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.

r f s The new sphere has a mass of m = m₀ and a radius of r > r₀.
r f s  The new sphere has a density of ρ = ρ₀ and a mass of m > m₀.
r f s  The new sphere has a radius of r < r₀ and a density of ρ = ρ₀.

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. Combination answers like 'f or s' are possible answers in some of the cases.

r f s r or s f or s The new sphere has a radius of r < r₀ and a mass of m = m₀.
r f s r or s f or s  The new sphere has a mass of m < m₀ and a radius of r = r₀.
r f s r or s f or s  The new sphere has a density of ρ < ρ₀ and a radius of r = r₀.

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. Combination answers like 'r or f or s' are possible answers in some of the cases.

r f s r or s f or s r or f or s The new sphere has a density of ρ > ρ₀ and a radius of r < r₀.
r f s r or s f or s r or f or s  The new sphere has a radius of r > r₀ and a mass of m < m₀.
r f s r or s f or s r or f or s  The new sphere has a density of ρ > ρ₀ and a mass of m < m₀.

Answer 1

Solution :

The Archimedes principle states that, "A body partially/wholly immersed in a fluid/any liquid experiences a force (bouyant) equal to the weight of the fluid displaced by the body".

(1) When the new sphere has a density of ρ > ρ₀ and a radius of r < r₀ : the water level RISES in the container.
Density of any body = Mass of the body/ Volume ( in kg/m³).
Now, as the density of the new sphere increases, it implies that its corresponding mass is also increasing significantly in the same volume. And now, according to Archimedes principle, the volume of water displaced is thus more with increased mass. This causes water level to rise.

(2) The new sphere has a radius of r > r₀ and a mass of m < m₀ : the water level FALLS in the container.
As the mass of the new sphere is lesser than the mass of the initial sphere, the corresponding volume of water displaced due to this mass is therefore, less too. This causes the fall in the water level.

(3) The new sphere has a density of ρ > ρ₀ and a mass of m < m₀ : The water level RISES in the container.
Here, the density and mass of the new sphere is greater than that of the initial sphere. This, therefore, is similar to part (a) resulting in greater displacement of water by the new sphere. Hence, the water level rises.