OCR A Level Physics A (H556)

Module 2 — Foundations in Physics

1. Specification Coverage

OCR Physics A Module 2 contains three major topic areas:

• 2.1 Physical quantities and units
• 2.2 Making measurements and analysing data
• 2.3 Nature of quantities

These foundations are used throughout the whole H556 qualification and can be assessed synoptically in questions from later modules.

SPEC: 2.1 — Physical quantities and units

SPEC: 2.1.1 — Physical quantities

You must understand that:

• physical quantities have a numerical value and a unit.

SPEC: 2.1.2 — S.I. units

You must know and apply:

• the Système Internationale (S.I.) base quantities and units required by OCR:
  • mass — kilogram, kg
  • length — metre, m
  • time — second, s
  • electric current — ampere, A
  • temperature — kelvin, K
  • amount of substance — mole, mol
• derived units formed from S.I. base units;
• expression of derived units in base-unit form;
• checking the homogeneity of physical equations using S.I. base units;
• prefixes and symbols:
  • pico, p, (10^{-12})
  • nano, n, (10^{-9})
  • micro, (\mu), (10^{-6})
  • milli, m, (10^{-3})
  • centi, c, (10^{-2})
  • deci, d, (10^{-1})
  • kilo, k, (10^{3})
  • mega, M, (10^{6})
  • giga, G, (10^{9})
  • tera, T, (10^{12})

SPEC: 2.2 — Making measurements and analysing data

SPEC: 2.2.1 — Measurements and uncertainties

You must be able to understand and apply:

• measurements and their associated uncertainties;
• absolute uncertainty;
• fractional uncertainty;
• percentage uncertainty;
• uncertainty when data are combined by:
  • addition
  • subtraction
  • multiplication
  • division
  • raising to powers
• graphical treatment of errors and uncertainties;
• error bars where appropriate;
• a line of best fit;
• a worst acceptable line;
• absolute and percentage uncertainty in a gradient/intercept where required;
• percentage difference;
• appropriate significant figures and decimal places;
• practical judgement about precision, accuracy, resolution, random and systematic effects.

SPEC: 2.3 — Nature of quantities

SPEC: 2.3.1 — Scalars and vectors

You must understand and apply:

• the distinction between scalar and vector quantities;
• examples of scalar and vector quantities;
• vector representation;
• addition and subtraction of vectors;
• finding the resultant of two or more coplanar vectors;
• resolving a vector into two perpendicular components;
• use of: [ F_x=F\cos\theta ] and [ F_y=F\sin\theta ] where the angle convention makes these expressions appropriate.

2. Core Content — Extreme Detail

2.1 SPEC: 2.1 — Physical Quantities and Units

2.1.1 SPEC: 2.1.1 — Physical quantities

A physical quantity is a measurable property that is represented by a numerical magnitude together with a unit.

DEFINITION — Physical quantity
A measurable property expressed by a numerical value and a unit.

For example:

[ 5.0,\text{m} ]

contains:

• numerical value: (5.0)
• unit: metre, (\text{m})

A number by itself is normally not a complete physical measurement.

For instance, writing:

[ v=12 ]

is incomplete if the speed is meant to be:

[ v=12,\text{m s}^{-1} ]

The unit is part of the physical meaning of the answer.

A* exam habit

Whenever you calculate a physical quantity:

1. identify the equation;
2. convert quantities to compatible units;
3. substitute numerical values;
4. calculate;
5. check the magnitude;
6. attach the correct unit;
7. round appropriately.

2.2 SPEC: 2.1.2 — S.I. Units

The Système Internationale, usually abbreviated S.I., provides a standardised international system of units.

DEFINITION — S.I. unit
A unit belonging to the internationally agreed Système Internationale system of measurement.

2.2.1 S.I. base quantities and units

For OCR Physics A, the six required base quantities are:

Critical OCR detail

The S.I. base unit for mass is the kilogram, not the gram.

Therefore:

[ 1,\text{g}=10^{-3},\text{kg} ]

and:

[ 1,\text{mg}=10^{-6},\text{kg} ]

A common calculation error is to substitute a mass stated in grams directly into a formula whose other quantities are expressed in S.I. units.

2.2.2 Derived quantities and derived units

A derived quantity is defined using combinations of other physical quantities.

A derived unit is produced by combining S.I. base units.

DEFINITION — Derived unit
A unit obtained by algebraic combination of S.I. base units.

Examples include:

• velocity
• acceleration
• force
• energy
• power
• pressure
• charge
• potential difference
• resistance
• density
• momentum

2.2.3 Velocity

[ v=\frac{\Delta x}{\Delta t} ]

Unit:

[ \text{m s}^{-1} ]

This means metres per second.

2.2.4 Acceleration

[ a=\frac{\Delta v}{\Delta t} ]

Unit:

[ \text{m s}^{-2} ]

2.2.5 Force and the newton

From Newton's second law:

[ F=ma ]

Base-unit form:

[ [F]=\text{kg}\times\text{m s}^{-2} ]

Therefore:

[ \boxed{1,\text{N}=1,\text{kg m s}^{-2}} ]

2.2.6 Work and energy: the joule

For a constant force acting in the direction of displacement:

[ W=Fs ]

So:

[ [W]=\text{N m} ]

and therefore:

[ \boxed{1,\text{J}=1,\text{N m}} ]

Using base units:

[ 1,\text{J}

1,\text{kg m}^{2}\text{s}^{-2} ]

2.2.7 Power: the watt

[ P=\frac{E}{t} ]

Therefore:

[ \boxed{1,\text{W}=1,\text{J s}^{-1}} ]

Base units:

[ 1,\text{W}

1,\text{kg m}^{2}\text{s}^{-3} ]

2.2.8 Pressure: the pascal

[ p=\frac{F}{A} ]

Therefore:

