Density/Buoyancy question

Question

A rubber ball with average density 900 kg/m^3 is floating in a container of water. Oil is slowly poured in the water until it completely covers the ball. When completely immersed in the oil, what fraction of the volume of the ball is submerged in the water?

Answer 1

To answer this question, you need to know the density of the oil. Since the oil covers the ball, we know it's density is less than that of the ball, but not how much.

If an object floats in a liquid, the fraction of it's volume submerged is related to the ratio of the densities of the material and liquid. V_sub = V_obj (ρ_obj / ρ_liquid)

The buoyant force the ball experiences due to the water is F_b = ρ_w g V_sub and this force is equal to the weight, w, of the ball (because it floats) or w = ρ_w g (ρ_b / ρ_w) V_b = ρ_b g V_b

It's easy to recognize the presence of the buoyant force when an object floats, but objects that sink also experience a buoyant force. Consequently, when the ball is covered by the oil, it experiences two buoyant forces - one due to the water, the other due to the oil. Part of the volume of the ball is in the water, call it V_w and part of the volume is in the oil, call it V_o and the sum of V_w + V_o = V_b , where V_b is the volume of the ball.

Write the force equation for both liquids; the buoyant force is still equal to the weight of the object.

F_b = w = F_w + F_o = ρ_w g V_w + ρ_o g V_o = ρ_b g V_b

Since you are interested in the volume in the water, solve for V_o = V_b - V_w and substitute. Since g appears in each term on both sides of the equality, it can be cancelled. Thus the equation reduces to:

ρ_w V_w + ρ_o (V_b - V_w) = ρ_b V_b

group like terms and solve for V_w:

(ρ_w - ρ_o) V_w = (ρ_b - ρ_o ) V_b --> V_w = V_b [ (ρ_b - ρ_o ) / (ρ_w - ρ_o)]

without knowing the density of the oil, it's not possible to reduce this farther.

Answer 2

To answer this question, you need to know the density of the oil. Since the oil covers the ball, we know it's density is less than that of the ball, but not how much.

If an object floats in a liquid, the fraction of it's volume submerged is related to the ratio of the densities of the material and liquid. V_sub = V_obj (ρ_obj / ρ_liquid)

The buoyant force the ball experiences due to the water is F_b = ρ_w g V_sub and this force is equal to the weight, w, of the ball (because it floats) or w = ρ_w g (ρ_b / ρ_w) V_b = ρ_b g V_b

It's easy to recognize the presence of the buoyant force when an object floats, but objects that sink also experience a buoyant force. Consequently, when the ball is covered by the oil, it experiences two buoyant forces - one due to the water, the other due to the oil. Part of the volume of the ball is in the water, call it V_w and part of the volume is in the oil, call it V_o and the sum of V_w + V_o = V_b , where V_b is the volume of the ball.

Write the force equation for both liquids; the buoyant force is still equal to the weight of the object.

F_b = w = F_w + F_o = ρ_w g V_w + ρ_o g V_o = ρ_b g V_b

Since you are interested in the volume in the water, solve for V_o = V_b - V_w and substitute. Since g appears in each term on both sides of the equality, it can be cancelled. Thus the equation reduces to:

ρ_w V_w + ρ_o (V_b - V_w) = ρ_b V_b

group like terms and solve for V_w:

(ρ_w - ρ_o) V_w = (ρ_b - ρ_o ) V_b --> V_w = V_b [ (ρ_b - ρ_o ) / (ρ_w - ρ_o)]

without knowing the density of the oil, it's not possible to reduce this farther.