A solid hemisphere of radius `R` is made to just sink in a liquid of density `rho`. Find a. the vertical thrust on the curved surface b. the vertical thrust on the flat surface c. the side thrust on the hemisphere. d. the total hydrostatic force acting on the hemisphere.

Explanation

a. Vertical thrust of the liquid is equal to weight of the liquid column above the curve (spherical) surface i.e.
`F_V=Vrhog`

where `V=` volume of the shaded portion
`=` Volume of the cylinder `-` volume of the hemisphere
`=(piR^(2))R=2/3piR^(3)=(piR^(3))/3`
Substituting `V` in the equation `F_(v)=Vrhog`, we have
`F_(v)=(piR^(3))/3rhog` (down)
b. The upward thrust on the base of the hemisphere is
`F_(V)^(')=V^(')rhog`

where `V'=` Volume of the liquid column above the base
`=(piR^(2))R=piR^(3)`. Then we have `F_(v)^(')=piR^(3)rhog` ( upward)
c. Side Thrust `F_(h)=0` (as discussed earlier)
d. The total hydrostatic force is
`F=F_(v)^(')-F_(v)`
`=piR^(3)rhog-(piR^(3))/3rhog=1/3piR^(2)rhog` (upward)