Units and Dimensions for Competitive Exams
In this article, Units and Dimensions which are important for general competitive exams like UPSC, SSC, State PSC are discussed.
| Physical Quantity | Unit | Symbol |
| Length | meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Intensity of light | candela | cd |
| Quantity of substance | mole | mol |
Supplementary Quantities
| Plane angle | Radian | rad |
| Solid angle | Steradian | sr |
Derived SI Units with Special Names
| Physical Quantity | SI Unit | Symbol |
| Frequency | hertz | Hz |
| Energy | joule | J |
| Force | newton | N |
| Power | watt | W |
| Pressure | pascal | Pa |
| Electric charge or quantity of electricity | coulomb | C |
| Electric potential difference and emf | volt | V |
| Electric resistance | ohm | Ω |
| Electric conductance | siemen | S |
| Electric capacitance | farad | F |
| Magnetic flux | weber | Wb |
| Inductance | henry | H |
| Magnetic flux density | tesla | T |
| Illumination | lux | Lx |
| Luminous flux | lumen | Lm |
Important Facts about Units
- Angstrom is the unit of length used to measure the wavelength of light. 1 Å = 10-10 m.
- Fermi is the unit of length used to measure nuclear distances. 1 Fermi = 10-15 meter.
- A light year is the unit of length for measuring astronomical distances.
- Light year = distance travelled by light in 1 year = 9.4605 × 1015 m.
- Astronomical unit = Mean distance between the sun and earth = 1.5 × 1011 m.
- Parsec = 3.26 light years = 3.084×1016 m.
- Barn is the unit of area for measuring scattering cross-section of collisions. 1 barn = 10-28 m2.
- Chronometer and metronome are time-measuring instruments. The quantity having the same unit in all the systems of units is time.
Important Conversions
- 1 bar = 106 dyne/cm2 = 105 Nm-2 = 105 pascal
- 76 cm of Hg = 1.013×106 dyne/cm2 = 1.013×105 pascal = 1.013 bar.
- 1 toricelli or torr = 1 mm of Hg = 1.333×103 dyne/cm2 = 1.333 millibar.
- 1 kmph = 5/18 ms-1
- 1 dyne = 10-5 N,
- 1 H.P = 746 watt
- 1 kilowatt hour = 36×105 J
- 1 kgwt = g newton
- 1 calorie = 4.2 joule
- 1 electron volt = 1.602×10-19 joule
- 1 erg = 10-7 joule
Important Physical Constants
- Velocity of light in vacuum (c) = 3 × 108 ms-1
- Velocity of sound in air at STP = 331 ms-1
- Acceleration due to gravity (g) = 9.81 ms-2
- Avogadro number (N) = 6.023 × 1023/mol
- Density of water at 4oC = 1000 kgm-3 or 1 g/cc.
- Absolute zero = -273.15oC or 0 K
- Atomic mass unit = 1.66 × 10-27 kg
- Quantum of charge (e) = 1.602 × 10-19 C
- Stefan’s constant = 5.67 × 10–8 W/m2/K4
- Boltzmann’s constant (K) = 1.381 × 10-23 JK-1
- One atmosphere = 76 cm Hg = 1.013 × 105 Pa
- Mechanical equivalent of heat (J) = 4.186 J/cal
- Planck’s constant (h) = 6.626 × 10-34 Js
- Universal gas constant (R) = 8.314 J/mol–K
- Permeability of free space (μ0) = 4π × 10-7 Hm-1
- Permittivity of free space (ε0) = 8.854 × 10-12 Fm-1
- The density of air at S.T.P. = 1.293 kg m-3
- Universal gravitational constant = 6.67 × 10-11 Nm2kg-2
What Are Dimensions?
Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.
Dimensional Formulas for Physical Quantities
| Physical Quantity | Unit | Dimensional Formula |
| Acceleration or acceleration due to gravity | ms–2 | LT–2 |
| Angle (arc/radius) | rad | MoLoTo |
| Angular displacement | rad | MoloTo |
| Angular frequency (angular displacement/time) | rads–1 | T–1 |
| Angular impulse (torque x time) | Nms | ML2T–1 |
| Angular momentum (Iω) | kgm2s–1 | ML2T–1 |
| Angular velocity (angle/time) | rads–1 | T–1 |
| Area (length x breadth) | m2 | L2 |
| Boltzmann’s constant | JK–1 | ML2T–2θ–1 |
| Bulk modulus(ΔP.VΔV) | Nm–2, Pa | M1L–1T–2 |
| Calorific value | Jkg–1 | L2T–2 |
| Coefficient of linear or areal or volume expansion | oC–1 or K–1 | θ–1 |
| Coefficient of surface tension (force/length) | Nm–1 or Jm–2 | MT–2 |
| Coefficient of thermal conductivity | Wm–1K–1 | MLT–3θ–1 |
| Coefficient of viscosity(F=ηAdvdx) | poise | ML–1T–1 |
| Compressibility (1/bulk modulus) | Pa–1, m2N–2 | M–1LT2 |
| Density (mass/volume) | kgm–3 | ML–3 |
| Displacement, wavelength, focal length | m | L |
| Electric capacitance (charge/potential) | CV–1, farad | M–1L–2T4I2 |