[ \boxed{1,\text{Pa}=1,\text{N m}^{-2}} ]

Base units:

[ 1,\text{Pa}

1,\text{kg m}^{-1}\text{s}^{-2} ]

2.2.9 Charge: the coulomb

[ Q=It ]

Therefore:

[ \boxed{1,\text{C}=1,\text{A s}} ]

2.2.10 Potential difference: the volt

[ V=\frac{W}{Q} ]

Therefore:

[ 1,\text{V}=1,\text{J C}^{-1} ]

Using base units:

[ 1,\text{V}

1,\text{kg m}^{2}\text{s}^{-3}\text{A}^{-1} ]

2.2.11 Resistance: the ohm

[ R=\frac{V}{I} ]

Therefore:

[ 1,\Omega

1,\text{V A}^{-1} ]

Base units:

[ 1,\Omega

1,\text{kg m}^{2}\text{s}^{-3}\text{A}^{-2} ]

2.2.12 Density

[ \rho=\frac{m}{V} ]

Unit:

[ \boxed{\text{kg m}^{-3}} ]

2.2.13 Momentum

[ p=mv ]

Unit:

[ \boxed{\text{kg m s}^{-1}} ]

Equivalent unit:

[ \text{N s} ]

because impulse is equal to change in momentum.

2.3 Checking Homogeneity of Physical Equations

OCR explicitly requires students to check the homogeneity of physical equations using S.I. base units.

DEFINITION — Homogeneous physical equation
A physical equation in which every term has the same base-unit dimensions.

This does not prove that an equation is physically correct, but if the units do not match, the equation must be incorrect.

2.3.1 Example: checking (v=u+at)

Left-hand side:

[ [v]=\text{m s}^{-1} ]

For (u):

[ [u]=\text{m s}^{-1} ]

For (at):

[ [a][t]

(\text{m s}^{-2})(\text{s})

\text{m s}^{-1} ]

All terms have units:

[ \text{m s}^{-1} ]

Therefore the equation is homogeneous.

2.3.2 Example: checking kinetic energy

[ E_k=\frac12mv^2 ]

The numerical factor (1/2) is dimensionless.

Units of the right-hand side:

[ \text{kg}\times(\text{m s}^{-1})^2 ]

[

\text{kg m}^{2}\text{s}^{-2} ]

This is the unit of energy:

[ \text{J} ]

Therefore the equation is homogeneous.

2.3.3 Example of an impossible equation

Suppose someone proposes:

[ s=ut+at ]

Units of (ut):

[ (\text{m s}^{-1})(\text{s})=\text{m} ]

Units of (at):

[ (\text{m s}^{-2})(\text{s})

\text{m s}^{-1} ]

The terms being added have different units.

Therefore the equation cannot be correct.

2.3.4 What homogeneity cannot tell you

Suppose:

[ E=mv^2 ]

The units are:

[ \text{kg m}^{2}\text{s}^{-2} ]

which are energy units.

However, the correct non-relativistic kinetic-energy expression is:

[ E_k=\frac12mv^2 ]

Unit checking alone cannot identify the missing dimensionless factor (1/2).

Examiner-safe statement

“The equation is dimensionally homogeneous, so it is not ruled out by its units, but unit consistency alone does not prove the equation is physically correct.”

2.4 S.I. Prefixes

A prefix represents a power-of-ten multiplier applied to a unit.

DEFINITION — S.I. prefix
A symbol placed before an S.I. unit to represent a decimal multiple or submultiple of that unit.

OCR requires:

OCR-specific warning

Do not overlook:

[ \boxed{\text{deci} = d = 10^{-1}} ]

It is explicitly included in OCR's Module 2 content.

2.4.1 Prefix conversions

Example:

[ 4.2,\text{mm}

4.2\times10^{-3},\text{m} ]

Example:

[ 3.5,\mu\text{s}

3.5\times10^{-6},\text{s} ]

Example:

[ 2.4,\text{GHz}

2.4\times10^9,\text{Hz} ]

Example:

[ 560,\text{pF}

560\times10^{-12},\text{F}

5.60\times10^{-10},\text{F} ]

2.4.2 Area and volume conversions

If:

[ 1,\text{cm}=10^{-2},\text{m} ]

then:

[ 1,\text{cm}^2

(10^{-2})^2,\text{m}^2

10^{-4},\text{m}^2 ]

and:

[ 1,\text{cm}^3

(10^{-2})^3,\text{m}^3

10^{-6},\text{m}^3 ]

Similarly:

[ 1,\text{mm}^2=10^{-6},\text{m}^2 ]

[ 1,\text{mm}^3=10^{-9},\text{m}^3 ]

Common lost mark

A student converts (2.0,\text{cm}^2) to:

[ 2.0\times10^{-2},\text{m}^2 ]

This is wrong because the conversion factor itself must be squared.

Correct:

[ 2.0,\text{cm}^2

2.0\times10^{-4},\text{m}^2 ]

2.5 Standard Form

DEFINITION — Standard form
A number written in the form (a\times10^n), where (1\leq|a|<10) and (n) is an integer.

Examples:

[ 4200000=4.2\times10^6 ]

[ 0.0000063=6.3\times10^{-6} ]

For multiplication:

[ (3.0\times10^5)(2.0\times10^{-3})

6.0\times10^2 ]

For division:

[ \frac{8.0\times10^7}{4.0\times10^2}

2.0\times10^5 ]

Powers-of-ten errors are among the most damaging mistakes in A-level Physics because they can produce answers that look numerically neat but are physically impossible.

2.6 SPEC: 2.2 — Making Measurements and Analysing Data

Experimental data are never perfectly exact. A physicist must distinguish between:

• the value obtained;
• the precision of the measuring process;
• the possible uncertainty in that value;
• systematic effects that may bias the result;
• random variation that produces scatter.

