A string of density `7.5 gcm^(-3)` and area of cross - section `0.2mm^(2)` is stretched under a tension of 20 N. When it is plucked at the mid-point, the speed of the transverse wave on the wire is

Options

A. `116ms^(-1)`
B. `40ms^(-1)`
C. `200ms^(-1)`
D. `80ms^(-1)`

Answer

A

Explanation

Here, `T=20N, A=0.2mm^(2)=0.2xx10^(-6)m^(2),rho=7.5g cm^(-3)=7.5xx10^(3)kgm^(-3)`
Mass per unit length of the wire,
`mu=Arho=0.2xx10^(-6)xx7.5xx10^(3)=1.5xx10^(-3)kgm^(-1)`
When a string is plucked at the midpoint, it vibrates with fundamental frequency.
Fundamental frequency of the stretched string is
`upsilon=(1)/(2L)sqrt((T)/(mu))=(1)/(2L)sqrt((20)/(1.5xx10^(-3)))`
Speed of transverse wave, `v=upsilon(2L)`
`therefore v=(1)/(2L)sqrt((20)/(1.5xx10^(-3)))xx2L~=116ms^(-1)`