A solid sphere is rotating about its diameter. Due to increase in room temperature, its volume increases by 0.5 %. If no external torgue acts, the angular speed of the sphere will :

Options

A. Increase by nearly `(1)/(2)%`
B. Increase by nearly `(1)/(3)%`
C. Decrease by nearly `(2)/(3)%`
D. Decrease by nearly `(1)/(3)%`

Answer

D

Explanation

`V=(4)/(3)pir^(3)`
`therefore (DeltaV)/(V)=(3Deltar)/(r)`
`(Deltar)/(r)=(1)/(3)(DeltaV)/(V)=(1)/(3)xx0.5=(1)/(6)%`
Since there is no external torque implies `Iomega` = constant
`(2)/(5)mr^(2)omega` = constant implies `r^(2)omega` = const.
Taking log on both sides,
`2logr+logomega=logc`
Differentiating, we have
`2.(1)/(r)dr+(1)/(omega)domega=0`
`(domega)/(omega)=-2(dr)/(r)=-2xx(1)/(6)=-(1)/(3)%`
`-ve` sign show that `omega` decreases