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?

#### Solution

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.