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Numbers, Units and Estimation · Lesson 3 of 6

Converting Units and Prefixes

15 min read4 ideas6 checks1 figure to playBy Asheshanand Acharya
What you will learn 0 / 4
Converting units and prefixes
Look around
250 units of what?

Look at a monthly electricity bill from your home. One line says something like "Units consumed: 250", and the charge is calculated from that number. Your science book measures energy in joules. The fan in your room is marked in watts. Three different words appear, yet the board, the book and the fan maker all speak about one thing.

you will turn the units on an electricity bill into joules, and turn the power and the hours of any appliance into units.

A number and a unit

Every measurement compares a quantity with a chosen standard of the same kind. That standard is the unit. A length of \(5\,\text{m}\) is five times the standard length called the metre. So a physical quantity has two parts, a number and a unit, and neither part means anything alone. A distance of "5" could be 5 mm or 5 km.

The same length can be written with different units: \(5\,\text{m} = 500\,\text{cm}\). The length itself has not changed. The centimetre is a hundred times smaller than the metre, so a hundred times more of them are needed. The smaller the unit, the larger the number.

RememberChanging the unit of a quantity
\[ \begin{gathered} Q = n_1u_1 = n_2u_2 \\ n_2 = n_1\,\dfrac{u_1}{u_2} \end{gathered} \]
Holds whenfor any quantity measured from a true zero, such as a length, a mass, a time, a speed or an energy. It fails for temperature scales with different zeros.
Symbols\(Q\) the quantity, \(n_1\) its number in the unit \(u_1\), \(n_2\) its number in the unit \(u_2\)
Units\(u_1/u_2\) is a pure number, for example \(1\,\text{km}/1\,\text{m} = 1000\)

Multiplying by one

The safest way to change a unit is to multiply by a fraction equal to one. Since \(1\,\text{km} = 1000\,\text{m}\), the fraction \(1000\,\text{m}/1\,\text{km}\) is equal to one. Multiplying by it changes the look of a number, never its value. Choose the fraction so that the unwanted unit cancels, like a common factor. For the length of a guitar string, \(64\,\text{cm} \times (1\,\text{m}/100\,\text{cm}) = 0.64\,\text{m}\).

Prefixes are powers of ten

A prefix in front of a unit multiplies it by a power of ten. The common ones, with lengths and energies from this book, are these.

  • femto (f) \(10^{-15}\): a nucleus is a few fm across. pico (p) \(10^{-12}\). nano (n) \(10^{-9}\): yellow light is 589 nm.
  • micro (\(\mu\)) \(10^{-6}\): a hair is about \(80\,\mu\text{m}\) thick. milli (m) \(10^{-3}\). centi (c) \(10^{-2}\).
  • kilo (k) \(10^{3}\). mega (M) \(10^{6}\): a power station gives 1000 MW. giga (G) \(10^{9}\). tera (T) \(10^{12}\).

Each prefix is read as its power of ten: \(589\,\text{nm} = 589 \times 10^{-9}\,\text{m} = 5.89 \times 10^{-7}\,\text{m}\).

CheckpointUnanswered

A human hair is about \(80\,\mu\text{m}\) thick. What is this in metres?

Squared and cubed units

A unit that is squared needs its factor squared too. One metre is 100 cm, so a square one metre on each side is 100 cm by 100 cm. Its area is \(100 \times 100 = 10^4\,\text{cm}^2\), as Fig 3.1 shows. A cubed unit needs its factor cubed: \(1\,\text{m}^3 = 10^6\,\text{cm}^3\). So a litre, \(1000\,\text{cm}^3\), is \(10^{-3}\,\text{m}^3\), and a density of \(1\,\text{g/cm}^3\) is \(1000\,\text{kg/m}^3\).

One square metre in square centimetresA square one metre on each side, which is 100 centimetres on each side, divided into a grid of 10 by 10 cells. One shaded cell is 10 centimetres by 10 centimetres, an area of 100 square centimetres. The grid has 100 such cells, so one square metre is 100 times 100, that is 10 000 square centimetres. The factor 100 of the metre is squared. 1 m = 100 cm one cell: 10 cm by 10 cm = 100 cm² 10 rows of 10 cells: 100 cells 1 m² = 100 × 100 cm² = 10 000 cm²
Fig 3.1A square one metre on each side, which is 100 cm on each side. The grid has 10 rows of 10 cells, and each cell is 10 cm by 10 cm, or \(100\,\text{cm}^2\). So \(1\,\text{m}^2 = 10^4\,\text{cm}^2\): the factor 100 of the metre is squared.
CheckpointUnanswered

A metal sheet has an area of \(2\,\text{m}^2\). What is its area in square centimetres?

