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. 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 mass of m > m₀ and a density of ρ < ρ₀.
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 mass of m < m₀ and a radius of r > r₀.

Use Archimedes's Principle.

Answer 1

(1) Archimedes' principle: It 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 displace".

Buoyant force = Weight of the fluid displaced.

Thus,

if mass increases more fluid will displaced, therefore water level rise. and if mass decreases less fluid will displace and there will be fall in water level.

density and volume is defined as;

CASE 1: mass increases and density decreases.

to decrease density volume will also increase;

mass and volume both increases rise in water level.

CASE 2: density decreases and radius increases.

if radius increases volume will increased from equation (2). Now from equation (1) mass remains same. Thus

no change in mass but volume decreases same or rise in water level.

CASE 3: mass decreases and radius increases.

mass decreases but volume increases fall or rise in water level.