1. Physical quantities and units

A measurement is written as number×unit, such as 3.0 m. The unit states the scale, while the number changes when the unit is changed.

Keep significant figures consistent with the measurement uncertainty and convert units before substituting into equations.

2.5 km is 2500 m; the physical distance is unchanged even though the numerical value differs.

A bare number is not a complete physical measurement, and adding quantities with incompatible units is meaningless.

Base quantities such as mass, length and time are defined independently; derived quantities such as speed, force and energy are built from them by equations.

Identify the quantity before choosing a formula, then check whether it is scalar or vector and whether the units match the physical meaning.

Speed is distance divided by time, so its SI unit is m s⁻¹; acceleration adds another factor of s⁻¹.

A derived quantity is not less fundamental as a measurement; its unit reveals how the underlying quantities combine.

The SI base units include kilogram (kg) for mass, metre (m) for length, second (s) for time, ampere (A), kelvin (K), mole (mol) and candela (cd).

Write symbols exactly, distinguish units from quantities and use the base set when checking an equation or converting a derived unit.

A force measured in newtons can be rewritten as kg m s⁻² using kg, m and s.

The symbol kg is already the base unit; prefixes such as g or ms do not replace the definition of the quantity.

Use the defining equation to express a derived unit in base units: N=kg m s⁻², J=kg m² s⁻² and W=kg m² s⁻³.

Treat unit algebra like algebra of powers, cancel common factors and check that the final dimensions match the quantity named.

Pressure is force/area, so Pa=N m⁻²=kg m⁻¹ s⁻².

A named unit such as joule is not dimensionless; replacing it with base units exposes the physical dimensions.

Every additive term in a valid equation must have the same dimensions, and both sides must match. Dimensions use base quantities such as [L], [T] and [M].

Do not use dimensional analysis to determine dimensionless numerical constants or prove an equation completely; it is a consistency test.

In s=ut+½at², both ut and at² have dimensions L, matching displacement s.

An equation can be dimensionally correct but physically wrong because of a missing factor or incorrect model.

Prefixes such as kilo (10³), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹) and pico (10⁻¹²) multiply the unit by a stated power of ten.

Convert the prefix before calculating, keep the exponent sign visible and distinguish a prefixed unit from a power of the base unit.

5.0 μm=5.0×10⁻⁶ m; 3.0 mm²=(3.0×10⁻³ m)²=9.0×10⁻⁶ m².

Squaring an area conversion squares the prefix factor too; converting 3 mm² as 3×10⁻³ m² is wrong.

A systematic error shifts readings in a consistent direction, while random error causes scatter between repeated readings. A zero error is a systematic offset from a faulty zero.

Repeat measurements and average to reduce random variation; calibration, zero checks or a changed method address systematic bias. Repeats cannot remove a constant offset.

A balance reading 0.20 g with an empty pan adds 0.20 g to every mass unless corrected.

A precise cluster can still be inaccurate if it is systematically shifted.

Precision is the agreement among repeated measurements; accuracy is closeness to an accepted or true value. A result may be precise but inaccurate, or accurate on average but imprecise.

Use repeated readings to assess spread and a reference value or calibration to assess bias. Report uncertainty alongside a measured value.

Readings 10.1,10.1,10.2 are precise; if the true value is 10.8 they are not accurate.

More decimal places do not create accuracy, and averaging removes random scatter but not systematic error.

For addition or subtraction, add absolute uncertainties. For multiplication, division or powers, add fractional or percentage uncertainties multiplied by the relevant power.

Keep units consistent, avoid overstating precision and round the uncertainty to an appropriate significant figure before matching the value’s decimal place.

For L=2.0±0.1 m and W=3.0±0.1 m, perimeter 2L+2W has absolute uncertainty 2(0.1)+2(0.1)=0.4 m.

Adding percentage uncertainties to an addition is wrong; the operation determines which uncertainty form is appropriate.

Mass, time, temperature and energy are scalars. Displacement, velocity, acceleration, force and momentum are vectors requiring direction as well as size.

Use scalar arithmetic only for scalars; vector components or geometry are needed when direction changes. A negative signed scalar can represent a component, not a vector by itself.

Speed is the magnitude of velocity; two velocities of 5 m s⁻¹ in opposite directions have equal speeds but a resultant velocity of zero.

A vector is not simply a scalar with an arrow drawn beside it; its direction affects addition and physical effect.

To add vectors, place the tail of one at the head of the other; the resultant joins the first tail to the final head. In components, add corresponding horizontal and vertical parts.

Preserve direction and scale in diagrams, then use Pythagoras or trigonometry only after resolving or forming the correct triangle.

Vectors (3,4) and (−1,2) add to (2,6), whose magnitude is √40.

Adding magnitudes ignores angle; opposite vectors can cancel even when each has a large magnitude.

Resolve a vector into perpendicular components Vx=Vcosθ and Vy=Vsinθ. Recover magnitude with √(Vx²+Vy²) and direction with a quadrant-aware tangent.

Choose axes before resolving, keep signs and use the angle measured from the correct axis. Components are not independent physical vectors unless the coordinate model says so.

A 10 N force at 30° above horizontal has components 10cos30° and 10sin30°.

Swapping sine and cosine changes which component is largest; the reference axis fixes the choice.