A uniform ring having mass m, radius R, cross section area of the wire A and young’s modulus Y is rotating with an angular speed `omega`(`omega` is small) on a smooth horizontal surface. Which of the following options is correct :

Options

A. Tension in the wire is `(mRomega^(2))/(2pi)`
B. Change in length of the wire is `(mR^(2) omega^(2))/(2A.Y)`
C. Change in radius of the ring is `(mR^(2)omega^(2))/(2piA.Y)`
D. elastic potential energy stored is `1/(4pi)((m^(2)omega^(4) R^(3))/(A.Y))`

Explanation

Let T be the tension in the string

`2Tsin (d theta)/2=m/(2piR)xxR.omega^(2).R d theta`
`T=(mRomega^(2))/(2pi)`
`Y=(T//A)/(Deltal//l)`
`(Deltal)/l =T/(Y.A) rArr Deltal=T/(Y.a)xxl=(m.Romega^(2))/(2pi)xx1/(Y.A)xx2piR=(mR^(2)omega^(2))/(Y.A)`
`(Deltal)/l=(DeltaR)/R=T/(Y.A)=(m.Romega^(2))/(2piA.Y)rArr DeltaR=(mR^(2)omega^(2))/(2piA.Y)`
`V=1/2 K.X^(2)=1/2((Y.A.)/l)xx(Deltal)^(2)=1/2(Y.A)/(2piR)xx((m.R^(2)omega^(2))/(Y.A))^(2)=1/(4pi)((m^(2)R^(3)omega^(4))/(Y.A))`