Question

A one-metre long stretched string having a mass of 40 g is attached to a tuning fork. The fork vibrates at 128 Hz in a direction perpendicular to the string. What should be the tension in the string if it is to vibrate in four loops?

Answer 1

Given:
Length of the stretched string (L) = 1.00 m
Mass of the string =40 g 
String is attached to the tuning fork that vibrates at the frequency (f) = 128 Hz
Linear mass density (m)
\[= \left( 40 \times {10}^{- 3} \right)  kg/m\]
No. of loops formed, (n) = 4

\[L = \frac{n\lambda}{2}\] 

\[ \Rightarrow \lambda = \frac{2L}{n} = \frac{2 \times 1}{4}\] 

\[ \Rightarrow \lambda = 0 . 5  m\] 

Wave  speed  \[ (v) = f\lambda = 128 \times 0 . 5\] 

\[ \Rightarrow v = 64  m/s\]

\[We  know: \] 

\[  v = \sqrt{\left( \frac{T}{m} \right)}\] 

\[ \Rightarrow T =  \nu^2 m\] 

\[ =  \left( 64 \right)^2  \times 40 \times  {10}^{- 3} \] 

\[ = 163 . 84 \approx 164  N\]
Hence, the tension in the string if it is to vibrate in four loops is 164 N.