 s.

height of one step ≈ 0.18 m, so rise h = 20 × 0.18 = 3.6 m
mass m ≈ 65 kg, g = 9.81 m s⁻²
energy gained = mgh = 65 × 9.81 × 3.6 ≈ 2300 J
power = energy / time = 2300 / 9 ≈ 260 W

A result of 20 W or 20 kW would tell you to recheck.

2.1.2 SI units

Base quantities

This specification uses six base quantities.

Symbol	Unit	Quantity
m	metre	length
kg	kilogram	mass
s	second	time
A	ampere	current
K	kelvin	temperature
mol	mole	amount of substance

Derived units

Other units are built from these. Momentum is mass × velocity, so its unit is kg m s⁻¹. Density is mass / volume, so its unit is kg m⁻³. Some derived units in base form:

Unit	In base units
pascal, Pa	kg m⁻¹ s⁻²
volt, V	kg m² s⁻³ A⁻¹
newton, N	kg m s⁻²
watt, W	kg m² s⁻³
joule, J	kg m² s⁻²

Derive these from defining equations. For the volt, use V = P / I, so kg m² s⁻³ divided by A.

Homogeneity

An equation is homogeneous if every term has the same base units. Check each term in a sum.

Worked example 1. The speed of a wave on a stretched string is v = √(T / μ), where T is tension (N) and μ is mass per unit length (kg m⁻¹). Is it homogeneous?

T / μ = (kg m s⁻²) / (kg m⁻¹) = m² s⁻²
√(m² s⁻²) = m s⁻¹, which is the unit of speed

It is homogeneous.

Worked example 2. A student writes s = ut + ½at. The term ut has unit m but at has unit m s⁻¹, so the equation is wrong. The correct term is ½at².

Homogeneity cannot detect a missing factor such as ½ or 2π. It only shows the units balance.

Prefixes

Factor	Name	Symbol
10⁻¹²	pico	p
10⁻⁹	nano	n
10⁻⁶	micro	μ
10⁻³	milli	m
10⁻²	centi	c
10⁻¹	deci	d
10³	kilo	k
10⁶	mega	M
10⁹	giga	G
10¹²	tera	T

Conversions.

450 nm = 450 × 10⁻⁹ m = 4.5 × 10⁻⁷ m
2.5 mm² = 2.5 × (10⁻³)² m² = 2.5 × 10⁻⁶ m²   (square the prefix too)
0.85 g cm⁻³ = 0.85 × 10⁻³ kg / 10⁻⁶ m³ = 850 kg m⁻³
72 km h⁻¹ = 72 × 1000 m / 3600 s = 20 m s⁻¹

Labelling axes and columns

Write a heading as quantity / unit, for example speed / m s⁻¹ or time / s. The unit then divides the quantity, so the entries below it are pure numbers. Do not put units on every cell.

2.2.1 Errors, precision and accuracy

Systematic and random errors

A systematic error shifts every reading by the same amount or the same proportion in one direction. A zero error is one example: a micrometer reads 0.02 mm with its jaws closed. Averaging does not remove it; correct for it.

A random error scatters readings unpredictably, for example reaction time with a stopwatch. Repeating and averaging reduces its effect.

Worked example. The micrometer above reads 1.46 mm on a wire, so the diameter is 1.46 − 0.02 = 1.44 mm.

Precision and accuracy

Readings are precise when they agree closely with each other. A result is accurate when it is close to the true value. A balance that reads 0.9 g too high every time is precise but not accurate. Precision depends on random error. Accuracy is damaged by systematic error too.

Uncertainty in a measurement

An uncertainty is stated as ± a value. For one scale reading, a common estimate is half the smallest division. For repeats, a common approach is half the range. The specification does not expect rigorous statistics.

Worked example. Five timings are 1.52, 1.58, 1.55, 1.49 and 1.56 s.

mean = 7.70 / 5 = 1.54 s
half range = (1.58 − 1.49) / 2 = 0.045 s
percentage uncertainty = 0.045 / 1.54 × 100 = 2.9 %

Write the result as 1.54 ± 0.05 s.

Percentage uncertainty is (absolute / value) × 100 and has no unit.

Combining uncertainties

Operation	Rule
a + b or a − b	add the absolute uncertainties
a × b or a / b	add the percentage uncertainties
aⁿ	multiply the percentage uncertainty by n

For a power, use the size of n. A square root is n = ½, so halve the percentage uncertainty.

Worked example (addition and subtraction). A liquid warms from 21.5 ± 0.5 °C to 64.0 ± 0.5 °C. The rise is 64.0 − 21.5 = 42.5 °C. The uncertainties add: ± 1.0 °C. That is 1.0 / 42.5 = 2.4 %. Subtracting similar numbers inflates the percentage uncertainty.

Worked example (products and powers). A solid cylinder has mass 214.6 ± 0.2 g, diameter 2.40 ± 0.02 cm and height 6.10 ± 0.05 cm. Find its density.

V = π d² h / 4 = π × 2.40² × 6.10 / 4 = 27.6 cm³
ρ = m / V = 214.6 / 27.6 = 7.78 g cm⁻³
% in m = 0.2 / 214.6 × 100 = 0.093 %
% in d = 0.02 / 2.40 × 100 = 0.833 %   (counts twice)
% in h = 0.05 / 6.10 × 100 = 0.820 %
total % = 0.093 + 2(0.833) + 0.820 = 2.58 %
absolute = 2.58 % of 7.78 = 0.20 g cm⁻³

So ρ = 7.78 ± 0.20 g cm⁻³. The diameter dominates because it is squared.