2.6.1 Accuracy

DEFINITION — Accuracy
The closeness of a measured value to the true or accepted value.

Accuracy is especially affected by systematic effects such as poor calibration.

2.6.2 Precision

DEFINITION — Precision
The closeness of agreement between repeated measured values.

High precision means the data have a small spread.

Precision does not guarantee accuracy.

Example:

True value:

[ 5.00,\text{V} ]

Readings:

[ 5.42,\ 5.41,\ 5.42,\ 5.41,\text{V} ]

These measurements are highly precise but inaccurate.

2.6.3 Resolution

DEFINITION — Resolution
The smallest change in the measured quantity that a measuring instrument can distinguish.

Examples:

• ruler marked every (1,\text{mm}): scale resolution (1,\text{mm});
• digital voltmeter displaying 4.327 V: display resolution (0.001,\text{V}).

A small numerical resolution does not guarantee an accurate result if the instrument is badly calibrated.

2.6.4 Random error

DEFINITION — Random error
Unpredictable variation between repeated measurements that produces scatter in the readings.

Possible causes:

• reaction-time variation;
• electrical fluctuations;
• environmental fluctuations;
• difficulty judging a scale;
• small uncontrolled variations in the apparatus.

Reducing the effect

• repeat;
• calculate a mean;
• measure over a larger interval where appropriate;
• improve the experimental method.

Examiner language

“Repeat the measurement and calculate a mean to reduce the effect of random variation.”

2.6.5 Systematic error

DEFINITION — Systematic error
An error caused by the method or measuring system that shifts results consistently away from the true value.

Examples:

• zero error;
• calibration error;
• constant background count not subtracted;
• consistent heat loss ignored by a model;
• fixed parallax due to an incorrect viewing position.

Crucial distinction

Repeating measurements does not remove a systematic bias.

Examiner language

“Repeating would not remove the systematic offset because each measurement is biased in the same direction.”

2.6.6 Zero error

DEFINITION — Zero error
A systematic error in which an instrument gives a non-zero reading when the true input is zero.

Example:

A micrometer reads:

[ +0.03,\text{mm} ]

when fully closed.

If the offset is stable, measured diameters are too large by (0.03,\text{mm}).

Corrected value:

[ d_{\text{true estimate}}

d_{\text{measured}}-0.03,\text{mm} ]

2.6.7 Parallax

Parallax occurs when the apparent position of a pointer or marker changes because the scale is viewed from the wrong direction.

Reduction methods:

• view perpendicular to the scale;
• use a mirror scale;
• align using a set square where suitable;
• use an appropriate sensor/data logger.

Do not simply label all parallax as random. A consistently wrong viewing angle can generate a systematic shift.

2.7 Uncertainty

A measured value may be represented as:

[ x\pm\Delta x ]

where:

• (x) is the measured result;
• (\Delta x) is the absolute uncertainty.

2.7.1 Absolute uncertainty

DEFINITION — Absolute uncertainty
The uncertainty expressed in the same units as the measured quantity.

Example:

[ L=(0.624\pm0.002),\text{m} ]

Absolute uncertainty:

[ \Delta L=0.002,\text{m} ]

2.7.2 Fractional uncertainty

DEFINITION — Fractional uncertainty
The absolute uncertainty divided by the measured value.

[ \boxed{ \text{fractional uncertainty}

\frac{\Delta x}{x} } ]

It has no unit.

Example:

[ x=2.50,\text{m} ]

[ \Delta x=0.02,\text{m} ]

Then:

[ \frac{\Delta x}{x}

\frac{0.02}{2.50}

0.008 ]

2.7.3 Percentage uncertainty

DEFINITION — Percentage uncertainty
The fractional uncertainty expressed as a percentage.

[ \boxed{ %\text{ uncertainty}

\frac{\Delta x}{x}\times100% } ]

Using the previous values:

[ \frac{0.02}{2.50}\times100

0.8% ]

2.7.4 Estimating uncertainty from repeated readings

For repeated measurements, a commonly appropriate estimate based on spread is:

[ \boxed{ \Delta x

\frac{x_{\max}-x_{\min}}{2} } ]

Example:

[ 2.41,\ 2.45,\ 2.43,\ 2.46,\ 2.42,\text{s} ]

Range:

[ 2.46-2.41=0.05,\text{s} ]

Half-range:

[ \Delta t

0.025,\text{s} ]

The mean is:

[ \bar{t}

\frac{2.41+2.45+2.43+2.46+2.42}{5}

2.434,\text{s} ]

A reasonable reported form is then chosen with consistent significant figures.

2.7.5 Uncertainty in a difference measurement

If a length is found from:

[ L=x_2-x_1 ]

and both scale readings have uncertainty:

[ \pm0.5,\text{mm} ]

then the absolute uncertainties add:

[ \Delta L

0.5+0.5

1.0,\text{mm} ]

This is important when using rulers, calipers and other instruments in which the result comes from two independent readings.

2.8 Combining Uncertainties

OCR explicitly requires uncertainty treatment when quantities are:

• added;
• subtracted;
• multiplied;
• divided;
• raised to powers.

2.8.1 Addition

If:

[ Q=A+B ]

then add the absolute uncertainties:

[ \boxed{ \Delta Q=\Delta A+\Delta B } ]

Example:

[ A=3.20\pm0.02,\text{m} ]

[ B=1.40\pm0.03,\text{m} ]

Then:

[ Q=4.60,\text{m} ]

and:

[ \Delta Q=0.02+0.03=0.05,\text{m} ]

Therefore:

[ \boxed{ Q=(4.60\pm0.05),\text{m} } ]

2.8.2 Subtraction

If:

[ Q=A-B ]

the absolute uncertainties still add:

[ \boxed{ \Delta Q=\Delta A+\Delta B } ]

Do not subtract uncertainties.

