Units & Measurements — Compound Errors & Deduction (Hard)

Quick Recap — Compound Errors & Deduction (Multi-step)

• Master rule: for $Q=\dfrac{a^p b^q}{c^r}$, $\dfrac{\Delta Q}{Q}=p\dfrac{\Delta a}{a}+q\dfrac{\Delta b}{b}+r\dfrac{\Delta c}{c}$ (all terms added in magnitude to get the maximum error).
• Classic experiments: Young's modulus $Y=\dfrac{MgL}{\pi r^2 \ell}\Rightarrow\%Y=\%M+\%L+2\%r+\%\ell$. Pendulum $g=\dfrac{4\pi^2 L}{T^2}\Rightarrow\%g=\%L+2\%T$. Resistivity $\rho=\dfrac{R\pi r^2}{L}\Rightarrow\%\rho=\%R+2\%r+\%L$.
• Dimensional deduction: write the unknown as a product of powers and match dimensions to fix the exponents (numerical constants are not obtained this way).
• Significant figures propagate through a multi-step calculation; round only at the end.

Worked mini-example. For $Y=\dfrac{MgL}{\pi r^2\ell}$ with errors $M:1\%$, $L:1\%$, $r:2\%$, $\ell:1\%$ (and $g$ exact): $\%Y=1+1+2(2)+1=7\%$.