Question

A small solid sphere of mass M₀, of radius R₀, and of uniform density ρ₀ is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that sphere is replaced by a new solid sphere of uniform density.
Options are:

R= Rises

F= Falls

U= Unchnaged

F or U

R or U

F or R or U
The new sphere has mass M > M₀ and density ρ = ρ₀
The new sphere has mass M = M₀ and radius R > R₀
The new sphere has radius R = R₀ and density ρ > ρ₀
The new sphere has radius R > R₀ and density ρ < ρ₀
The new sphere has density ρ = ρ₀ and radius R < R₀
The new sphere has radius R < R₀ and mass M = M₀

Answer 1

The new sphere has mass M > M0 and density ρ = ρ₀   

Rise : as the mass of new sphere is greater , it will displace more amount of water. hence the water level will rise.

The new sphere has mass M = M0 and radius R > R0

Unchanged : since the mass of new sphere is same , it will displace same amount of water.

The new sphere has radius R = R0 and density ρ > ρ0

Rise : since density of new sphere is greater for the same radius, it displace more water. so water level rise.

The new sphere has radius R > R0 and density ρ < ρ0

F or R or U

The new sphere has density ρ = ρ0 and radius R < R0

fall : this is because for the same density , the radius of new sphere is smaller, hence smaller volume and smaller mass . for smaller mass , it will displace lesser amount of water.

The new sphere has radius R < R0 and mass M = M0

Unchanged : Since the mass of spheres are same