2.8.3 Multiplication

If:

[ Q=AB ]

add fractional or percentage uncertainties:

[ \boxed{ \frac{\Delta Q}{Q}

\frac{\Delta A}{A} + \frac{\Delta B}{B} } ]

or:

[ \boxed{ %\Delta Q

%\Delta A+%\Delta B } ]

2.8.4 Division

If:

[ Q=\frac{A}{B} ]

again add fractional or percentage uncertainties:

[ \boxed{ \frac{\Delta Q}{Q}

\frac{\Delta A}{A} + \frac{\Delta B}{B} } ]

Do not subtract the denominator's uncertainty.

2.8.5 Powers

For:

[ Q=A^n ]

the percentage uncertainty is approximately:

[ \boxed{ %\Delta Q

|n|\times%\Delta A } ]

Example:

[ A=\pi r^2 ]

If:

[ %\Delta r=1.5% ]

then:

[ %\Delta A=2(1.5)=3.0% ]

2.8.6 Mixed example

For:

[ Q=\frac{AB^2}{C^3} ]

the percentage uncertainty is:

[ \boxed{ %\Delta Q

%\Delta A + 2(%\Delta B) + 3(%\Delta C) } ]

The signs in the algebraic expression do not cause uncertainty contributions to cancel.

2.9 Percentage Difference

OCR specifically includes percentage difference in graphical/data treatment.

Percentage difference is commonly used to compare two measured or experimental values when neither is being treated as the exact accepted value.

A standard form is:

[ \boxed{ %\text{ difference}

\frac{|A-B|} {(A+B)/2} \times100% } ]

Example

Two experimental values are:

[ A=4.80 ]

and:

[ B=5.10 ]

Difference:

[ |A-B|=0.30 ]

Mean:

[ \frac{4.80+5.10}{2}=4.95 ]

Therefore:

[ %\text{ difference}

\frac{0.30}{4.95}\times100 ]

[ \boxed{ 6.06%\approx6.1% } ]

Percentage difference vs percentage error

Do not confuse percentage difference with comparison to an accepted value.

If an accepted/reference value (x_{\text{ref}}) is supplied and the question asks for percentage error/difference from accepted value, the relevant form may be:

[ \frac{|x_{\text{measured}}-x_{\text{ref}}|} {x_{\text{ref}}}\times100% ]

Always follow the wording and definitions supplied in the question.

2.10 Significant Figures and Decimal Places

Measurements must not imply unjustified precision.

DEFINITION — Significant figures
Digits that express the meaningful precision of a number, beginning with the first non-zero digit.

Good calculation practice

• retain extra digits during intermediate calculations;
• round only at the end;
• use a final precision appropriate to the input data and uncertainty;
• include the unit.

Example:

Calculator output:

[ 4.73826194,\text{m} ]

If the uncertainty is:

[ 0.03,\text{m} ]

a suitable result is:

[ \boxed{ (4.74\pm0.03),\text{m} } ]

rather than reporting eight decimal places.

2.11 Graphical Analysis

Graphs are used to:

• determine relationships;
• identify linear trends;
• obtain gradients;
• determine intercepts;
• compare with theoretical equations;
• estimate uncertainties.

2.11.1 Plotting

Use:

• suitable scales;
• labelled axes;
• units;
• clearly marked points;
• appropriate error bars;
• a thin line/curve of best fit.

Avoid:

• tiny scales using only a small part of the graph;
• thick blobs;
• freehand wandering lines;
• point-to-point joining unless instructed.

2.11.2 Line of best fit

DEFINITION — Line of best fit
A line representing the overall trend of the measured data rather than being forced through every individual point.

A best-fit line should have a sensible distribution of points above and below it, taking uncertainty into account.

Do not automatically force the line through the origin.

2.11.3 Gradient

For a straight-line graph:

[ m=\frac{\Delta y}{\Delta x} ]

Choose two points:

• on the best-fit line;
• far apart;
• giving a large triangle.

The unit of the gradient is:

[ \boxed{ \frac{\text{vertical-axis unit}} {\text{horizontal-axis unit}} } ]

2.11.4 Worst acceptable line

OCR uses the concept of a worst line.

DEFINITION — Worst acceptable line
A line with the greatest plausible difference in gradient from the best-fit line while remaining consistent with the uncertainty/error bars of the data.

The exact direction chosen depends on which line gives the largest deviation in gradient from the best-fit value.

If:

[ m_{\text{best}} ]

is the best-fit gradient and:

[ m_{\text{worst}} ]

is the worst acceptable gradient, then an estimate of the absolute uncertainty in the gradient can be:

[ \boxed{ \Delta m

|m_{\text{best}}-m_{\text{worst}}| } ]

Percentage uncertainty:

[ \boxed{ %\Delta m

\frac{|m_{\text{best}}-m_{\text{worst}}|} {|m_{\text{best}}|} \times100% } ]

Follow any specific method stated in the examination question.

2.11.5 Error bars

DEFINITION — Error bar
A graphical indication of the uncertainty interval associated with a plotted measurement.

If:

[ y=5.0\pm0.3 ]

then the vertical uncertainty interval is:

[ 4.7\leq y\leq5.3 ]

If the horizontal quantity also has significant uncertainty, horizontal error bars may also be shown.

2.11.6 Intercepts

For:

[ y=mx+c ]

the vertical intercept is:

[ c ]

The units of (c) are the same as the vertical-axis quantity.

A non-zero intercept may indicate:

• a real physical offset;
• a systematic error;
• an inadequacy in the assumed model;
• another term in the physical relationship.

Do not automatically label every non-zero intercept a zero error without evidence.

2.11.7 Direct proportionality

A linear graph is not necessarily directly proportional.

If:

[ y\propto x ]

then:

[ y=kx ]

and a graph of (y) against (x) should be a straight line through the origin, within uncertainty.

