Question

A 121-cm-long, 4.00 g string oscillates in its m = 3 mode with a frequency of 180 Hz and a maximum amplitude of 5.00 mm. What are (a) the wavelength and (b) the tension in the string?

Answer 1

Answer:

0.8067 m

69.696 N

Explanation:

m = Mode = 3

M = Mass of string = 4 g

f = Frequency = 180 Hz

l = Length of string = 121 cm

Length of the string is given by

$l=m\dfrac{\lambda}{2}\\\Rightarrow \lambda=2\dfrac{l}{m}\\\Rightarrow \lambda=2\times \dfrac{1.21}{3}\\\Rightarrow \lambda=0.8067\ m$

The wavelength is 0.8067 m

Linear density is given by

$\mu=\dfrac{M}{l}\\\Rightarrow \mu=\dfrac{4\times 10^{-3}}{1.21}$

Speed of the wave

$v=f\lambda\\\Rightarrow v=180\times 2\times \dfrac{1.21}{3}\\\Rightarrow v=145.2\ m/s$

Speed of wave is given by

$v=\sqrt{\dfrac{T}{\mu}}\\\Rightarrow T=\mu v^2\\\Rightarrow T=\dfrac{4\times 10^{-3}}{1.21}\times 145.2^2\\\Rightarrow T=69.696\ N$

Tension is given by 69.696 N