Question

A solid, homogeneous sphere with a mass of m₀, a radius of r₀ and a density of ?₀ 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 radius of r < r₀ and a density of ? > ?₀.
r f s r or s f or s r or f or s  The new sphere has a density of ? > ?₀ 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 radius of r < r₀ and a mass of m > m₀.

Answer 1

All that you should keep in mind is that the amount of water displaced equals the weight of the sphere (as long as it floats)

case 1: clearly, the volume is increasing but the mass remains same because of the increased density. Clearly then, more water needs to be displaced to keep it afloat. If it doesn't float but sinks, also more water will be displaced because the radius (volume) of the sphere is the increases, so going under displaces more water than staying afloat-> r

case 3: more mass, so more weight, more amount of water displaced -> r
case 2: less mass, so less weight, less water displaced-> f