A uniform ring of radius `R` and made up of a wire of cross - sectional radius `r` is rotated about its axis with a frequency `f`. If density of the wire is `rho ` and young's modulus is `Y` . Find the fractional change in radius of the ring .

Explanation

`2T sin d theta=(dm)R omega^(2)`
For small angles, `sin d theta ~~ d theta`
`2T sin d theta = (dm) R omega^(2)`
For small angles , sin `d theta ~~ d theta`
`:. 2 T(d theta ) = (2 R d theta )( pi r^(2)) (rho) (R) (2 pi f) `
` T = (4 pi ^(3) f ^(2) R^(2) rho ) `
Now `Delta l = (Tl)/( AY) `
` (Delta l )/l = (T)/(AY) = (T)/(pi r^(2) Y)`
` l = 2 pi R`
` Delta l = 2 pi(Delta R)`
`:. (Delta l )/ l = (Delta R)/R = (T)/(pir^(2) Y )`
` = (4 pi ^(3) f^(2) R(2) r^(2) rho)/(pir^(2)Y)`