Since half of the log is floating above the water, the buoyant force must equal the gravitational force on the log for the net force to equal .

When the ball is stably submerged (no longer accelerating), the buoyant force will equal the gravitational force.

Let the mass of the ball be .

The buoyancy force is equal to the weight of the material the ball displaces, which is .

 is the volume of the material the ball displaces.

We need the mass of the ball in terms of its density, which is .

From equation ,

Solving for  we get

.

This is the entire volume of the ball, so the whole ball is submerged. 

Here is a more complicated route to the solution that would be necessary if the density of the water was greater than the ball:

Using the geometry of a spherical cap, you'll find the volume as a function of the depth  is . Setting this equal to volume displaced we get

In order for these to equal

This can also be done conceptually by realizing in order for the buoyant force to equal the gravitation force, the whole ball must be submerged since the density of the ball and water are equivalent. 

Note*: Any answer greater than  would also be acceptable since the ball would still be completely submerged, and the buoyancy force would still equal the gravitation force. However, the only answer choice greater than or equal to  is , so that is the answer.

In this question, we're told that three balls of differing densities are submerged in water. We're then asked to determine the relative rates at which the balls will rise to the surface.

To begin, it's important to realize that whenever an object is submerged under water, there are two main forces acting upon it. One of these is the gravitational force, which points downwards. The other main force is the buoyant force, which acts upwards on the object. Because these two forces act in opposite directions, they can either cancel each other out partially or even completely.

Ball A, which has a density greater than water's, will not float to the surface but will rather sink to the bottom. This is because the gravitational force of ball A will be greater than the buoyant force from the displaced water. Consequently, there will be a net force downward.

For balls B and C, both of them have a density less than water. Hence, each of them will experience a net upward force and will thus rise to the surface. However, the magnitude of the upward force that each experiences will be different. This net upward force will, in turn, influence the degree to which each ball accelerates and will thus affect the rate at which they rise to the surface.

Ball B has a greater density than ball C. As a result, its gravitational force constitutes a greater downward force compared to ball C. The significance of this is that the net upward force for ball B will be lower than that of ball C. Thus, ball C will rise to the surface faster than ball B because it is enacted on by a greater net upward force.

So all in all, ball B and C will both float to the top. Ball C will reach the surface faster than ball B. Ball A, however, will sink to the bottom.