1.1 Physical Quantities
Measurement is a method used to determine the value of a physical quantity. Accurate measurements are essential because they enable us to make sound decisions in daily life and scientific endeavors.
Learning Objectives
- Explain what a physical quantity is and state it in terms of magnitude and unit.
- Identify and define base quantities and their respective S.I. units and symbols.
- Define derived quantities and express them in terms of base quantities and S.I. base units.
- Distinguish between scalar quantities and vector quantities with appropriate examples.
Relationship Between Base and Derived Quantities Flowchart
See how derived quantities such as force and pressure break down into base quantities and their S.I. base units.
From Derived Quantity to Base Units
\(F = m \times a\)
\(m, l, t\)
\(P = \dfrac{F}{A}\)
\(\mathrm{kg\,m^{-1}\,s^{-2}}\)
\(\rho = \dfrac{m}{V}\)
\(\mathrm{kg\,m^{-3}}\)
Scalar vs Vector Quantities
Scalar
- Magnitude only
- Distance, speed, mass, energy, time, volume, density
Vector
- Magnitude + direction
- Displacement, velocity, force, acceleration, momentum
Short Explanation
What is a Physical Quantity?
A physical quantity is a measurable attribute. It must always be stated as a numerical magnitude accompanied by its appropriate unit. For example, length = \(100\ \mathrm{m}\), where \(100\) is the magnitude and \(\mathrm{m}\) is the unit.
Metric vs Imperial Units
Metric units include centimetres (\(\mathrm{cm}\)), metres (\(\mathrm{m}\)) and kilopascals (\(\mathrm{kPa}\)). Imperial units include inches (\(\mathrm{in}\)), pounds per square inch (\(\mathrm{psi}\)), gallons, miles, feet and yards. In 1999, the Mars Climate Orbiter was lost because one engineering group used imperial units while another used metric units.
Base Quantities
A base quantity is a physical quantity that cannot be derived from another physical quantity. There are seven base quantities in the International System of Units (S.I.).
Derived Quantities
A derived quantity is related to base quantities through a mathematical formula involving multiplication, division, or both.
| Base Quantity | Symbol | S.I. Unit | Unit Symbol |
|---|---|---|---|
| Length | \(l\) | metre | \(\mathrm{m}\) |
| Mass | \(m\) | kilogram | \(\mathrm{kg}\) |
| Time | \(t\) | second | \(\mathrm{s}\) |
| Thermodynamic temperature | \(T\) | kelvin | \(\mathrm{K}\) |
| Electric current | \(I\) | ampere | \(\mathrm{A}\) |
| Luminous intensity | \(I_v\) | candela | \(\mathrm{cd}\) |
| Amount of substance | \(n\) | mole | \(\mathrm{mol}\) |
| Derived Quantity | Formula | In Terms of Base Quantities | S.I. Base Unit | Special Unit |
|---|---|---|---|---|
| Area | \(A = l^2\) | \(l \times l\) | \(\mathrm{m^2}\) | — |
| Volume | \(V = l^3\) | \(l \times l \times l\) | \(\mathrm{m^3}\) | — |
| Density | \(\rho = \dfrac{m}{V}\) | \(\dfrac{m}{l^3}\) | \(\mathrm{kg\,m^{-3}}\) | — |
| Velocity | \(v = \dfrac{l}{t}\) | \(\dfrac{l}{t}\) | \(\mathrm{m\,s^{-1}}\) | — |
| Acceleration | \(a = \dfrac{v}{t}\) | \(\dfrac{l}{t^2}\) | \(\mathrm{m\,s^{-2}}\) | — |
| Force | \(F = m \times a\) | \(\dfrac{m \times l}{t^2}\) | \(\mathrm{kg\,m\,s^{-2}}\) | newton, \(\mathrm{N}\) |
| Momentum | \(p = m \times v\) | \(\dfrac{ml}{t}\) | \(\mathrm{kg\,m\,s^{-1}}\) | — |
| Pressure | \(P = \dfrac{F}{A}\) | \(\dfrac{\mathrm{kg\,m\,s^{-2}}}{\mathrm{m^2}}\) | \(\mathrm{kg\,m^{-1}\,s^{-2}}\) | pascal, \(\mathrm{Pa}\) |
| Energy / Work | \(W = F \times l\) | \(\dfrac{ml^2}{t^2}\) | \(\mathrm{kg\,m^2\,s^{-2}}\) | joule, \(\mathrm{J}\) |
| Electric charge | \(Q = I \times t\) | \(I \times t\) | \(\mathrm{A\,s}\) | coulomb, \(\mathrm{C}\) |
Try to Answer First
Answer in your mind, then press “Check Answer”.
What is the difference between a base quantity and a derived quantity?
Check Answer
State the derived unit of force in terms of S.I. base units and give its special name.
Check Answer
A car travels at \(80\ \mathrm{km\,h^{-1}}\) due east. Is this speed or velocity, and is it a scalar or vector quantity?
Check Answer
Common Mistakes
- Writing a physical quantity without its unit, for example writing length as just “\(100\)” instead of “\(100\ \mathrm{m}\)”.
- Confusing scalar and vector counterparts, such as treating speed (scalar) and velocity (vector), or distance (scalar) and displacement (vector) as identical terms.
- Incorrectly using derived units instead of S.I. base units when asked to express a quantity in base units, for example stating the base unit of force as “\(\mathrm{N}\)” instead of “\(\mathrm{kg\,m\,s^{-2}}\)”.
- Mixing up imperial and metric units without conversion, for example mixing feet and metres.
Concept Misunderstandings
All units with names after scientists, like newton, joule and pascal, are base units.
These are derived units with specific names given in honour of scientists. They can all be broken down into combinations of the 7 S.I. base units.
Temperature is a derived quantity because it is calculated using degrees Celsius.
Thermodynamic temperature is one of the seven base quantities, and its standard S.I. unit is the kelvin (\(\mathrm{K}\)).
A vector quantity only needs a direction.
A vector quantity requires both magnitude and direction to be completely specified.
Summary
- Measurement determines the value of a physical quantity, which consists of magnitude and unit.
- There are seven base quantities: length (\(\mathrm{m}\)), mass (\(\mathrm{kg}\)), time (\(\mathrm{s}\)), thermodynamic temperature (\(\mathrm{K}\)), electric current (\(\mathrm{A}\)), luminous intensity (\(\mathrm{cd}\)), and amount of substance (\(\mathrm{mol}\)).
- Derived quantities are derived from base quantities via multiplication and/or division formulas.
- Scalar quantities have magnitude only, for example distance, mass and energy.
- Vector quantities have both magnitude and direction, for example displacement, velocity, force and momentum.
Short Activity
Objective: Identify base quantities, derived quantities, scalar quantities and vector quantities.
A. Objective Quiz
1 Which of the following is an S.I. base quantity?
2 Which pair contains only vector quantities?
B. Fill in the Blanks
3 In terms of S.I. base units, the unit of acceleration is expressed as .
Expected answer: \(\mathrm{m\,s^{-2}}\) (or \(\mathrm{m/s^2}\)).
4 Physical quantities that have both magnitude and direction are known as quantities.
Expected answer: vector.
C. Matching / Drag and Drop
Drag each derived quantity to the correct formula in terms of base quantities or S.I. base unit representation. If using a phone, tap the answer first, then tap the matching box.
