A thin ring of radius `R` is made of a material of density `rho` and Young's modulus `Y`. If the ring is rotated about its centre in its own plane with angular velocity `omega` , find the small increase in its radius.

Explanation

Consider an element PQ of length dl. Let T be the tension and A the area of cross section of the wire.
`"Mass of element dm"="Volume"xx"density"`
`=A(dl)phi`
The component of T, towards the centre provides the necessary centripetal force
`therefore2Tsin(d theta)/(2)=(dm)Romega^(2)" ...(i)"`

For small angle, `sin(d theta)/(2)~~(d theta)/(2)~~(((dl)/(R)))/(2)`
Substitution in equation (i), we have
`2T(d theta)/(2)=(dm)R omega^(2)`
`T.(dl)/(R)=A(dl)phi R omega^(2)`
`T=Aphi omega^(2)R^(2)`
Tension in the ring is `A phi omega^(2)R^(2)`
`(DeltaR)/(R)=(T)(AY)=(phi omega^(2)R^(2))/(Y)rArr DeltaR=(phiomega^(2)R^(3))/(Y)`