Two stretched strings of same material are vibrating under the force vibration having sae tension in fundamental mode. The ratio of their frequencies is 1:4 and ratio of the length of the vibrating segments is 1:8 the, the ratio of the radii of the strings is

Options

A. `2:1`
B. `20:1`
C. `12:5`
D. `32:1`

Answer

D

Explanation

The frequency of fundamental note is given by
`v=(1)/(2lr)sqrt((T)/(pid))`
the fundamental frequency of first string
`v=(1)/(2l_(1)r_(1))sqrt((T)/(pid))` . . . (i)
Fundamental frequency of second string,
`v_(2)=(1)/(2l_(2)r_(2))sqrt((T)/(pid))` . . . (ii)
dividing eqs. (i) by (ii),
`(v_(1))/(v_(2))=(1)/(l_(1)r_(1))xx(l_(2)r_(2))/(r_(1)))`
`=(1)/(4)=((8)/(1))xx((r_(2))/((r_(1)))[(becausel_(1):l_(2)=1:4),(v_(1):v_(2)=1:2)]`
`(r_(2))/(r_(1))=(1)/(32)` or `r_(1):r_(2)=32:1`