If:

[ y=mx+c ]

with a definite non-zero (c), the relationship is linear but not directly proportional.

Examiner-safe wording

“The data are consistent with direct proportionality because the graph is a straight line passing through the origin within experimental uncertainty.”

2.12 Practical Improvements

OCR practical questions reward specific physics, not generic phrases.

Weak:

“Use better equipment.”

Strong:

“Use a micrometer instead of a metre rule to measure the wire diameter because the micrometer has a smaller resolution, reducing the uncertainty in diameter.”

Weak:

“Do more repeats.”

Strong:

“Repeat the timing for 20 oscillations several times and calculate a mean, reducing the effect of random variation.”

2.12.1 Increase the measured interval

Percentage uncertainty is:

[ \frac{\Delta x}{x}\times100% ]

If (\Delta x) remains approximately constant, increasing (x) reduces percentage uncertainty.

Example:

Timing one oscillation:

[ t=1.5,\text{s} ]

with uncertainty:

[ 0.2,\text{s} ]

gives:

[ \frac{0.2}{1.5}\times100 \approx13% ]

Timing 20 oscillations:

[ t=30,\text{s} ]

with approximately the same stopwatch reaction contribution gives:

[ \frac{0.2}{30}\times100 \approx0.67% ]

Then divide by 20 to obtain the period.

2.12.2 Measuring small diameters

For a wire, measuring diameter accurately matters because cross-sectional area is:

[ A=\frac{\pi d^2}{4} ]

Therefore:

[ %\Delta A \approx 2(%\Delta d) ]

A modest percentage uncertainty in diameter doubles in its contribution to area.

Good practice:

• use a micrometer;
• check zero error;
• measure diameter at several positions;
• rotate the wire where appropriate;
• calculate a mean.

This accounts for possible non-uniformity or non-circularity.

2.13 SPEC: 2.3 — Nature of Quantities

2.13.1 Scalars

DEFINITION — Scalar quantity
A physical quantity that has magnitude only.

Examples include:

• mass;
• time;
• temperature;
• distance;
• speed;
• energy;
• work;
• power;
• density;
• pressure;
• electric charge.

A scalar does not require a direction for its full specification.

Example:

[ 20,\text{J} ]

is a complete energy statement.

2.14 Vectors

DEFINITION — Vector quantity
A physical quantity that has both magnitude and direction.

Examples include:

• displacement;
• velocity;
• acceleration;
• force;
• momentum;
• electric field strength;
• gravitational field strength.

Example:

[ 12,\text{m s}^{-1}\text{ east} ]

contains both magnitude and direction.

2.14.1 Vector notation

Vectors may be represented by arrows.

The arrow:

• points in the vector's direction;
• has a length proportional to magnitude when drawn to scale.

A vector may also be represented using bold notation or an arrow above a symbol depending on context.

2.15 Distance vs Displacement

DEFINITION — Distance
The total scalar path length travelled.

DEFINITION — Displacement
The vector change in position from the initial point to the final point.

Example:

A runner completes one full (400,\text{m}) lap.

Distance:

[ 400,\text{m} ]

Displacement:

[ 0,\text{m} ]

because the final position equals the initial position.

2.16 Speed vs Velocity

DEFINITION — Speed
The scalar rate of change of distance.

DEFINITION — Velocity
The vector rate of change of displacement.

A speed of:

[ 20,\text{m s}^{-1} ]

is scalar.

A velocity of:

[ 20,\text{m s}^{-1}\text{ north} ]

is vector.

2.17 Adding Vectors

Vectors must be added using both magnitude and direction.

2.17.1 Head-to-tail method

To add:

[ \vec{A}+\vec{B} ]

1. draw (\vec{A});
2. place the tail of (\vec{B}) at the head of (\vec{A});
3. draw the resultant from the tail of (\vec{A}) to the head of (\vec{B}).

DEFINITION — Resultant vector
A single vector that has the same overall effect as two or more vectors acting together.

2.17.2 Perpendicular vectors

If two vectors of magnitudes (A) and (B) are perpendicular:

[ R=\sqrt{A^2+B^2} ]

Direction can be found using:

[ \tan\theta=\frac{B}{A} ]

provided (\theta) is defined relative to the (A)-direction.

Example

A velocity has components:

[ v_x=6.0,\text{m s}^{-1} ]

[ v_y=8.0,\text{m s}^{-1} ]

Magnitude:

[ v=\sqrt{6.0^2+8.0^2} ]

[ \boxed{ v=10.0,\text{m s}^{-1} } ]

Direction:

[ \tan\theta=\frac{8.0}{6.0} ]

[ \theta=53.1^\circ ]

2.18 Subtracting Vectors

Vector subtraction can be treated as addition of the negative vector:

[ \vec{A}-\vec{B}

\vec{A}+(-\vec{B}) ]

The vector (-\vec{B}) has:

• the same magnitude as (\vec{B});
• opposite direction.

2.19 Resolving Vectors

DEFINITION — Resolving a vector
Replacing one vector with two or more component vectors that have the same combined effect as the original vector.

OCR specifically requires resolution into two perpendicular components.

Suppose a force (F) acts at an angle (\theta) above the horizontal.

Horizontal component:

[ \boxed{ F_x=F\cos\theta } ]

Vertical component:

[ \boxed{ F_y=F\sin\theta } ]

This assumes (\theta) is measured from the horizontal.

2.19.1 Do not memorise sine/cosine blindly

The reliable rule is:

• component adjacent to the stated angle: [ F\cos\theta ]
• component opposite the stated angle: [ F\sin\theta ]

If the angle is instead measured from the vertical, the assignments swap.

2.19.2 Example

A force of:

[ F=50,\text{N} ]

acts at:

[ 30^\circ ]

above the horizontal.

