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

The Correct Option is B

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 \(T = m' g\)
\(\Rightarrow v = k\sqrt{\frac{TL}{M}} = k \sqrt{\frac{m'gL}{M}}\)
\(\Rightarrow \frac{\Delta v}{v} = \frac{1}{2} \frac{\Delta m'}{m'} + \frac{1}{2} \frac{\Delta L}{L} + \frac{1}{2} \frac{\Delta M}{M}\)
\(= \frac{1}{2}\times \frac{0.1}{5} + \frac{1}{2} \times \frac{1\times 10^{-3}}{1} + \frac{1}{2} \times \frac{0.1}{3}\)
\(= 0.01 + 0.0005 + 0.016\)
\(= 0.0271 = 2.7 \%\)