Graphs, error bars and percentage difference

Plot the data with error bars. A ± 2 mm uncertainty gives a bar 4 mm long. Draw the line of best fit with points balanced about it. A worst line is the steepest or shallowest straight line that still passes through all the bars.

Worked example. Weights are hung on a spring and each extension x is read from a ruler with uncertainty ± 2 mm. Force F is taken as exact.

F / N	1.0	2.0	3.0	4.0	5.0
x / mm	12	25	36	49	60

best-fit gradient = 12.0 mm N⁻¹
steepest line: (1.0, 10) to (5.0, 62), gradient = 52 / 4 = 13.0 mm N⁻¹
shallowest line: (1.0, 14) to (5.0, 58), gradient = 44 / 4 = 11.0 mm N⁻¹
uncertainty in gradient = (13.0 − 11.0) / 2 = 1.0 mm N⁻¹ (8.3 %)

Both worst lines pass inside every bar. Since k = F / x, the gradient of x against F is 1/k. So k = 1000 / 12.0 = 83 N m⁻¹, with the same 8.3 % uncertainty, giving k = 83 ± 7 N m⁻¹. The intercept uncertainty comes from the intercepts of the best and worst lines in the same way.

Percentage difference. This compares your result with a reference value. A common form is

percentage difference = |measured − reference| / reference × 100

The specification names the term without giving a formula, so state the one you use. For the cylinder density 7.78 g cm⁻³ against a reference of 7.87 g cm⁻³, the difference is 0.09 / 7.87 × 100 = 1.1 %. This is smaller than the 2.6 % uncertainty, so the result agrees with the reference within its uncertainty.

2.3.1 Scalars and vectors

A scalar is fully described by a size. A vector needs a size and a direction.

Scalars	Vectors
distance, speed, mass, time	displacement, velocity, acceleration
energy, power, work	force, weight, momentum
temperature, charge, density	field strength

Someone who jogs once round a 400 m track has covered 400 m but ends with zero displacement.

Adding and subtracting vectors

Add vectors tip to tail. Draw the first, start the second at its tip, and the resultant joins the start of the first to the tip of the second. To subtract, add the reversed vector: a − b = a + (−b).

Worked example (change in velocity). A car travels at 18 m s⁻¹ north and then turns to travel at 18 m s⁻¹ east. Find the change in velocity.

Δv = v(final) − v(initial) = (east 18) + (south 18)
magnitude = √(18² + 18²) = 25.5 m s⁻¹
direction = 45° south of east

The speed is unchanged but the velocity is not.

Vector triangle for two coplanar vectors

For perpendicular vectors use Pythagoras and tangent. A boat steers at 2.4 m s⁻¹ across a river that flows at 0.9 m s⁻¹.

speed = √(2.4² + 0.9²) = 2.56 m s⁻¹
angle to the intended line = tan⁻¹(0.9 / 2.4) = 20.6°

For other angles, draw the triangle and use the cosine and sine rules, or draw it to scale. Two forces of 12 N and 7 N act at a point with 50° between their lines of action. Placing them tip to tail makes the interior angle 130°, because 180° − 50° = 130°.

R² = 12² + 7² − 2(12)(7) cos 130° = 193 + 168 cos 50° = 301.0
R = 17.3 N
sin α / 7 = sin 130° / 17.35, so α = 18.0° from the 12 N force

Check by components: along the 12 N line 12 + 7 cos 50° = 16.50 N, and perpendicular 7 sin 50° = 5.36 N. Then √(16.50² + 5.36²) = 17.3 N.

For a scale drawing, choose a scale such as 1 cm to 2 N, use a protractor for angles, and state the scale.

Resolving into perpendicular components

A vector F at angle θ to the x-axis has components Fx = F cos θ and Fy = F sin θ. They act independently.

Worked example. A sledge is pulled by a rope with tension 85 N at 28° above the horizontal.

horizontal = 85 cos 28° = 75.1 N
vertical = 85 sin 28° = 39.9 N

Worked example (slope). A 12 kg block sits on a 35° slope. Its weight is 12 × 9.81 = 117.7 N. The weight makes an angle of 35° with the line perpendicular to the slope, so:

component along the slope = 117.7 sin 35° = 67.5 N
component perpendicular to the slope = 117.7 cos 35° = 96.4 N

Sine goes with the component opposite the angle, cosine with the component adjacent to it.

Common errors

• Giving no unit, or treating 1 mm² as 10⁻³ m².
• Adding percentage uncertainties for a sum, or absolute uncertainties for a product.
• Forgetting that a squared quantity counts its percentage uncertainty twice.
• Taking a gradient from data points rather than points on the line.
• Using sin where cos belongs after misreading which angle is given.
• Treating weight, momentum or velocity as scalars.

Next, try the practice questions. The practical skills guide shows these methods in experiments.

Official syllabus

OCR, Advanced GCE in Physics A (H556) specification, Version 3.0 (March 2026), for first assessment in 2017. Sections 2.1.1, 2.1.2, 2.2.1 and 2.3.1.