A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45Hz. The mass of the wire is `3.5xx10^(-2)`kg and its linear mass density is `4.0xx10^(-2)kgm^(-1)`. What is (a) the speed of a transverse wave on the string , and (b) the tension in the string?

Explanation

(a) Mass of the wire, m `= 3.5 xx 10^(-2) kg`
Linear mass density,
`mu=(m)/(l)=4.0xx10^(-2) kg m^(-1)`
Frequency of vibration, v=45 Hz
`therefore ` Length of the wire , `l=(m)/(mu)=(3.5xx10^(-2))/(4.0xx10^(-2))=0.875 m`
The wavelength of the stationary wave (λ) is related to the length of the wire by the relation:
`lambda=(2l)/(n)`
Where, n=number of nodes in the wire
For fundamental , n=1
`lambda=2l`
`lambda=2xx0.875 =1.75` m
The speed of the transverse wave in the string is given as:
`v=vlambda=45xx1.75=78.75 m//s`
(b) The tension produced in the string is given by the relation
`T=v^(2)mu`
`=(78.75)^(2)xx4.0xx10^(-2)=248.06 N`