Physics
Units and Measurements
Units and Measurements Formula Sheet
All key Units and Measurements formulas — SI units, dimensional formulas, error propagation, significant figures & least count for JEE Main and Advanced quick revision.
Every must-know relation from this chapter on one page — SI base/derived units, dimensional formulas, error propagation, significant figures, and least count. Use it for last-minute revision before JEE Main and Advanced.
SI Base Units
| Quantity | SI Base Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric Current | ampere | A |
| Temperature | kelvin | K |
| Amount of Substance | mole | mol |
| Luminous Intensity | candela | cd |
Memory hook: MMMATTL
Mass, Meter, Mole, Ampere, Temperature, Time, Luminous intensity. The mole is exactly $6.02214076\times10^{23}$ entities; the second is $9{,}192{,}631{,}770$ periods of Cs-133 radiation.
SI Derived Units (most-used)
| Quantity | Unit | In Base Units | Formula |
|---|---|---|---|
| Force | newton (N) | $\text{kg·m·s}^{-2}$ | $ma$ |
| Pressure | pascal (Pa) | $\text{kg·m}^{-1}\text{s}^{-2}$ | $F/A$ |
| Work / Energy | joule (J) | $\text{kg·m}^2\text{s}^{-2}$ | $Fs$ |
| Power | watt (W) | $\text{kg·m}^2\text{s}^{-3}$ | $E/t$ |
| Frequency | hertz (Hz) | $\text{s}^{-1}$ | $1/T$ |
| Charge | coulomb (C) | $\text{A·s}$ | $It$ |
| Voltage | volt (V) | $\text{kg·m}^2\text{s}^{-3}\text{A}^{-1}$ | — |
| Resistance | ohm ($\Omega$) | $\text{kg·m}^2\text{s}^{-3}\text{A}^{-2}$ | — |
| Capacitance | farad (F) | $\text{kg}^{-1}\text{m}^{-2}\text{s}^{4}\text{A}^{2}$ | — |
| Magnetic field | tesla (T) | $\text{kg·s}^{-2}\text{A}^{-1}$ | — |
| Magnetic flux | weber (Wb) | $\text{kg·m}^2\text{s}^{-2}\text{A}^{-1}$ | — |
SI Prefixes
| Prefix | Symbol | Factor | Prefix | Symbol | Factor |
|---|---|---|---|---|---|
| tera | T | $10^{12}$ | deci | d | $10^{-1}$ |
| giga | G | $10^{9}$ | centi | c | $10^{-2}$ |
| mega | M | $10^{6}$ | milli | m | $10^{-3}$ |
| kilo | k | $10^{3}$ | micro | $\mu$ | $10^{-6}$ |
| hecto | h | $10^{2}$ | nano | n | $10^{-9}$ |
| deca | da | $10^{1}$ | pico | p | $10^{-12}$ |
| femto | f | $10^{-15}$ |
High-yield prefix trap
When a prefixed unit is raised to a power, the power applies to the prefix too: $1\ \text{cm}^3 = (10^{-2}\ \text{m})^3 = 10^{-6}\ \text{m}^3$, not $10^{-2}\ \text{m}^3$.
Important Non-SI Units & Conversions
| Quantity | Non-SI Unit | SI Equivalent |
|---|---|---|
| Length | 1 angstrom (Å) | $10^{-10}$ m |
| Length | 1 light year | $9.46\times10^{15}$ m |
| Length | 1 parsec | $3.08\times10^{16}$ m |
| Mass | 1 atomic mass unit (u) | $1.66\times10^{-27}$ kg |
| Time | 1 year | $3.156\times10^{7}$ s ($\approx\pi\times10^{7}$ s) |
| Energy | 1 electron volt (eV) | $1.6\times10^{-19}$ J |
| Energy | 1 calorie | $4.186$ J |
| Energy | 1 kWh | $3.6\times10^{6}$ J |
| Pressure | 1 atmosphere (atm) | $1.013\times10^{5}$ Pa |
| Pressure | 1 bar | $10^{5}$ Pa |
| Power | 1 horsepower (hp) | $746$ W |
| Force | 1 dyne | $10^{-5}$ N |
| Conversion | Factor |
|---|---|
| km/h $\to$ m/s | $\times\,5/18$ |
| m/s $\to$ km/h | $\times\,18/5$ |
| litre $\to$ m³ | $\times\,10^{-3}$ |
| cm³ $\to$ m³ | $\times\,10^{-6}$ |
Dimensional Formulas
Seven fundamental dimensions: $[M]$ mass, $[L]$ length, $[T]$ time, $[I]$ or $[A]$ current, $[\theta]$ or $[K]$ temperature, $[mol]$ amount, $[cd]$ or $[J]$ luminous intensity.