The seven si base units

Seven base units

Scientists of every country use one system of units, the SI, from its French name Système International. It chooses seven base quantities and one unit for each. Length is measured in metres (m), mass in kilograms (kg) and time in seconds (s). Electric current is in amperes (A), temperature in kelvin (K), amount of substance in moles (mol) and luminous intensity in candela (cd).

Every other unit is a derived unit, built from the base units by the equation that defines its quantity. Speed is distance over time, so its unit is m/s. Force is mass times acceleration, so the newton is \(1\,\text{N} = 1\,\text{kg m s}^{-2}\). Work is force times distance, so the joule is \(1\,\text{J} = 1\,\text{N m}\), and the watt is \(1\,\text{W} = 1\,\text{J/s}\).

Units named after people keep a small letter when written as words and a capital as symbols. We write "newton" but N, and "joule" but J. A symbol is never made plural and never takes a full stop, so we write 5 kg, never 5 kgs. A space separates the number from the unit.

Story · The General Conference on Weights and Measures, 20 May 2019
The kilogram that was retired

For 130 years the kilogram was one metal cylinder of platinum and iridium, kept under three glass bells near Paris. Every balance in the world was compared with it through a chain of copies. Yet careful weighings showed that its copies drifted by tens of micrograms from the master. On 20 May 2019 a new definition came into force. The kilogram is now fixed by an exact value of the Planck constant, a number of nature that cannot drift. Any well-equipped laboratory can now realise the kilogram without visiting Paris.

Kilometres per hour and metres per second

Kilometres per hour and metres per second

A speedometer reads in km/h, but the equations of motion need m/s. Convert both parts of the unit at once. One kilometre is 1000 m and one hour is 3600 s, so \(1\,\text{km/h} = 1000\,\text{m}/3600\,\text{s} = \tfrac{5}{18}\,\text{m/s}\). The reverse is \(1\,\text{m/s} = 3.6\,\text{km/h}\). A car at \(72\,\text{km/h}\) moves at \(72 \times \tfrac{5}{18} = 20\,\text{m/s}\).

One speed on two scalesTwo scales for the same speed, one above the other: kilometres per hour above, from 0 to 144, and metres per second below, from 0 to 40. A slider sets the speed in metres per second, and a pointer crosses both scales at once. At 10 metres per second the upper scale reads 36 kilometres per hour, at 20 it reads 72, at 25 it reads 90 and at 35 it reads 126. The upper number is always 3.6 times the lower, because one metre per second is 3.6 kilometres per hour. 0 10 20 30 40 m/s 0 36 72 108 144 km/h v = 20 m/s = 72 km/h 20 m/s × 3.6 = 72 km/h
SimOne speed on two scales. Move the slider. The pointer crosses the m/s scale below and the km/h scale above at the same time. The upper number is always 3.6 times the lower one, at every speed, because both scales measure one speed.

The two scales in the figure above are both straight scales that start at zero. So one factor, 3.6, converts every speed. This is true of every unit change that the formula of this lesson covers. It is not true of temperature scales, whose zeros differ.

Checkpoint · numericUnanswered

A Rajdhani Express runs at \(126\,\text{km/h}\) on a straight stretch. What is its speed in m/s?

m/s± 2 %
The kilowatt hour on the bill

The unit on the bill

The electricity board sells energy, and its unit is the kilowatt-hour. One unit is the energy that a \(1\,\text{kW}\) appliance uses in one hour. In joules, \(1\,\text{kWh} = 1000\,\text{W} \times 3600\,\text{s} = 3.6 \times 10^6\,\text{J}\). The name contains "watt", but the kilowatt-hour is an energy, a power multiplied by a time.

ExampleThis is the room heater of Electric Current and Resistance, lesson 5, done with the tool

A \(2\,\text{kW}\) room heater runs for 3 hours a day. How many units does it use in a day, and how many joules is that? What does it cost at Rs 5 per unit, for a day and for a winter month of thirty days?

Try it on paper first. The solution is one tap away when you are ready.

The phenomenon is answered. "Units consumed: 250" means 250 kWh. That is \(250 \times 3.6 \times 10^6 = 9.0 \times 10^8\,\text{J}\) of energy in the month. The fan's watts and the hours it runs give its share of those units.

CheckpointUnanswered

The kilowatt-hour on an electricity bill is a unit of which quantity?

CheckpointUnanswered

How many joules make one unit of electricity?

Checkpoint · numericUnanswered

A \(750\,\text{W}\) mixer-grinder runs for 20 minutes every day. How many units does it use in 30 days?

kWh± 2 %
Common mistake

Two conversions need care. A squared or cubed unit needs its factor squared or cubed, as the square metre showed. And temperature scales do not start at the same zero. So \(20\,°\text{C}\) is not \(20 \times 1.8\,°\text{F}\): the Fahrenheit reading is \(1.8 \times 20 + 32 = 68\,°\text{F}\). A change of 1 °C is a change of 1 K, but a temperature of \(20\,°\text{C}\) is \(293\,\text{K}\).

