Units & Measurements — Formula & Dimensions Sheet

UNITS & MEASUREMENTS — FORMULA & DIMENSIONS SHEET SI BASE UNITS length m · mass kg · time s · current A · temperature K · amount mol · luminous intensity cd COMMON DIMENSIONAL FORMULAE • Velocity $[L\,T^{-1}]$ · Acceleration $[L\,T^{-2}]$ • Force $[M\,L\,T^{-2}]$ · Momentum & Impulse $[M\,L\,T^{-1}]$ • Work, Energy, Torque $[M\,L^{2}\,T^{-2}]$ · Power $[M\,L^{2}\,T^{-3}]$ • Pressure, Stress $[M\,L^{-1}\,T^{-2}]$ · Density $[M\,L^{-3}]$ • Frequency $[T^{-1}]$ · Charge $[A\,T]$ • Gravitational constant $G\ [M^{-1}\,L^{3}\,T^{-2}]$ · Planck's constant $h\ [M\,L^{2}\,T^{-1}]$ • Surface tension $[M\,T^{-2}]$ · Coefficient of viscosity $[M\,L^{-1}\,T^{-1}]$ DIMENSIONLESS strain, angle, solid angle, refractive index, relative density, Poisson's ratio, trigonometric ratios. ERROR PROPAGATION $$Z = A \pm B \;\Rightarrow\; \Delta Z = \Delta A + \Delta B$$ $$Z = AB\ \text{or}\ \tfrac{A}{B} \;\Rightarrow\; \dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$$ $$Z = \dfrac{A^{p} B^{q}}{C^{r}} \;\Rightarrow\; \dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B} + r\dfrac{\Delta C}{C}$$ Percentage error $=\dfrac{\Delta Z}{Z}\times 100\%$. SIGNIFICANT FIGURES Add/Subtract → least DECIMAL PLACES. Multiply/Divide → least SIGNIFICANT FIGURES. LEAST COUNT $\text{Vernier: LC}=1\,\text{MSD}-1\,\text{VSD}$ · $\text{Screw gauge: LC}=\dfrac{\text{pitch}}{\text{divisions}}$ Reading $=$ main-scale reading $+$ (coinciding division $\times$ LC) $-$ zero error. HANDY CONSTANTS / CONVERSIONS $1\ \text{Å}=10^{-10}\ \text{m}$ · $1\ \text{ly}\approx 9.46\times10^{15}\ \text{m}$ · $1\ \text{AU}\approx 1.496\times10^{11}\ \text{m}$ $1\ \text{atm}\approx 1.013\times10^{5}\ \text{Pa}$ · $1\ \text{eV}=1.6\times10^{-19}\ \text{J}$