A cylindrical container with radius $R$ is filled with two

immiscible fluids with densities ${}_1$ and ${}_2,$ respectively, as

shown in the figure. The container has a flat bottom wall

and vertical side walls and spins with a constant angular

velocity $.$ Assume that the fluids have identical

volumes $V_1=V_2=V,$ and that surface tension effects can be

neglected. Find the magnitude of the critical angular velocity ${}_c,$ at

which the container bottom will be exposed. No liquid is

spilled from the tank.