Horizontal:

[ F_x=50\cos30^\circ ]

[ \boxed{ F_x=43.3,\text{N} } ]

Vertical:

[ F_y=50\sin30^\circ ]

[ \boxed{ F_y=25.0,\text{N} } ]

2.20 Equilibrium and Vector Components

Although detailed force equilibrium is developed further in Module 3, vector resolution is the mathematical foundation.

For an object in equilibrium:

[ \sum F_x=0 ]

and:

[ \sum F_y=0 ]

Resolving forces into perpendicular directions allows complex force systems to be analysed using ordinary algebra.

This is one of the most important synoptic applications of Module 2.

3. Exact OCR Exam Language

These are original examiner-safe structures that model the precision expected in OCR Physics A answers.

3.1 Define

Give a precise meaning without unnecessary narrative.

Example:

“A vector quantity has both magnitude and direction.”

3.2 State

Give the required fact directly.

Example:

“The S.I. unit of electric current is the ampere, A.”

3.3 Describe

State the observable pattern or sequence.

For a graph:

“As (x) increases, (y) increases linearly.”

Do not explain why unless asked.

3.4 Explain

Use:

physical cause → mechanism → consequence

Example:

“Timing a larger number of oscillations increases the measured time interval while the absolute timing uncertainty remains approximately similar, so the fractional and percentage uncertainties are reduced.”

3.5 Show

Demonstrate the mathematical reasoning.

For homogeneity:

“The right-hand side has units (\text{kg}\times\text{m s}^{-2}=\text{kg m s}^{-2}=\text{N}), matching the left-hand side, so the equation is homogeneous.”

3.6 Determine

Extract or calculate using the supplied information.

Show enough working that the method is clear.

3.7 Calculate

Use:

1. equation;
2. consistent units;
3. substitution;
4. numerical result;
5. appropriate precision;
6. unit.

3.8 Suggest

Make a context-specific proposal.

Weak:

“Use better equipment.”

Strong:

“Use a micrometer to measure the wire diameter because its smaller resolution reduces the uncertainty in diameter.”

3.9 Justify

Claim + evidence + physics.

“The second method is preferable because its percentage uncertainty is smaller, so the resulting value is more precise.”

3.10 Compare

Explicitly address both quantities.

“Speed is a scalar quantity with magnitude only, whereas velocity is a vector quantity with both magnitude and direction.”

3.11 Evaluate

Use evidence and limitations.

Good structure:

“The data support the model because… However… This discrepancy may arise from… The effect would be… Therefore…”

Avoid generic comments disconnected from the data.

4. Common Misconceptions & Lost Marks

4.1 Forgetting deci

OCR includes:

[ \boxed{d=10^{-1}} ]

Do not use an AQA-only prefix list for OCR.

4.2 Gram as an S.I. base unit

❌ gram

✅ kilogram

4.3 Checking units but claiming the equation is proven

A homogeneous equation is not necessarily correct.

Unit checking can disprove an inconsistent equation but cannot prove that all numerical factors and physical assumptions are correct.

4.4 Squared unit conversions

❌:

[ 1,\text{cm}^2=10^{-2},\text{m}^2 ]

✅:

[ 1,\text{cm}^2=10^{-4},\text{m}^2 ]

4.5 Cubed unit conversions

❌:

[ 1,\text{cm}^3=10^{-2},\text{m}^3 ]

✅:

[ 1,\text{cm}^3=10^{-6},\text{m}^3 ]

4.6 Precision = accuracy

False.

• precision → spread/agreement;
• accuracy → closeness to true value.

4.7 Repeating to remove systematic error

False.

Repeating mainly reduces the effect of random variation.

A systematic error requires calibration, correction or a change to the method.

4.8 “Human error”

Too vague.

Identify:

• reaction time;
• parallax;
• scale-reading uncertainty;
• inconsistent endpoint judgement.

4.9 Subtracting uncertainty when quantities are subtracted

For:

[ Q=A-B ]

absolute uncertainties still add.

4.10 Subtracting uncertainty during division

For:

[ Q=A/B ]

percentage/fractional uncertainties add.

4.11 Forgetting the exponent in uncertainty

If:

[ Q=x^3 ]

then:

[ %\Delta Q\approx3(%\Delta x) ]

4.12 Wrong percentage-difference denominator

When comparing two experimental values, OCR-style percentage difference commonly uses their mean in the denominator:

[ \frac{|A-B|}{(A+B)/2}\times100% ]

Do not automatically divide by one of the values unless the question defines one as the reference/accepted value.

4.13 Joining graph points

A line of best fit represents the trend.

Do not simply join data points unless instructed.

4.14 Forcing the graph through the origin

A theoretical expectation does not justify ignoring the actual plotted evidence unless the question or physical argument supports doing so.

4.15 Straight line = directly proportional

False unless the line also passes through the origin within uncertainty.

4.16 Tiny gradient triangle

Use coordinates far apart on the line to reduce percentage reading uncertainty.

4.17 Gradient without units

Gradient units are:

[ \frac{\text{vertical unit}}{\text{horizontal unit}} ]

4.18 Worst line drawn outside error bars

A worst acceptable line must still be consistent with the uncertainty represented by the data.

4.19 Scalar/vector confusion

Common errors:

• speed called a vector;
• distance called a vector;
• velocity called a scalar;
• force direction omitted.

4.20 Adding vector magnitudes directly

If vectors are not parallel in the same direction, do not simply add their magnitudes.

Direction matters.

4.21 Wrong sine/cosine component

Do not memorise “horizontal = cos” in every problem.

Identify whether the component is adjacent or opposite to the angle actually shown.

4.22 Missing vector direction

A vector answer is incomplete if the direction is required but omitted.

For example:

[ 10,\text{N} ]

is only a magnitude.

