Question

A solid, homogeneous sphere with a mass of m0, a radius of r0 and a density of ρ0 is placed in a container of water. Initially the sphere floats and the water level is marked on the side of the container. What happens to the water level, when the original sphere is replaced with a new sphere which has different physical parameters? Notation: r means the water level rises in the container, f means falls, s means stays the same.

1. The new sphere has a density of ρ = ρ0 and a mass of m < m0.

2. The new sphere has a mass of m = m0 and a radius of r > r0.

3. The new sphere has a density of ρ = ρ0 and a radius of r > r0.

Answer 1

Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid.

so if new sphere has

Fb = _L *Volume submerged *g

and gravitational force = m*g = m0*g

the first sphere is floatings so  

mass = density*4/3*pi*r^3

Fb >  m0*g = ρ0 **4/3*pi*r^3

first case

ρ = ρ0 and m <m0.

so m*g <m0*g  

so water level will falls​

case 2

m = m0 but  r > r0

so gravitatinal force m0 *g

water level  stays the same.

case 3

ρ = ρ0 ​

r0<r

r0/r <1

m0/ 4/3*pi*r0^3 =m/ 4/3*pi*r^3​

m0/m = (ro/r)^3 <1

m0<m

water level rises