Mechanics
| Quantity | Dimensional Formula |
|---|---|
| Area | $[L^2]$ |
| Volume | $[L^3]$ |
| Density | $[ML^{-3}]$ |
| Velocity / Wave velocity | $[LT^{-1}]$ |
| Acceleration / $g$ | $[LT^{-2}]$ |
| Momentum / Impulse | $[MLT^{-1}]$ |
| Force / Weight | $[MLT^{-2}]$ |
| Work / Energy / Torque | $[ML^2T^{-2}]$ |
| Power | $[ML^2T^{-3}]$ |
| Pressure / Stress | $[ML^{-1}T^{-2}]$ |
| Angular velocity / Frequency | $[T^{-1}]$ |
| Angular acceleration | $[T^{-2}]$ |
| Moment of inertia | $[ML^2]$ |
| Angular momentum | $[ML^2T^{-1}]$ |
| Surface tension / Spring constant | $[MT^{-2}]$ |
| Viscosity | $[ML^{-1}T^{-1}]$ |
Same-dimension family (JEE favourite)
Work, Energy, and Torque all share $[ML^2T^{-2}]$ — but torque’s unit is N·m, not joule. Surface tension and spring constant both have $[MT^{-2}]$. Momentum and impulse both have $[MLT^{-1}]$.
Thermal
| Quantity | Dimensional Formula |
|---|---|
| Heat | $[ML^2T^{-2}]$ |
| Specific heat | $[L^2T^{-2}K^{-1}]$ |
| Latent heat | $[L^2T^{-2}]$ |
| Thermal conductivity | $[MLT^{-3}K^{-1}]$ |
| Stefan’s constant | $[MT^{-3}K^{-4}]$ |
| Boltzmann constant | $[ML^2T^{-2}K^{-1}]$ |
| Gas constant $R$ | $[ML^2T^{-2}K^{-1}\text{mol}^{-1}]$ |
Electricity & Magnetism
| Quantity | Dimensional Formula |
|---|---|
| Charge | $[IT]$ |
| Voltage / EMF | $[ML^2T^{-3}I^{-1}]$ |
| Resistance | $[ML^2T^{-3}I^{-2}]$ |
| Capacitance | $[M^{-1}L^{-2}T^{4}I^{2}]$ |
| Electric field | $[MLT^{-3}I^{-1}]$ |
| Magnetic field | $[MT^{-2}I^{-1}]$ |
| Magnetic flux | $[ML^2T^{-2}I^{-1}]$ |
| Inductance | $[ML^2T^{-2}I^{-2}]$ |
| Permittivity $\varepsilon_0$ | $[M^{-1}L^{-3}T^{4}I^{2}]$ |
| Permeability $\mu_0$ | $[MLT^{-2}I^{-2}]$ |
Special Constants & Modern Physics
| Quantity | Dimensional Formula |
|---|---|
| Gravitational constant $G$ | $[M^{-1}L^{3}T^{-2}]$ |
| Planck’s constant $h$ | $[ML^2T^{-1}]$ |
| Work function | $[ML^2T^{-2}]$ |
| Decay constant / Half-life | $[T^{-1}]$ / $[T]$ |
| Avogadro’s number | $[\text{mol}^{-1}]$ |
Dimensionless quantities
$$\boxed{[\text{Dimensionless}] = [M^0L^0T^0] = [1]}$$Strain, angle (and $\sin\theta$, etc.), relative density, refractive index, coefficient of friction, Poisson’s ratio, and Reynolds number $Re=\rho v L/\eta$ are all dimensionless.
Dimensional Analysis
Principle of Homogeneity
$$\boxed{\text{If } A = B + C, \text{ then } [A] = [B] = [C]}$$
Only terms with identical dimensions can be added or subtracted. Arguments of $\sin$, $\cos$, $\log$, and $e^x$ must be dimensionless.
Deriving a relation — assume a power law $Q = k\,A^a B^b C^c$, write each dimensional formula, equate powers of M, L, T, and solve.
Unit conversion between systems (for a quantity of dimensions $[M^aL^bT^c]$):
$$\boxed{n_2 = n_1 \left(\frac{M_1}{M_2}\right)^a \left(\frac{L_1}{L_2}\right)^b \left(\frac{T_1}{T_2}\right)^c}$$Standard derived results from this chapter:
| Derivation | Result |
|---|---|
| Simple pendulum period | $T = k\sqrt{l/g}$ — independent of mass |
| Speed of sound in gas | $v = k\sqrt{P/\rho}$ |
| Speed of water waves | $v = k\sqrt{\lambda g}$ |
| Stokes’ law (viscous force) | $F = k\,\eta r v$ |
| Centripetal force | $F = k\,mv^2/r$ |
| Oscillation of liquid drop | $T = k\sqrt{r^3\rho/S}$ |
What dimensional analysis CANNOT do
It cannot find dimensionless constants (e.g. the $2\pi$ in $T=2\pi\sqrt{l/g}$), cannot distinguish quantities with the same dimensions (work vs torque), cannot handle equations involving addition/subtraction, and cannot fix numerator vs denominator — physics decides that.
