Two stretched wires made of the same material have lengths, diameters and tensions, each in the ratio 1 : 2 . The first wire emits a fundamental tone of frequency 200 Hz . What is the funcamental frequency of the 2nd wire ?

Explanation

If d is the diameter of a wire , the mass per unit
length is ` m = (pi d^(2))/(4) `
where ` rho = ` density of the material of the wire. It is the same for the two wires,
Now, the fundamental frequency,
`n = (1) /(2l) sqrt((T)/(m))= (1)/(2l) sqrt((T * 4)/(pi d^(2) rho))= (1) /(ld)sqrt((T)/(pi rho))`
So, for the two wires,
`(n_(1))/(n_(2)) = (l_(2))/(l_(1)) * (d_(2))/(d_(1))* sqrt((T_(1))/(T_(2)))`
or , `n_(2) = n_(1) * (l_(1))/(l_(2)) * (d_(1))/(d_(2)) * sqrt((T_(2))/(T_(2)))= 200 xx (1)/(2) xx (1)/(2) xx sqrt((2)/(1))`
[Here, `n_(1) = 200 Hz` ]
` :. n_(2) = 50 sqrt(2) = 70 . 7 ` Hz .