Question

A wire of length $l=6\pm 0.06$cm and radius $r=0.5\pm 0.005$cm and mass $m=0.3\pm 0.003$ gm. Maximum percentage error in density is?

• A. $4$
• B. $2$
• C. $1$
• D. $6.8$

Answer 1

The correct option is A $4$

$\begin{array}{c}\text{Given,}\\ l=6cm\\ dl=0.06cm\\ r=0.5cm\\ dr=0.005cm\\ m=0.3gm\\ dm=0.003gm\\ \text{density}\left(\rho \right)=\frac{\text{Mass}cm}{\text{Volume}}\text{(v)}\end{array}$

$\rho =\frac{m}{\pi r^{2}l}$

$\text{Taking log}$

$\log \rho =\log m-\log \pi -2\log \gamma$

$\text{Differentiating:}$

$\frac{d\rho }{\rho }=\frac{\delta m}{m}-\frac{2\delta r}{r}-\frac{\delta l}{l}$

$\begin{array}{cc}\text{Error always added,}& \\ \frac{\delta \rho }{\rho }=& \frac{0.003}{0.3}+\frac{2\times 0.005}{0.5}+\frac{0.06}{6}\\ =& 0.04\end{array}$

$\begin{array}{cc}\text{\%error in}\rho & =\frac{\delta \rho \times 100}{\rho }\\ & =4\%\end{array}$