A uniform material rod of length L is rotated in a horizontal plane about a vertical axis through one of its ends. The angular speed of rotation is w. Find increase in length of the rod. It is given that density and Young’s modulus of the rod are r and Y respectively.

Answer

`03.00`

Explanation

`Y=(FL)/(Al)`
Here, F =centrifugal force due to rotation.
`:. L=(FL)/(AY)`
From the figure, considering small elongation dx of length segment x `dl=(dF xx x)/(AY) ….(i)`
where, dF =centrifugal force of differential mass dM
but `dF =dM omega^2x=(rho Adx)omega^2x`
`=rhoAomega^2xdx`
Substuiting in equation (i),
`:. dl=((rhoAomega^2xdx)xxx)/(AY)`
Intergrating over limits,
`:. int_0^ldl=(rhoomega^2)/Yint_0^L=rhoomega^2)/(3Y)[L^3-0)`
`:. l=(rho omega^2L^3)/(3Y)`
`implies n=3`