EduNinja

1.1 Physical quantities

Every physical quantity is written as a numerical magnitude together with a unit. In 7.0 N, 7.0 is the magnitude and N is the unit of force.

The magnitude tells how many of the chosen unit make up the quantity. Changing unit changes the number but not the physical quantity: 2.5 km and 2500 m describe the same distance.

A number alone or a unit alone is not a complete physical quantity. The unit does not have to be an SI base unit, and the magnitude does not have to be written in scientific notation.

A reasonable estimate is a reasoned approximate value, not a random guess. It combines a familiar scale or physical relationship with a suitable unit and sensible order of magnitude.

Identify the quantity and unit.
Choose familiar values or a relationship that links the quantity to known values.
Round to simple numbers and calculate.
Check whether the order of magnitude is physically plausible.

To estimate an adult’s weight, use a mass of about 70 kg and g ≈ 10 N kg⁻¹. From W = mg, W ≈ 70 × 10 = 700 N = 7 × 10² N. A value near 7 N is too small for an adult, while 7 × 10⁴ N is too large.

An estimate should communicate scale, not false precision. Extra decimal places do not make an estimate more reliable.

1.2 SI units

An SI base quantity is defined independently rather than from other quantities. For this syllabus, recall exactly these five mappings:

Base quantity	SI base unit	Symbol
mass	kilogram	kg
length	metre	m
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K

Use kilogram, not gram, as the base unit of mass; kelvin, not degree Celsius, for temperature; and ampere, not coulomb, for current. Unit symbols are case-sensitive.

A derived unit is a product or quotient of base units. Start from the equation that defines the quantity, replace each quantity by its SI unit, then simplify the powers.

$F=ma \quad\Rightarrow\quad 1\,\mathrm{N}=1\,\mathrm{kg\,m\,s^{-2}}$

Pressure is force per area, p = F/A. Therefore Pa = N m⁻² = (kg m s⁻²)m⁻² = kg m⁻¹ s⁻². The final expression contains only SI base units.

A named unit such as N, J or Pa is not dimensionless. Replace it through a defining equation before cancelling factors or comparing units.

A physical equation is homogeneous when every term that is added or subtracted has the same SI base units, and the units on the left-hand side match those on the right-hand side.

Write the SI base units of each quantity.
Include powers and products exactly as they appear.
Simplify each term separately.
Compare all additive terms and both sides.

For s = ut + ½at²: s has unit m; ut has (m s⁻¹)(s) = m; and at² has (m s⁻²)(s²) = m. Every term has unit m, so the equation is homogeneous.

Homogeneity is necessary but not sufficient. It cannot detect a wrong dimensionless number such as 2 instead of ½, so a homogeneous equation may still use the wrong physics.

A prefix multiplies a base or derived unit by an exact power of ten. Prefix symbols are case-sensitive.

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

For length, 5.0 μm = 5.0 × 10⁻⁶ m. For area, the factor is squared: 3.0 mm² = 3.0(10⁻³ m)² = 3.0 × 10⁻⁶ m².

Do not change the case of a symbol: m means milli but M means mega. When a unit is squared or cubed, raise the prefix factor to the same power.

1.3 Errors and uncertainties

A systematic error biases measurements in a consistent way, so results tend to be too high or too low. A random error causes unpredictable variation, producing scatter among repeated readings.

Repeating and averaging readings reduces the effect of random error. It does not remove systematic error; that requires a zero check, calibration, correction or a changed method.

A zero error is systematic: if a balance reads +0.20 g when empty, its uncorrected readings are shifted upward by 0.20 g. By contrast, reading an analogue scale from slightly different angles can make readings scatter.

A systematic error need not add the same fixed amount; faulty calibration may make the bias vary with the reading. The defining feature is consistent bias, not simply a constant offset.

Precision describes how closely repeated measurements agree with one another. Accuracy describes how close a measured value is to the true or accepted value.