Errors in Measurement
Mean and absolute error
For $n$ readings $a_1,\dots,a_n$:
$$\boxed{\bar{a} = \frac{a_1 + a_2 + \cdots + a_n}{n}} \qquad \Delta a_i = |a_i - \bar{a}|$$$$\boxed{\Delta a_{\text{mean}} = \frac{\Delta a_1 + \Delta a_2 + \cdots + \Delta a_n}{n}} \qquad a = \bar{a} \pm \Delta a_{\text{mean}}$$Relative & percentage error
$$\boxed{\text{Relative error } \delta a = \frac{\Delta a}{a}} \qquad \boxed{\text{Percentage error} = \frac{\Delta a}{a}\times 100\%}$$Error propagation
| Operation | Formula | Error rule |
|---|---|---|
| Sum | $Z = A + B$ | $\Delta Z = \Delta A + \Delta B$ |
| Difference | $Z = A - B$ | $\Delta Z = \Delta A + \Delta B$ |
| Product | $Z = A \times B$ | $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ |
| Quotient | $Z = A / B$ | $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ |
| Power | $Z = A^n$ | $\frac{\Delta Z}{Z} = n\,\frac{\Delta A}{A}$ |
| Constant multiple | $Z = kA$ | $\Delta Z = k\,\Delta A$ |
General function $Z = \dfrac{A^a B^b}{C^c D^d}$:
$$\boxed{\frac{\Delta Z}{Z} = a\frac{\Delta A}{A} + b\frac{\Delta B}{B} + c\frac{\Delta C}{C} + d\frac{\Delta D}{D}}$$
The three rules in one line
Add/Subtract $\to$ add absolute errors. Multiply/Divide $\to$ add relative errors. Powers $\to$ multiply relative error by the power. Errors always add, never subtract — even for $A - B$.
Least count & instrumental error
$$\boxed{\text{Instrumental error} = \pm(\text{Least Count})}$$$$\text{Vernier LC} = 1\,\text{MSD} - 1\,\text{VSD} = \frac{\text{Main Scale Division}}{\text{No. of Vernier divisions}}$$$$\text{Screw gauge LC} = \frac{\text{Pitch}}{\text{No. of circular scale divisions}}$$| Instrument | Least Count |
|---|---|
| Meter scale | $1\,\text{mm} = 0.1\,\text{cm}$ |
| Vernier calipers | $0.01\,\text{cm} = 0.1\,\text{mm}$ |
| Screw gauge | $0.001\,\text{cm} = 0.01\,\text{mm}$ |
| Stopwatch (analog / digital) | $0.1\,\text{s}$ / $0.01\,\text{s}$ |
When LC is not given
Take half the smallest division as the error — e.g. a mm-marked ruler has error $\pm 0.5\,\text{mm}$.
Accuracy = closeness to true value (set by systematic errors, fixed by calibration). Precision = closeness of readings to each other (set by random errors, reduced by averaging multiple readings).
Significant Figures
Counting rules
| Digit type | Rule |
|---|---|
| Non-zero digits | Always significant |
| Captive zeros (between non-zeros) | Always significant |
| Leading zeros | Never significant |
| Trailing zeros | Significant only if a decimal point is present |
| Exact numbers (counts, definitions) | Infinite significant figures |
Operation rules
| Operation | Result keeps |
|---|---|
| Addition / Subtraction | Least number of decimal places |
| Multiplication / Division | Fewest significant figures |
| Powers / Roots | Same sig figs as the original number |
Sig-fig exam traps
Use scientific notation to kill ambiguity ($4500$ could be 2–4 sig figs; $4.50\times10^3$ is unambiguously 3). Keep one extra digit in intermediate steps and round only at the end. Exact numbers and $\pi$ never limit the result.
Banker’s rounding: when the dropped digit is exactly 5 (with nothing nonzero after), round to the nearest even digit — e.g. $12.45\to12.4$, $12.55\to12.6$.
Related Topics
- SI Units — base & derived units in depth
- Dimensions — full dimensional formula tables
- Dimensional Analysis — derivations & unit conversion
- Errors — worked error-propagation problems
- Significant Figures — rounding & operation rules