Question

A physical quantity $P$ is related to four observables $a,b,c,d$ as follows:
$P=\frac{a^3{b}^2}{(\sqrt{c}\cdot d)}$
The percentage errors in the measurement of $a,\dot{b},c$ and $d$ are $1\%,3\%,4\%$ and $2\%$ respectively. The percentage error in the quantity $P$ is

• A. $7\%$
• B. $9\%$
• C. $11\%$
• D. $13\%$

Answer 1

Answer: (D)

$P=a^3{b}^2{c}^{-\frac{1}{2}}{d}^{-1}$
$\ln P=3\ln a+2\ln b-\frac{1}{2}\ln c-1\ln d$
$\left|\frac{\Delta P}{P}\right|=3\left|\left(\frac{\Delta a}{a}\right)\right|+2\left|\left(\frac{\Delta b}{b}\right)\right|+\frac{1}{2}\left|\frac{\Delta c}{c}\right|+\mid \frac{\Delta d}{d}$
$\Rightarrow$ Percentage error in $P$
$=3(1\%)+2(3\%)+\frac{1}{2}(4\%)+2\%=(3+6+2+2)\%$
$=13\%$