Measurement/Unit

• Measurement means comparing an unknown physical quantity with a fixed standard.
• A unit is a definite standard quantity used to measure another quantity of the same kind.
• Every physical measurement is written as:

Physical quantity = Numerical value × Unit

Example:

Length = 5 metre

Here, 5 is the numerical value and metre is the unit.

• A good system of units should be:

  • well defined;
  • easily reproducible;
  • universally accepted;
  • independent of place and time.

• The major systems discussed in this chapter are:

  • CGS
  • FPS
  • MKS
  • SI

The SI system is the internationally accepted modern system of measurement. It was formally adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960.

CGS System

• C = centimetre
• G = gram
• S = second

FPS System

• Often called the British system.

• Uses:

  • foot
  • pound
  • second

1 pound ≈ 453.59237 g

MKS System

• Uses:

  • metre
  • kilogram
  • second

SI System

• SI stands for Système International d'Unités / International System of Units.
• It is based on seven fundamental quantities.

Since 20 May 2019, several SI units are defined using fixed values of fundamental physical constants.

• The metre is defined using the speed of light in vacuum.
• Speed of light is fixed as:

c = 299,792,458 m/s

• Thus, one metre is the distance travelled by light in vacuum during:

1 / 299,792,458 second

• Earlier, kilogram was defined using a platinum-iridium prototype kept near Paris.
• Since 2019, it is defined using the fixed value of the Planck constant:

h = 6.62607015 × 10⁻³⁴ J·s

• One second is defined using radiation associated with the caesium-133 atom.
• The fixed frequency is:

9,192,631,770 Hz

• The ampere is the SI unit of electric current.
• It is defined using the elementary electric charge:

e = 1.602176634 × 10⁻¹⁹ coulomb

• The kelvin is the SI unit of thermodynamic temperature.
• It is defined using the Boltzmann constant:

k = 1.380649 × 10⁻²³ J K⁻¹

• One mole contains exactly:

6.02214076 × 10²³ elementary entities

• This number is called the Avogadro constant.

The entities may be:

• atoms;
• molecules;
• ions;
• electrons;
• other specified particles.

• These are independent physical quantities.
• Their units are called base units.

Examples:

• length;
• mass;
• time;
• electric current.

• These are obtained mathematically from base quantities.

Examples:

Speed = distance / time

Force = mass × acceleration

Pressure = force / area

• Force changes or tends to change the state of motion of an object.

Force = Mass × Acceleration

SI Unit

newton (N)

1 N = 1 kg·m·s⁻²

CGS Unit

dyne

• Work is done when a force produces displacement.

Work = Force × Displacement

SI Unit

joule (J)

1 J = 1 N·m

• Joule is also the SI unit of energy and heat.

• Power is the rate of doing work.

Power = Work / Time

SI Unit

watt (W)

1 W = 1 J/s

Common Multiple

1 kilowatt = 10³ W

1 megawatt = 10⁶ W

Horsepower

1 horsepower ≈ 746 W

• Frequency is the number of cycles or oscillations per second.

SI Unit

hertz (Hz)

1 Hz = 1 cycle/second

Computer Processor

• CPU clock frequency is commonly expressed in GHz.

1 GHz = 10⁹ Hz

• Pressure is force acting per unit area.

Pressure = Force / Area

SI Unit

pascal (Pa)

1 Pa = 1 N/m²

Common Pressure Units

Stress

• Stress is force per unit area.

Stress = Force / Area

• SI unit: pascal

Strain

• Strain is the ratio of change in dimension to original dimension.

Strain = Change in dimension / Original dimension

• Since it is a ratio of similar quantities, strain has no unit.

• Resistance depends on:

  • material;
  • length;
  • cross-sectional area.

• Resistivity is an intrinsic property of the material.

ρ = RA / l

where:

• ρ = resistivity
• R = resistance
• A = cross-sectional area
• l = length

Tesla

1 T = 1 Wb/m²

also,

1 T = 1 N/(A·m)

Becquerel

1 Bq = 1 nuclear decay per second

Curie

• Curie (Ci) is an older non-SI unit of radioactivity.

Metric Units

• Symbol: μm
• Earlier commonly called micron.

1 μm = 10⁻⁶ m

• Commonly used for:

  • cells;
  • bacteria;
  • microscopic structures.

Conversion

1 μm = 10⁻⁴ cm = 1/1000 mm

1 nm = 10⁻⁹ m

Used for measuring:

• molecules;
• very small electronic components;
• wavelengths;
• nanostructures.

Useful Conversion

1 metre = 10⁹ nm

1 nm = 10⁻⁷ cm

• Symbol: Å

1 Å = 10⁻¹⁰ m

Used historically for:

• wavelengths;
• atomic dimensions;
• interatomic distances;
• crystal spacing.

Relation

1 Å = 0.1 nm

1 fm = 10⁻¹⁵ m

• Used mainly for:

  • nuclear dimensions;
  • proton and neutron scales.

Among micron, nanometre, angstrom and fermi, the smallest is fermi.

• Symbol: au
• Mainly used for distances within the Solar System.
• It represents approximately the mean Earth–Sun distance.

1 au = 149,597,870,700 m

≈ 1.496 × 10¹¹ m

≈ 149.6 million km

• A light-year is a unit of distance, not time.
• It is the distance light travels in vacuum in one year.

