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 $3.0 \,g$ , length of string is $1 \,m$ and mass of string is $5 \,g$ . If possible error in measuring mass is $0.1 \,g$ and that of length is $1 \,mm$ , the percentage error in measurement of velocity is

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

Answer 1

Answer: (B)

$2.7\%$

Answer 2

The correct option is(B): 2.7%.

According to the question,
$v\propto\sqrt{\frac{T}{\mu}} = k \sqrt{\frac{T}{\mu}}$
As $\mu = \frac{M}{L}$ and $T = m' g$
$\Rightarrow v = k\sqrt{\frac{TL}{M}} = k \sqrt{\frac{m'gL}{M}}$
$\Rightarrow \frac{\Delta v}{v} = \frac{1}{2} \frac{\Delta m'}{m'} + \frac{1}{2} \frac{\Delta L}{L} + \frac{1}{2} \frac{\Delta M}{M}$
$= \frac{1}{2}\times \frac{0.1}{5} + \frac{1}{2} \times \frac{1\times 10^{-3}}{1} + \frac{1}{2} \times \frac{0.1}{3}$
$= 0.01 + 0.0005 + 0.016$
$= 0.0271 = 2.7 \%$