Units and Measurements formulas
Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 1, the special cases questions are built on, and diagrams where they help.
By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE
Most used formulasOther formulas and cases
1Units
| Base quantity | SI unit | Base quantity | SI unit |
|---|---|---|---|
| Length | metre (m) | Temperature | kelvin (K) |
| Mass | kilogram (kg) | Luminous intensity | candela (cd) |
| Time | second (s) | Amount of substance | mole (mol) |
| Electric current | ampere (A) | Plane / solid angle | radian / steradian |
Numerical value $\propto\dfrac1{\text{unit}}$. Converting a quantity with dimensions $[M^aL^bT^c]$ between systems. E.g. $1$ N $=10^5$ dyne, $1$ J $=10^7$ erg.
- Length: 1 Å $=10^{-10}$ m, 1 fermi $=10^{-15}$ m, 1 AU $=1.496\times10^{11}$ m, 1 ly $=9.46\times10^{15}$ m, 1 parsec $=3.26$ ly $=3.08\times10^{16}$ m.
- 1 kmph $=\frac5{18}$ m/s; 1 kWh $=3.6\times10^6$ J; 1 HP $=746$ W; 1 cal $=4.18$ J; 1 eV $=1.6\times10^{-19}$ J; 1 u $=1.66\times10^{-27}$ kg; 1 atm $=1.013\times10^5$ Pa; 1 T $=10^4$ G.
- Prefixes: da $10^1$, h $10^2$, k $10^3$, M $10^6$, G $10^9$, T $10^{12}$, P $10^{15}$, E $10^{18}$; d $10^{-1}$, c $10^{-2}$, m $10^{-3}$, µ $10^{-6}$, n $10^{-9}$, p $10^{-12}$, f $10^{-15}$, a $10^{-18}$.
2Dimensions
Homogeneity: only quantities of the same dimensions can be added, subtracted or equated; arguments of $\sin$, $\log$, $e^x$ must be dimensionless. In $P=\dfrac{a}{b}$ with $a+ct^2$: $[a]=[ct^2]$.
| Dimensions | Quantities |
|---|---|
| $[MLT^{-1}]$ | momentum, impulse |
| $[MLT^{-2}]$ | force, weight, tension, thrust |
| $[ML^2T^{-2}]$ | work, energy, torque, heat |
| $[ML^2T^{-1}]$ | angular momentum, Planck's constant $h$, angular impulse |
| $[ML^{-1}T^{-2}]$ | pressure, stress, modulus of elasticity, energy density |
| $[MT^{-2}]$ | surface tension, spring constant, surface energy |
| $[ML^2T^{-3}]$ | power; $[MT^{-3}]$: intensity, solar constant |
| $[ML^{-1}T^{-1}]$ | coefficient of viscosity $\eta$ |
| $[M^{-1}L^3T^{-2}]$ | gravitational constant $G$ |
| $[L^2T^{-2}]$ | latent heat, gravitational potential |
| $[L^2T^{-2}K^{-1}]$ / $[ML^2T^{-2}K^{-1}]$ | specific heat / heat capacity, entropy, $k_B$ |
| $[MLT^{-3}K^{-1}]$ | thermal conductivity; $[MT^{-3}K^{-4}]$: Stefan's constant $\sigma$ |
| $[T]$ | $RC$, $L/R$, $\sqrt{LC}$ |
| $[T^{-1}]$ | frequency, $\omega$, velocity gradient, decay constant |
| $[L^{-1}]$ | wave number, lens power, Rydberg constant |
| $[ML^2T^{-3}A^{-1}]$ | potential, emf; $[ML^2T^{-3}A^{-2}]$: resistance |
| $[M^{-1}L^{-3}T^4A^2]$ | permittivity $\varepsilon_0$; $[MLT^{-2}A^{-2}]$: permeability $\mu_0$ |
| $[MT^{-2}A^{-1}]$ | magnetic field $B$; $[ML^2T^{-2}A^{-1}]$: flux |
| $[M^{-1}L^{-2}T^4A^2]$ | capacitance; $[ML^2T^{-2}A^{-2}]$: inductance |
| $[M^0L^0T^0]$ | strain, angle, $\mu$, refractive index, $K$, Poisson's ratio, $\chi$ |
Useful combinations. With $c$, $h$, $G$ as base units: $[M]=h^{1/2}c^{1/2}G^{-1/2}$. Van der Waals: $[a]=[ML^5T^{-2}]$, $[b]=[L^3]$.
Deriving a relation: equate powers of M, L, T. Limitations: cannot find the constant $k$; fails for sums, trigonometric/log/exponential functions; needs ≤ 3 unknown powers.
3Errors
Mean (true value), absolute error, mean absolute error, relative error ($\times100$ = percentage error). Smaller percentage error → higher accuracy.
| Operation | Maximum error |
|---|---|
| $Z=A\pm B$ | $\Delta Z=\Delta A+\Delta B$ (absolute errors add) |
| $Z=AB$ or $A/B$ | $\dfrac{\Delta Z}{Z}=\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}$ (relative errors add) |
| $Z=\dfrac{A^pB^q}{C^r}$ | $\dfrac{\Delta Z}{Z}=p\dfrac{\Delta A}{A}+q\dfrac{\Delta B}{B}+r\dfrac{\Delta C}{C}$ |
| $Z=A^n$ | $\dfrac{\Delta Z}{Z}=n\dfrac{\Delta A}{A}$ |
- Series resistors: $\Delta R=\Delta R_1+\Delta R_2$. Parallel: $\dfrac{\Delta R'}{R'^2}=\dfrac{\Delta R_1}{R_1^2}+\dfrac{\Delta R_2}{R_2^2}$.
- Error in a measured quantity with a formula like $I=I_0(e^{V/V_T}-1)$: differentiate, $\Delta I\approx\dfrac{dI}{dV}\Delta V$.
- Systematic (instrumental, zero error, method, personal) — one direction, can be corrected. Random — both directions; reduced by averaging ($\propto1/n$). Gross — carelessness.
- Accuracy: closeness to true value; precision: closeness of readings to each other (resolution).
4Significant figures
- All non-zero digits count; zeros between them count; leading zeros (0.00308) do not; trailing zeros after a decimal point count (30.00 → 4); trailing zeros without a decimal point do not (2030 → 3). Powers of 10 do not count.
- × and ÷: keep the least number of significant figures. + and −: keep the least number of decimal places.
- Rounding: drop digit > 5 → raise; < 5 → keep; exactly 5 → raise only if the previous digit is odd (4.7253 → 4.72, 4.7153 → 4.72).
5Measuring instruments
$n$ VSD = $(n-1)$ MSD. Zero error: subtract positive, add negative zero error (correction = −error).
Pitch = distance moved per rotation. Backlash error from loose screw threads.
Searle's method; error taken as the least count of each instrument. Pendulum $g=\dfrac{4\pi^2L}{T^2}$: $\dfrac{\Delta g}{g}=\dfrac{\Delta L}L+2\dfrac{\Delta T}T$.
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