Consider an expanding sphere of instantaneous radius ? whose total mass remains constant. The expansion is such that the instantaneous density `rho` remains uniform throughout the volume. The rate of fractional change in density `((dp)/(rhodt))` is constant. The velocity v of any point on the surface of the expanding sphere is proportional to

Options

A. R
B. `R^(3)`
C. `(1)/(R )`
D. `R^(2//3)`

Answer

a

Explanation

Total mass of the object = Volume `xx` Density M =`rhoV`
Differentiationg both sides.
`rho (dV)/(dt) +V (drho)/(dt) = 0`
`rho (dV)/(dt) = - V (drho)/(dt)`
`(1)/(v) (dV)/(dt) = - (1)/(rho) (drho)/(dt)`
As it is given the question that `(1)/(rho) (drho)/(dt)` is a constant let it be equal to K .
`(1)/(V) (dV)/(dt) = k `
`(dV)/(dt) = - KV " ".....(i)`
The instantaneous volume of the sphere is given by ` V = (4)/(3) piR^(3)`
`(dV)/(dt) = 4 pi R^(2) (dR)/(dt)`
From eqution (i)
`4piR^(2) (dR)/(dt) = - KV`
`4 piR^(2) (dR)/(dt) = - K ((4)/(3) piR^(3))`
`(dR)/(dt) = -(K)/(3) R " "......(ii)`
The velocity of any point on the sphere will be .
`v = (dR)/(dt)`
Thus , `v = (K)/(3)R`
Hence , the correct option is (a).