Units & Measurements — Errors & Significant Figures (Medium)

Quick Recap — Errors & Significant Figures

• Combining errors (relative/percentage add): for a sum or difference $Z=A\pm B$, the absolute errors add ($\Delta Z=\Delta A+\Delta B$). For a product/quotient the relative errors add.
• Powers: if $Z=\dfrac{A^p B^q}{C^r}$ then $\dfrac{\Delta Z}{Z}=p\dfrac{\Delta A}{A}+q\dfrac{\Delta B}{B}+r\dfrac{\Delta C}{C}$.
• Examples: density $\rho=\dfrac{m}{V}\Rightarrow\%\rho=\%m+\%V$; from a sphere $\%\rho=\%m+3\%r$; kinetic energy $\%KE=\%m+2\%v$; $g=\dfrac{4\pi^2 L}{T^2}\Rightarrow\%g=\%L+2\%T$.
• Significant figures: in $\times$ and $\div$ keep the fewest significant figures; in $+$ and $-$ keep the fewest decimal places.
• Least count: vernier $=$ 1 MSD $-$ 1 VSD; screw gauge $=\dfrac{\text{pitch}}{\text{no. of divisions}}$.

Worked mini-example. For $g=\dfrac{4\pi^2 L}{T^2}$ with $L$ measured to $0.5\%$ and $T$ to $0.5\%$: $\%g=\%L+2\%T=0.5+2(0.5)=1.5\%$.