A complete vector result may need:

[ 10,\text{N at }35^\circ\text{ above the horizontal} ]

5. Worked Exam-Style Questions

Question 1 — S.I. Units and Homogeneity [5 marks]

A student proposes that the energy (E) stored in a moving object is:

[ E=mv^2 ]

(a) Express the units of (mv^2) in S.I. base units. [2 marks]

[ [mv^2]

\text{kg}\times(\text{m s}^{-1})^2 ]

[

\boxed{ \text{kg m}^{2}\text{s}^{-2} } ]

Mark allocation:

• M1 squares velocity unit correctly;
• A1 obtains (\text{kg m}^2\text{s}^{-2}).

(b) Explain what this tells you about the proposed equation. [3 marks]

Model answer:

“(\text{kg m}^2\text{s}^{-2}) is the S.I. base-unit form of the joule, so both sides have energy units and the equation is homogeneous. However, this does not prove the equation is physically correct because dimensional homogeneity cannot determine dimensionless numerical factors. For kinetic energy the correct expression contains the factor (1/2).”

Mark allocation:

• B1 identifies joule/energy units;
• B1 states equation is homogeneous;
• B1 explains homogeneity does not prove full physical correctness.

Question 2 — Uncertainty in a Derived Quantity [6 marks]

The resistivity (\rho) of a wire is calculated from:

[ \rho=\frac{RA}{L} ]

where:

[ A=\frac{\pi d^2}{4} ]

Measurements have percentage uncertainties:

[ R:\ 1.0% ]

[ d:\ 1.5% ]

[ L:\ 0.5% ]

Determine the percentage uncertainty in (\rho).

Model answer

Since:

[ A\propto d^2 ]

the percentage uncertainty in (A) is:

[ 2(1.5%)=3.0% ]

Since:

[ \rho=\frac{RA}{L} ]

percentage uncertainties add:

[ %\Delta\rho

1.0+3.0+0.5 ]

[ \boxed{ %\Delta\rho=4.5% } ]

Mark allocation:

• M1 identifies (A\propto d^2);
• A1 area uncertainty (=3.0%);
• M1 uses addition of percentage uncertainties for multiplication/division;
• A1 includes resistance contribution;
• A1 includes length contribution;
• A1 obtains (4.5%).

Question 3 — Percentage Difference [4 marks]

Two groups measure the same physical constant.

Group A obtains:

[ 6.35 ]

Group B obtains:

[ 6.61 ]

Calculate the percentage difference between the two results.

Model answer

Difference:

[ |6.61-6.35|

0.26 ]

Mean:

[ \frac{6.61+6.35}{2}

6.48 ]

Percentage difference:

[ \frac{0.26}{6.48}\times100

4.012... ]

[ \boxed{ 4.0% } ]

Mark allocation:

• B1 absolute difference (0.26);
• B1 mean (6.48);
• M1 correct percentage-difference structure;
• A1 (4.0%).

Question 4 — Graphical Uncertainty [6 marks]

A graph has a best-fit gradient:

[ m_{\text{best}}=3.42,\text{N m}^{-1} ]

The worst acceptable line has gradient:

[ m_{\text{worst}}=3.11,\text{N m}^{-1} ]

(a) Determine the absolute uncertainty in the gradient. [2 marks]

[ \Delta m

|3.42-3.11| ]

[ \boxed{ \Delta m=0.31,\text{N m}^{-1} } ]

(b) Determine the percentage uncertainty. [2 marks]

[ %\Delta m

\frac{0.31}{3.42}\times100 ]

[ \boxed{ 9.1% } ]

(c) Explain what is meant by the “worst acceptable line”. [2 marks]

Model answer:

“It is a line that remains consistent with the uncertainty of the plotted data but gives the greatest plausible deviation in gradient from the best-fit line.”

Question 5 — Vector Resolution [6 marks]

A cable exerts a tension of:

[ 180,\text{N} ]

at an angle of:

[ 35^\circ ]

above the horizontal.

(a) Calculate the horizontal component. [2 marks]

[ F_x=F\cos\theta ]

[ F_x=180\cos35^\circ ]

[ \boxed{ F_x=147,\text{N} } ]

(b) Calculate the vertical component. [2 marks]

[ F_y=F\sin\theta ]

[ F_y=180\sin35^\circ ]

[ \boxed{ F_y=103,\text{N} } ]

(c) Explain why these two components can replace the original force in a calculation. [2 marks]

Model answer:

“The perpendicular components have a vector sum equal to the original (180,\text{N}) force. They therefore have the same combined mechanical effect as the original vector.”

Question 6 — A/A* Experimental Evaluation [6 marks]

A student measures the period of a pendulum by timing one oscillation with a handheld stopwatch.

Suggest and explain improvements that reduce the uncertainty in the result.

Model answer

“The student should time a large number of complete oscillations, such as 20, and divide the measured total time by 20. The total measured time is much larger while the absolute reaction-time contribution from starting and stopping remains of a similar order, so the percentage uncertainty is smaller. The timing should be repeated several times and a mean calculated to reduce the effect of random variation. The student should use a fixed fiducial marker and begin and end timing as the bob passes the same point in the same direction so that the definition of a complete oscillation is consistent.”

Indicative points:

• time multiple oscillations;
• divide total by number of oscillations;
• larger measured interval;
• similar absolute reaction-time contribution;
• lower percentage uncertainty;
• repeat/mean and/or fixed reference point with consistent direction.

6. Links to Other OCR Topics

6.1 Module 1 — Development of Practical Skills

Module 2 data-handling skills directly support:

• planning experiments;
• recording measurements;
• assessing uncertainty;
• processing data;
• graphing;
• evaluating procedures;
• Practical Endorsement work.

6.2 Module 3 — Forces and Motion

Vectors are fundamental to:

• displacement;
• velocity;
• acceleration;
• force;
• momentum;
• projectiles;
• moments/equilibrium;
• Newton's laws.