| Electric conductance (1/resistance) | Ohm–1 or mho or siemen | M–1L–2T3I2 |
| Electric conductivity (1/resistivity) | siemen/metre or Sm–1 | M–1L–3T3I2 |
| Electric charge or quantity of electric charge (current x time) | coulomb | IT |
| Electric current | ampere | I |
| Electric dipole moment (charge x distance) | Cm | LTI |
| Electric field strength or intensity of electric field (force/charge) | NC–1, Vm–1 | MLT–3I–1 |
| Electric resistance | ohm | ML2T–3I–2 |
| Emf (or) electric potential (work/charge) | volt | ML2T–3I–1 |
| Energy (capacity to do work) | joule | ML2T–2 |
| Energy density(energyvolume) | Jm–3 | ML–1T–2 |
| Entropy(ΔS=ΔQ/T) | Jθ–1 | ML2T–2θ–1 |
| Force (mass x acceleration) | newton (N) | MLT–2 |
| Force constant or spring constant (force/extension) | Nm–1 | MT–2 |
| Frequency (1/period) | Hz | T–1 |
| Gravitational potential (work/mass) | Jkg–1 | L2T–2 |
| Heat (energy) | J or calorie | ML2T–2 |
| Illumination (Illuminance) | lux (lumen/metre2) | MT–3 |
| Impulse (force x time) | Ns or kgms–1 | MLT–1 |
| Inductance (L)(energy=12LI2)orcoefficient of self-induction | henry (H) | ML2T–2I–2 |
| Intensity of gravitational field (F/m) | Nkg–1 | L1T–2 |
| Intensity of magnetization (I) | Am–1 | L–1I |
| Joule’s constant or mechanical equivalent of heat | Jcal–1 | MoLoTo |
| Latent heat (Q = mL) | Jkg–1 | MoL2T–2 |
| Linear density (mass per unit length) | kgm–1 | ML–1 |
| Luminous flux | lumen or (Js–1) | ML2T–3 |
| Magnetic dipole moment | Am2 | L2I |
| Magnetic flux | weber (Wb) | ML2T–2I–1 |
| Magnetic induction | NI–1m–1 or T | MT–2I–1 |
| Magnetic pole strength (unit: ampere–meter) | Am | LI |
| Modulus of elasticity (stress/strain) | Nm–2, Pa | ML–1T–2 |
| Moment of inertia (mass x radius2) | kgm2 | ML2 |
| Momentum (mass x velocity) | kgms–1 | MLT–1 |
| Permeability of free space | Hm–1 or NA–2 | MLT–2I–2 |
| Permittivity of free space | Fm–1 or C2N–1m–2 | M–1L–3T4I2 |
| Planck’s constant (energy/frequency) | Js | ML2T–1 |
| Poisson’s ratio (lateral strain/longitudinal strain) | –– | MoLoTo |
| Power (work/time) | Js–1 or watt (W) | ML2T–3 |
| Pressure (force/area) | Nm–2 or Pa | ML–1T–2 |
| Pressure coefficient or volume coefficient | oC–1 or θ–1 | θ–1 |
| Pressure head | m | MoLTo |
| Radioactivity | disintegrations per second | MoLoT–1 |
| Ratio of specific heats | –– | MoLoTo |
| Refractive index | –– | MoLoTo |
| Resistivity or specific resistance | Ω–m | ML3T–3I–2 |
| Specific conductance or conductivity (1/specific resistance) | siemen/metre or Sm–1 | M–1L–3T3I2 |
| Specific entropy (1/entropy) | KJ–1 | M–1L–2T2θ |
| Specific gravity (density of the substance/density of water) | –– | MoLoTo |
| Specific heat (Q = mst) | Jkg–1θ–1 | MoL2T–2θ–1 |
| Specific volume (1/density) | m3kg–1 | M–1L3 |
| Speed (distance/time) | ms–1 | LT–1 |
| Stefan’s constant | Wm–2θ–4 | MLoT–3θ–4 |
| Strain (change in dimension/original dimension) | –– | MoLoTo |
| Stress (restoring force/area) | Nm–2 or Pa | ML–1T–2 |
| Surface energy density (energy/area) | Jm–2 | MT–2 |
| Temperature | oC or θ | MoLoToθ |
| Temperature gradient | oCm–1 or θm–1 | MoL–1Toθ |
| Thermal capacity (mass x specific heat) | Jθ–1 | ML2T–2θ–1 |
| Time period | second | T |
| Torque or moment of force (force x distance) | Nm | ML2T–2 |
| Universal gas constant (work/temperature) | Jmol–1θ–1 | ML2T–2θ–1 |
| Universal gravitational constant | Nm2kg–2 | M–1L3T–2 |
| Velocity (displacement/time) | ms–1 | LT–1 |
| Velocity gradient (dv/dx) | s–1 | T–1 |
| Volume (length x breadth x height) | m3 | L3 |
| Water equivalent | kg | MLoTo |
| Work (force x displacement) | J | ML2T–2 |
Quantities Having the Same Dimensional Formula
- Impulse and momentum.
- Work, torque, the moment of force, energy.
- Angular momentum, Planck’s constant, rotational impulse.
- Stress, pressure, modulus of elasticity, energy density.
- Force constant, surface tension, surface energy.
- Angular velocity, frequency, velocity gradient.
- Gravitational potential, latent heat.
- Thermal capacity, entropy, universal gas constant and Boltzmann’s constant.
- Force, thrust.
- Power, luminous flux.
Also, refer:
