Question

A body of density `rho` floats with volume `V_(1)` of its total volume `V` immersed in a liquid of density `rho_(1)` and with the remainder of volume `V_(2)` immersed in another liquid of density `rho_(2)` where `rho_(1)gtrho_(2)`. Find the volume immersed in two liquids. (`V_(1)` and `V_(2)`).

Answer 1

The mass of the body = `V rho`
When the body floats , the weight of the body mst be equal to the weight of the displaced volume of the two liquids , i.e.
`V rho g = V_(1) rho_(1) g + V_(2) rho_(2) g`
or ` Vrho_(1) + V_(2) rho_(2)` …(i)
But, `V = V_(1) + V_(2)` ..(ii)
Multiplying (ii) by `rho_(2)` , we get
`V rho_(2) = V_(1) rho_(2) + V_(2) rho_(2)` ...(iii)
from (i) and (iii) , we get
`V (rho-rho_(2)) = V_(1) (rho_(1)-rho_(2))`
or `V_(1) = ((rho-rho_(2))/(rho_(1)-rho_(2)))V`
Multipying (ii) by `rho_(1)` , we get
`Vrho_(1) = V_(1)rho_(1) + V_(2)rho_(1)` ..(iv)
From (i) and (iv) , we get
`V(rho-rho_(1)) = V_(2) (rho_(2)-rho_(1))`
or , ` V_(2) =((rho-rho_(1))/(rho_(2)-rho_(1)))V = ((rho_(1)-rho)/(rho_(1)-rho_(2)))V`.