Question

A physical quantity $X$ is given by $X=\frac{2 k^{3} l^{2}}{m \sqrt{n}}$ The percentage error in the measurements of $k, l, m$ and $n$ are $1 \%, 2 \%, 3 \%$ and $4 \%$ respectively. The value of $X$ is uncertain by

• A. 8%
• B. 10%
• C. 12%
• D. None of these

Answer 1

Answer: (C)

Given : $X =\frac{2 k ^{3} 1^{2}}{ m \sqrt{ n }}$
$\therefore \frac{\Delta X }{ X }=3 \frac{\Delta k }{ k }+2 \frac{\Delta 1}{1}+\frac{\Delta m }{ m }+\frac{1}{2} \frac{\Delta n }{ n }$
Percentage error in $X$
$\frac{\Delta X }{ X } \times 100=\left(3 \frac{\Delta k }{ k }+2 \frac{\Delta 1}{1}+\frac{\Delta m }{ m }+\frac{1}{2} \frac{\Delta n }{ n }\right) \times 100$
$=3 \times 1 \%+2 \times 2 \%+3 \%+\frac{1}{2} \times 4 \%$
$=3 \%+4 \%+3 \%+2 \%=12 \%$
Hence, the value of $X$ is uncertain by $12 \%$.