Resolution gives:

[ F_x=F\cos\theta ]

[ F_y=F\sin\theta ]

and equilibrium frequently requires:

[ \sum F_x=0 ]

[ \sum F_y=0 ]

6.3 Module 3 — Materials

Measurements and uncertainty are heavily tested when determining:

• spring constant;
• stress;
• strain;
• Young modulus.

For a wire:

[ A=\frac{\pi d^2}{4} ]

so diameter uncertainty is doubled in its contribution to area uncertainty.

6.4 Module 4 — Electrons, Waves and Photons

S.I. units and uncertainty recur in:

[ Q=It ]

[ R=\frac{V}{I} ]

[ P=IV ]

[ v=f\lambda ]

Wave experiments also require uncertainty in distances, fringe spacing, wavelengths and gradients.

6.5 Module 5 — Newtonian World and Astrophysics

Vector quantities include:

• gravitational field strength;
• velocity;
• acceleration.

Orders of magnitude and prefixes become essential in astrophysical scales.

Graphical analysis is important in oscillations and thermal experiments.

6.6 Module 6 — Particles and Medical Physics

S.I. prefixes are critical for:

• nuclear sizes;
• particle energies;
• capacitances;
• electric-field quantities;
• magnetic quantities;
• medical-imaging scales.

Experimental uncertainties and data interpretation remain synoptic.

7. Summary Notes

7.1 Physical quantities

A physical quantity has:

• a numerical value;
• a unit.

7.2 OCR S.I. base units

• mass → kilogram, kg
• length → metre, m
• time → second, s
• electric current → ampere, A
• temperature → kelvin, K
• amount of substance → mole, mol

7.3 Derived units

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

[ 1,\text{J}=1,\text{kg m}^{2}\text{s}^{-2} ]

[ 1,\text{W}=1,\text{kg m}^{2}\text{s}^{-3} ]

[ 1,\text{Pa}=1,\text{kg m}^{-1}\text{s}^{-2} ]

[ 1,\text{C}=1,\text{A s} ]

7.4 Homogeneity

A valid physical equation must be homogeneous:

• every added/subtracted term must have the same units;
• left and right sides must have matching base-unit dimensions.

Homogeneity can identify an impossible equation.

It cannot prove an equation is physically correct.

7.5 OCR prefixes

[ \text{T}=10^{12} ]

[ \text{G}=10^9 ]

[ \text{M}=10^6 ]

[ \text{k}=10^3 ]

[ \text{d}=10^{-1} ]

[ \text{c}=10^{-2} ]

[ \text{m}=10^{-3} ]

[ \mu=10^{-6} ]

[ \text{n}=10^{-9} ]

[ \text{p}=10^{-12} ]

OCR-specific reminder:

[ \boxed{\text{deci}=10^{-1}} ]

7.6 Area/volume conversion

[ 1,\text{cm}^2=10^{-4},\text{m}^2 ]

[ 1,\text{cm}^3=10^{-6},\text{m}^3 ]

[ 1,\text{mm}^2=10^{-6},\text{m}^2 ]

[ 1,\text{mm}^3=10^{-9},\text{m}^3 ]

7.7 Measurement language

Accuracy: closeness to true/accepted value.

Precision: closeness of agreement between repeated measurements.

Resolution: smallest change an instrument can distinguish.

Random error: unpredictable scatter between readings.

Systematic error: consistent bias caused by method/instrument.

Zero error: non-zero instrument reading for a zero true input.

7.8 Uncertainty

Absolute:

[ x\pm\Delta x ]

Fractional:

[ \boxed{ \frac{\Delta x}{x} } ]

Percentage:

[ \boxed{ \frac{\Delta x}{x}\times100% } ]

Repeated readings, where appropriate:

[ \boxed{ \Delta x= \frac{x_{\max}-x_{\min}}{2} } ]

7.9 Combining uncertainties

Addition/subtraction:

[ \boxed{ \text{add absolute uncertainties} } ]

Multiplication/division:

[ \boxed{ \text{add fractional or percentage uncertainties} } ]

Power:

[ Q=x^n ]

[ \boxed{ %\Delta Q

|n|(%\Delta x) } ]

7.10 Percentage difference

For two experimental values:

[ \boxed{ %\text{ difference}

\frac{|A-B|} {(A+B)/2}\times100% } ]

If one value is explicitly an accepted/reference value, follow the comparison method requested in the question.

7.11 Graphs

• sensible scale;
• labelled axes;
• units;
• precise plotting;
• error bars where required;
• thin best-fit line/curve;
• no automatic forcing through origin;
• use a large triangle for gradient.

Gradient:

[ \boxed{ m=\frac{\Delta y}{\Delta x} } ]

OCR worst-line estimate:

[ \boxed{ \Delta m

|m_{\text{best}}-m_{\text{worst}}| } ]

Percentage gradient uncertainty:

[ \boxed{ %\Delta m

\frac{\Delta m}{|m_{\text{best}}|} \times100% } ]

7.12 Scalars

Scalar = magnitude only.

Examples:

• mass
• distance
• speed
• time
• temperature
• energy
• power
• density
• pressure
• charge

7.13 Vectors

Vector = magnitude + direction.

Examples:

• displacement
• velocity
• acceleration
• force
• momentum
• field strength

7.14 Vector addition

Use:

• head-to-tail construction;
• Pythagoras for perpendicular components;
• trigonometry for direction.

For perpendicular components:

[ R=\sqrt{A^2+B^2} ]

[ \tan\theta=\frac{\text{opposite}}{\text{adjacent}} ]

7.15 Resolving vectors

If (\theta) is measured from the horizontal:

[ \boxed{ F_x=F\cos\theta } ]

[ \boxed{ F_y=F\sin\theta } ]

Always inspect the diagram:

• adjacent component → cosine;
• opposite component → sine.