1 light-year ≈ 9.461 × 10¹⁵ m

• A parsec is another astronomical unit of distance.
• It is the distance at which 1 astronomical unit subtends an angle of 1 arcsecond.

1 parsec ≈ 3.086 × 10¹⁶ m

1 parsec ≈ 3.262 light-years

Order of Astronomical Distances

1 AU < 1 light-year < 1 parsec

Large Multiples

Small Multiples

Temperature

Common temperature scales:

• Celsius
• Fahrenheit
• Kelvin

SI Unit

kelvin (K)

Celsius–Kelvin Relation

K = °C + 273.15

approximately,

K = °C + 273

• Heat is a form of energy transferred because of a temperature difference.

SI Unit

joule

Other traditional units include:

• calorie
• kilocalorie
• erg

• A calorie is a non-SI unit of energy.
• Food energy is commonly expressed as kilocalories.

1 calorie ≈ 4.184 joule

1 kilocalorie = 1000 calories

Nautical Mile

• Unit of distance used in marine and air navigation.

1 nautical mile = 1852 m

Knot

• Unit of speed.

1 knot = 1 nautical mile per hour

• Mach number compares the speed of an object with the local speed of sound.

Mach number = Object speed / Speed of sound

• Mach 1 → speed of sound
• Mach > 1 → supersonic

• Decibel (dB) is a logarithmic unit commonly used to express:

  • sound level;
  • signal ratios;
  • intensity ratios.

• It is not an SI base unit.

• Dioptre (D) is used for optical power of a lens.

Power of lens = 1 / focal length in metre

1 D = 1 m⁻¹

Example:

A lens of focal length 0.5 m has power:

P = 1/0.5 = 2 D

• Cusec means cubic foot per second.
• It measures volume flow rate, especially water flow.

1 cusec ≈ 28.317 litres/second

• A petroleum barrel is a unit of volume.

1 barrel of oil ≈ 158.987 litres

≈ 159 litres

• Also:

1 barrel = 42 US gallons

• Dobson Unit (DU) measures the total amount of ozone in a vertical atmospheric column.

1 DU corresponds to 0.01 mm of pure ozone at standard conditions

• NTU stands for Nephelometric Turbidity Unit.
• It measures the turbidity of water.

Higher suspended particles → higher NTU.

• The amount of solar radiation received per unit area near Earth's mean orbital distance is approximately:

1361 W/m²

≈ 1.36 kW/m²

• In many examination options, this is rounded to approximately:

1.4 kW/m²

The source chapter also introduces important scientific measuring instruments.

Joule vs Watt

Newton vs Pascal

Ohm vs Ohm-Metre

Light-Year vs Year

Nautical Mile vs Knot

Celsius vs Calorie

Becquerel vs Gray vs Sievert

metre → millimetre → micrometre → nanometre → angstrom → femtometre

Key Relations

1 AU ≈ 1.496 × 10¹¹ m

1 light-year ≈ 9.461 × 10¹⁵ m

1 parsec ≈ 3.262 light-years

• SI has seven base quantities.
• Electric current, not force or velocity, is a fundamental quantity.
• Force is a derived quantity.
• SI unit of force is newton.
• SI unit of work and energy is joule.
• SI unit of power is watt.
• 1 watt = 1 joule per second.
• SI unit of pressure is pascal.
• 1 pascal = 1 N/m².
• SI unit of frequency is hertz.
• GHz is a unit of frequency and is commonly used for processor clock speed.
• Ampere measures electric current.
• Volt measures potential difference.
• Ohm measures resistance.
• Ohm-metre measures resistivity.
• Farad, not “Faraday”, is the SI unit of capacitance.
• Tesla measures magnetic flux density.
• Weber measures magnetic flux.
• Becquerel is the SI unit of radioactivity.
• Curie is a non-SI radioactivity unit.
• Gray measures absorbed radiation dose.
• Sievert represents dose equivalent/effective radiation dose.
• Light-year is a distance, not time.
• Parsec is a distance, not time.
• 1 parsec ≈ 3.26 light-years.
• Astronomical unit is based on the Earth–Sun distance.
• Angstrom = 10⁻¹⁰ m.
• Nanometre = 10⁻⁹ m.
• Micrometre = 10⁻⁶ m.
• Fermi/femtometre = 10⁻¹⁵ m.
• Among micron, nanometre, angstrom and fermi, fermi is the smallest.
• Strain has no unit because it is a ratio.
• Mach number is dimensionless.
• 1 horsepower ≈ 746 W.
• Nautical mile measures distance.
• Knot measures speed.
• 1 nautical mile = 1.852 km.
• Decibel represents a logarithmic sound/signal level.
• Celsius measures temperature, not heat.
• SI unit of heat is joule.
• Calorie is a non-SI energy unit.
• Cusec measures flow rate, usually water flow.
• 1 barrel of petroleum ≈ 159 litres.
• Dobson unit measures total atmospheric ozone.
• NTU measures turbidity of water.
• Dioptre measures power of a lens.
• Ammeter → current.
• Barometer → atmospheric pressure.
• Anemometer → wind speed.
• Calorimeter → heat.
• Altimeter → altitude.
• Hygrometer → humidity.
• Sphygmomanometer → blood pressure.
• Seismograph/seismometer → seismic waves.
• Odometer → distance travelled.
• Speedometer → speed.