Question

The frequency $(f)$ of a wire oscillating with a length $\ell$, in $p$ loops, under a tension $T$ is given by $f=\frac{p}{2\ell }\sqrt{\frac{T}{\mu }}$ where $\mu =$ linear density of the wire. If the error made in determining length, tension and linear density be $1\%,-2\%$ and $4\%$, then find the percentage error in the calculated frequency.

• A. $-4\%$
• B. $-2\%$
• C. $-1\%$
• D. $-5\%$

Answer 1

Answer: (A)

Given $f=\frac{p}{2\ell }\sqrt{\frac{T}{\mu }}$. Taking log of both sides
$\log f=\log \left(\frac{p}{2}\right)-\log \ell +\frac{1}{2}\log T-\frac{1}{2}\log \mu$
Differentiating partially on both sides,
$\frac{df}{f}=0-\frac{d\ell }{\ell }+\frac{1}{2}\frac{dT}{T}-\frac{1}{2}\frac{d\mu }{\mu }$
or $\frac{df}{f}\times 100=\left(-\frac{d\ell }{\ell }\times 100\right)+\left(\frac{1}{2}\frac{dT}{T}\times 100\right)-\left(\frac{1}{2}\frac{d\mu }{\mu }\times 100\right)$
$=(-1)+\frac{1}{2}(-2)-\frac{1}{2}(4)$
$=-1-1-2=-4\%$