Judge precision from the spread or uncertainty of repeated readings. Judge accuracy by comparison with a true or accepted value, or by evidence of systematic bias.

If the true value is 10.8 cm, readings 10.1 cm, 10.1 cm and 10.2 cm are precise because they cluster tightly, but inaccurate because the cluster is far from 10.8 cm.

Extra decimal places show resolution, not guaranteed precision or accuracy. A tight cluster may still be systematically shifted from the true value.

For a sum or difference, add absolute uncertainties. For a product or quotient, add percentage uncertainties. If a measured quantity is raised to a power, multiply its percentage uncertainty by the magnitude of that power.

Derived form	Uncertainty rule
Q = x + y or x − y	ΔQ = Δx + Δy
Q = xy or x/y	%ΔQ = %Δx + %Δy
Q = xⁿ	%ΔQ =

A temperature rise is (100.0 ± 0.5) °C − (40.0 ± 0.5) °C. The rise is 60.0 °C and its absolute uncertainty is 0.5 + 0.5 = 1.0 °C, so the result is (60.0 ± 1.0) °C. Its percentage uncertainty is (1.0/60.0) × 100% = 1.7%.

Choose the rule from the mathematical operation, not from the units. Exact numerical constants do not contribute measurement uncertainty; a power changes the percentage contribution of the measured quantity.

1.4 Scalars and vectors

A scalar quantity is fully described by magnitude and unit. A vector quantity also requires direction; changing the direction changes the vector even when its magnitude is unchanged.

Scalars include distance, speed, mass, time, temperature and energy. Vectors include displacement, velocity, acceleration, force, weight and momentum.

A car travelling 5.0 m s⁻¹ east has speed 5.0 m s⁻¹ but velocity 5.0 m s⁻¹ east. The distance travelled is scalar; displacement from the starting point is vector.

A minus sign may encode direction along a chosen axis, but it does not by itself make a quantity a vector. The physical definition of the quantity determines whether direction is required.

For head-to-tail addition, move a vector parallel to itself so its tail meets the previous head. The resultant points from the first tail to the final head. Translation does not change a vector's magnitude or direction.

To calculate A − B, reverse B to make −B, then add A + (−B). In components, add or subtract corresponding x- and y-components with their signs.

A force of 3.0 N east plus 4.0 N north has components (3.0, 4.0) N. Its resultant magnitude is 5.0 N and its direction is tan⁻¹(4/3) = 53° north of east.

Do not add magnitudes unless the vectors act along the same direction. Opposite directions require signs; other angles require geometry or components.

Two perpendicular components add vectorially to reproduce the original vector. Choose and label the axes first; each component is the projection of the vector onto one axis.

For a vector of magnitude V at angle θ measured from the positive x-axis: Vx = V cos θ and Vy = V sin θ. Give each component the sign set by its direction along the chosen axes.

A 10.0 N force at 30° above the horizontal has Fx = 10.0 cos 30° = 8.66 N horizontally and Fy = 10.0 sin 30° = 5.00 N upward.

Cosine gives the component adjacent to the stated angle; it is not automatically the horizontal component. If θ is measured from the vertical, the vertical component is V cos θ.

1. Physical quantities and units

Exam analysis

Most tested topics

1.1 Physical quantities

A physical quantity combines a numerical magnitude with a unit

Estimate a physical quantity from familiar scales and relationships

1.2 SI units

The five required SI base quantities have fixed unit names and symbols

Derive a unit by applying the quantity's defining relationship

Check homogeneity by reducing every term to SI base units

SI prefixes rescale units by exact powers of ten

1.3 Errors and uncertainties

Systematic error shifts results; random error scatters them

Precision is agreement; accuracy is closeness to the true value

The operation determines how uncertainties combine

1.4 Scalars and vectors

Scalars have magnitude; vectors have magnitude and direction

Add vectors head-to-tail; subtract by reversing the vector

Resolve a vector along two chosen perpendicular axes