1. Tardigrade
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  4. A physical quantity P is related to four observables a, b, c, d as follows: P=(a3 b2/(√c ⋅ d)) The percentage errors in the measurement of a, dotb, c and d are 1 %, 3 %, 4 % and 2 % respectively. The percentage error in the quantity P is

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Q. 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

Physical World, Units and Measurements

Solution:

$ 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 $P$
$=3(1 \%)+2(3 \%)+\frac{1}{2}(4 \%)+2 \%=(3+6+2+2) \%$
$=13 \%$