Question

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

#1.1 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 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.
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.

Please help

Answer 1

1) Here, we shall be using Archimedes Principle which states that the object floats if the water displacement is balanced by the weight of the object.

For first case m> m0. Since mass increases and hence the weight increases so the water displacement will be more resulting in increase in water level. So answer is r.

For Case 2: Density increases that means mass per unit volume increases. But radius is same so the volume will remain same. The only possibility to raise the density is by increasing the mass so it behaves like the upper case. The object will float only if the water level rises so answer is r.

For Case 3: As discussed, here the mass remains same but the radius must decrease to increase the resultant density. Here, same mass will displace the water in the lower volume leading to rise in water level. Hence, ans is r.

1.1) For CaseI radius is reducing and m is increasing both are leading to increase in density and greater density means more displacement in water and hence, water level will rise, Ans is r.

For Case II It is possible if only volume increases , that will either lead to fall or same water level. Ans is f s.

For CaseIII Here, m must increase as the volume decreases so the for greater density water level will rise. Ans is r.