Question

A physical quantity $S$ is given by $S=\frac{a^2{b}^3}{c\sqrt{d}}$.
If errors of measurements in $a,b,c,d$ are $4\%,2\%,3\%,1\%$ respectively, find the percentage error in the value of $S$.

• A. $7.5\%$.
• B. $17.5\%$.
• C. $27.5\%$.
• D. $10.5\%$.

Answer 1

The correct option is B $17.5\%$.

$\text{Given- a plysical quantity}s=\frac{a^2\cdot b^3}{c\cdot \sqrt{d}}$

$\begin{array}{c}\text{using error analysis method -}\\ \text{we have}s=a^2\cdot b^3\cdot c^{-1}\cdot d^{-1/2}\end{array}$

$\begin{array}{c}\text{Taking logarithm both sidus}\\ \log s=2\cdot \log a+3\cdot \log b-\log c-\frac{1}{2}\cdot \log d\end{array}$

$\begin{array}{c}\text{differentiate both Sides-}\\ \frac{ds}{s}=2\frac{da}{a}+3\frac{db}{b}-\frac{dc}{c}-\frac{1}{2}\frac{d\left(d\right)}{d}\end{array}$

$\text{we have}\frac{da}{a}=4\%\frac{db}{b}=\frac{210}{2\%}\frac{dc}{c}=3\%\frac{d\left(d\right)}{d}=1\%$

$\begin{array}{c}=17.5\%\\ \text{Persentage error in}S.\end{array}$