A solid sphere of radius `R` has a concentric cavity of radius `(R)/(3)` inside it. The sphere is found to just float in water with the highest point of it touching the water surface. Find the specific gravity of the material of the sphere.

Explanation

`(V_("cavity"))/(V_(S))=(V_(S)-V_("meta"))/(V_(S))=1-(V_("meta"))/(V_(S))-(1)`
`(V_(S)="Total volume of the sphere")`
According to Archimedes. principle Weight of body = Weight of displaced liquid
`mg=V_(S) d_(w)g rArr V_("metal")g=V_(S)d_(w)g`
`(d_(S)="Density of Solid matreial, "d_(w)="Density of Water")`
`(V_("metal"))/(V_(S))=(d_(w))/(d_(s))-(2)," From equation 1 and 2"`
`therefore (V_("Cavity"))/(V_(S))=1-(d_(w))/(d)=1-(1)/((d)/(d_(w)))=1-(1)/(S.G)`
`therefore ((4)/(3)pi((R)/(3))^(3))/((4)/(3)piR^(3))=1-(1)/(S.G) rArr (1)/(27)=1-(1)/(S.G)`
`rArr (1)/(S.G)=1-(1)/(27)rArr S.G=(27)/(26)`