A sphere of radius R has a concentric spherical cavity of radius r. The relative density of the material of the sphere is `sigma`. It just floats when placed in tank full of water. The value of `R/r` is

Options

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

Answer

A

Explanation

Buoyancy force due to water, on a sphere of radius R,
`" "F_(w)=rho_(w)gV_(s)`
where, `" "rho_(w)=` density of water
`" "V_(s)=`volume of sphere
`therefore " "F_(w)=rho_(w)g(4/3piR^(3)) " "(because V_(s)=4/3piR^(3))`
weight of the sphere,
`" "F_(s)=rho_(s)g(4/3pi(R^(3)-r^(3)))`
where `rho_(s)=` density of material of sphere
According to Archimedes. principle for floating,
`therefore " "F_(w)=F_(s)`
`rArr rho_(w)g(4/3piR^(3))=rho_(s)g(4/3pi(R^(3)-r^(3)))`
`rArr rho_(w)R^(3)=rho_(s)(R^(3)-r^(3))`
`rArr R^(3)(rho_(s)-rho_(w))=r^(3)rho_(s)`
`rArr R/r=(rho_(s)/(rho_(s)-rho_(w)))^(1//3) " or "R/r=(1/(1-(rho_(w))/rho_(s)))^(1//3)`
`because` Relative density of material of the sphere, `sigma=rho_(s)/rho_(w)`
or `" "R/r=(1/(1-1/sigma))^(1//3)=(sigma/(sigma-1))^(1//3)`