SI Base Quantities

$\text{Length: } [M^0 L^1 T^0] \quad \text{Mass: } [M^1 L^0 T^0] \quad \text{Time: } [M^0 L^0 T^1]$

Key Derived Quantities

$F = ma \quad [M^1 L^1 T^{-2}] \quad \text{(Newton)}$

$W = Fd \quad [M^1 L^2 T^{-2}] \quad \text{(Joule)}$

$P = \frac{W}{t} \quad [M^1 L^2 T^{-3}] \quad \text{(Watt)}$

$p_\text{pressure} = \frac{F}{A} \quad [M^1 L^{-1} T^{-2}] \quad \text{(Pascal)}$

$p_\text{momentum} = mv \quad [M^1 L^1 T^{-1}]$

$L_\text{angular} = mvr \quad [M^1 L^2 T^{-1}]$

$\tau = rF\sin\theta \quad [M^1 L^2 T^{-2}]$

$S = \frac{F}{l} \quad [M^1 L^0 T^{-2}] \quad \text{(Surface tension)}$

$\eta = \frac{F}{A(dv/dx)} \quad [M^1 L^{-1} T^{-1}] \quad \text{(Viscosity)}$

Unit Conversion (Dimensional Method)

$n_2 = n_1 \left(\frac{M_1}{M_2}\right)^a \left(\frac{L_1}{L_2}\right)^b \left(\frac{T_1}{T_2}\right)^c$

Error Propagation

$Z = A \pm B \implies \Delta Z = \Delta A + \Delta B \quad \text{[absolute]}$

$Z = A \times B \text{ or } \frac{A}{B} \implies \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \quad \text{[relative]}$

$Z = A^n \implies \frac{\Delta Z}{Z} = n \cdot \frac{\Delta A}{A}$

$Z = \frac{A^a \cdot B^b}{C^c} \implies \frac{\Delta Z}{Z} = a\frac{\Delta A}{A} + b\frac{\Delta B}{B} + c\frac{\Delta C}{C}$

$\% \text{ error} = \frac{\Delta Z}{Z} \times 100\%$

Pendulum (Dimensional Derivation)

$T = k\sqrt{\frac{L}{g}} \quad \text{where } k = 2\pi \text{ (cannot be found dimensionally)}$