To solve this problem, we need to consider the forces acting on the small ball as it falls through the liquid. The two main forces are the gravitational force and the buoyant force.


The gravitational force acting on the small ball can be calculated using the formula Fg = mg, where m is the mass of the ball and g is the acceleration due to gravity.


The buoyant force acting on the small ball can be calculated using the formula Fb = ρfluid * V * g, where ρfluid is the density of the liquid at a particular depth, V is the volume of the ball, and g is the acceleration due to gravity.


The volume of the ball can be calculated using the formula V = (4/3) * π * r^3, where r is the radius of the ball.


For the small ball to be in equilibrium, the gravitational force and the buoyant force must be equal. Therefore, we can set up the following equation:

mg = ρfluid * V * g


We can substitute the expression for V into the equation:

mg = ρfluid * (4/3) * π * r^3 * g

Next, we substitute the expression for ρfluid into the equation:

mg = ρ0 * 2 * (α * β * h) * (4/3) * π * r^3 * g

The mass of the ball, m, can be expressed in terms of its density and volume:

m = ρ0 * V

Substituting this into the equation, we get:

ρ0 * V * g = ρ0 * 2 * (α * β * h) * (4/3) * π * r^3 * g

Simplifying the equation, we find:

V = 8 * (α * β * h) * r^3 / 3


In conclusion, the volume of the ball can be determined by using the equation V = 8 * (α * β * h) * r^3 / 3, where α and β are constants and h is the depth of the liquid. This equation allows us to calculate the volume of the ball at any given depth. By knowing the volume, we can then calculate the gravitational force and the buoyant force acting on the ball. To achieve equilibrium, these two forces must be equal.