Question

Velocity of wave in a string is given by $v=\sqrt{\frac{T}{\mu }}$, where T is lension and $\mu$ is mass per unit length $\left(m/\ell \right)$. If $T=(30\pm 0.9)N,$ $m=(2.5\pm 0.01)N,$gm and $\ell =(1\pm 0.01)$, then calculate % error in v.

• A. 2.2 %
• B. 1.8 %
• C. 1.5 %
• D. 3.0 %

Answer 1

The correct option is A 2.2 %

$\begin{array}{c}\text{Given;}v=\sqrt{\frac{T}{\mu }}-\left(i\right)\\ T=(30\pm 0.9)N\\ m=(2.5\pm 0.01)kg=\\ l=(1\pm 0.01)m\\ {\text{Taking log in eq}}^n\text{(i) bothside}\\ \log v=\log \sqrt{\frac{I}{\mu }}\\ \log v=\frac{1}{2}\log T-\frac{1}{2}\log M+\frac{1}{2}\log l\end{array}$

$\begin{array}{c}\text{Differentiating both side:}\\ \frac{\delta v}{v}=\frac{1}{2}\frac{\delta I}{T}-\frac{1}{2}\frac{\delta m}{m}+\frac{1}{2}\frac{\delta l}{l}\\ \text{Error is always added}\\ \frac{\delta v}{v}=\frac{1}{2}\frac{\delta T}{T}+\frac{1}{2}\frac{\delta m}{m}+\frac{1}{2}\frac{sl}{l}\\ \text{\%error:}\end{array}$

$\begin{array}{cc}\frac{\delta v}{v}\times {100}^{}& =\frac{1}{2}\frac{\delta T}{T}\times 100\%+\frac{1}{2}\frac{\delta m}{m}\times 100\%+\frac{1}{2}\frac{\delta \lambda }{l}\times 100\%\\ & =\frac{1}{2}\times \frac{0.9}{30}\times 100+\frac{1}{2}\frac{0.01}{2.5}\times 100+\frac{1}{2}\times \frac{0.01}{1}\times 100\%.\\ & =\frac{3}{2}\%+\frac{1}{5}\%+\frac{1}{2}\%\\ & =2+0.2\\ & =2.2\%\text{Ans.}\end{array}$