Question

The velocity of a transverse wave in a string is directly proportional to $\sqrt{T}$ and inversely proportional to $\sqrt{\mu }$ . In a measurement, the mass applied at the end of string is $3.0g$ , length of string is $1m$ and mass of string is $5g$ . If possible error in measuring mass is $0.1g$ and that of length is $1mm$ , the percentage error in measurement of velocity is

• A. $4.5\%$
• B. $2.7\%$
• C. $2.1\%$
• D. $3.7\%$

Answer 1

Answer: (B)

According to the question,
$v\propto \sqrt{\frac{T}{\mu }}=k\sqrt{\frac{T}{\mu }}$
As $\mu =\frac{M}{L}$ and $T=m^{\prime }g$
$\Rightarrow v=k\sqrt{\frac{TL}{M}}=k\sqrt{\frac{m^{\prime }gL}{M}}$
$\Rightarrow \frac{\Delta v}{v}=\frac{1}{2}\frac{\Delta m^{\prime }}{m^{\prime }}+\frac{1}{2}\frac{\Delta L}{L}+\frac{1}{2}\frac{\Delta M}{M}$
$=\frac{1}{2}\times \frac{0.1}{5}+\frac{1}{2}\times \frac{1\times 10^{-3}}{1}+\frac{1}{2}\times \frac{0.1}{3}$
$=0.01+0.0005+0.016$
$=0.0271=2.7\%$