In the world
Where this tool works in physics

Every numerical answer in this book passes through a unit conversion. Motion problems turn km/h into m/s. The waves chapter turns a guitar string of 64 cm and 2.0 g into metres and kilograms before it finds the speed. The electricity chapter prices energy in kilowatt-hours, and the modern physics chapters measure energy in electron volts, \(1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}\). Calorimetry changes calories into joules, and optics measures the power of a lens in dioptres, with the focal length in metres.

Exam traps
  1. Multiplying km/h by 3.6 to get m/s.
    A metre per second is the larger unit, so the number must shrink. Divide km/h by 3.6, or multiply by \(5/18\).
  2. Converting \(1\,\text{m}^2\) to \(100\,\text{cm}^2\).
    Square the factor as well as the unit: \(1\,\text{m}^2 = (100\,\text{cm})^2 = 10^4\,\text{cm}^2\).
  3. Calling the kilowatt-hour a unit of power.
    It is power times time, an energy: \(1\,\text{kWh} = 3.6 \times 10^6\,\text{J}\).
  4. Writing 5 Kg, 5 kgs or 5 k.g.
    The symbol is kg, in small letters, with no plural and no full stop.

Where this tool works in the book

The physics that leans on this lesson. Each card opens the topic where the tool is at work.

Motion in One Dimension
Lesson 2 · Distance and SpeedEqual distances give the harmonic meanWhen equal distances are covered at two speeds, the average speed is the harmonic mean, which is always less than the arithmetic mean.
Relative Velocity in One Dimension
Lesson 2 · Relative Velocity on a LineRelative velocity on a lineThe velocity of A relative to B is \(v_{AB} = v_A - v_B\).
Dynamics of Rotation
Lesson 4 · Rotational Kinetic Energy and the Work of a TorquePower delivered by a torqueA torque \(\tau\) acting on a body turning at \(\omega\) delivers power \(P = \tau\omega\), the rotational twin of \(P = Fv\).
Fluid Flow
Lesson 2 · The Equation of ContinuityVolume flow rateThe volume flow rate \(Q\) is the volume of fluid that crosses a section each second.
Viscosity
Lesson 1 · Friction Inside a FluidUnits of viscosityThe SI unit of viscosity is the pascal second, the CGS unit is the poise (0.1 Pa s), and the dimensions of \(\eta\) are \(\text{ML}^{-1}\text{T}^{-1}\).
Temperature and Thermal Expansion
Lesson 2 · Scales, Conversions and Faulty ThermometersCelsius fahrenheit and kelvinCelsius, Fahrenheit and Kelvin share the fixed points with different numbers: C/5 = (F − 32)/9 = (K − 273)/5.
Calorimetry and Change of State
Lesson 1 · Heat and Specific HeatHeat and its unitsHeat is energy that flows from a hotter body to a colder one.
Propagating Waves on a String
Lesson 2 · Tension, Mass and the Hanging RopeFinding mu and the wave speedThe mass per unit length of a uniform string is its mass over its length, \(\mu = m/L\), in SI units.
Electric Potential, Energy and the Dipole
Lesson 2 · Electric PotentialPotential difference and the electron voltThe potential difference between two points is the work per coulomb to move a charge between them.
Capacitance and Dielectrics
Lesson 1 · Capacitance and Its GeometriesCapacitance of an isolated sphereAn isolated conducting sphere of radius \(R\) has \(V = kQ/R\), so \(C = 4\pi\varepsilon_0 R = R/k\).
Electric Current and Resistance
Lesson 5 · Joule Heating and Electric PowerThe electricity bill in unitsElectricity is sold in kilowatt-hours, the 'units' of the bill: \(1\,\text{kWh} = 3.6 \times 10^6\,\text{J}\).
Spherical Surfaces and Lenses
Lesson 3 · Thin Lenses and the Lens Maker's FormulaPower of a lensThe power of a lens is \(P = 1/f\) with \(f\) in metres, measured in dioptres; converging lenses have positive power, and opticians write prescriptions in dioptres.
Photons and the Photoelectric Effect
Lesson 1 · Light That Frees ElectronsThe electron voltOne electron volt is the energy an electron gains in crossing a potential difference of one volt, \(1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}\).
The Nucleus, Binding Energy and Nuclear Reactions
Lesson 2 · Mass Defect and Binding EnergyBinding energyThe binding energy \(B = \Delta m\, c^2\) is the energy needed to pull a nucleus apart into free nucleons, and the energy given out when it forms.
Diodes, Transistors and Logic Gates
Lesson 3 · Diodes and Light: Photodiode, LED, Solar CellBand gap and colourA photon can make an electron-hole pair only if \(h\nu \ge E_g\), and a recombining pair gives a photon of energy about \(E_g\).