In this question, we're presented with a situation in which two balls of different densities but equal volumes are held underneath the surface of a container of water. Then, each ball is released and allowed to accelerate up to the surface. The question is to determine how the acceleration of each ball compares to the other. In order to answer this, let's start by imagining all of the forces acting on the submerged ball.

In the x-direction, the forces acting to the left of the ball are exactly equal to the forces acting on the right of the ball. Therefore, all of the forces acting in the x-direction cancel out, resulting in a net force of zero in the x-direction.

In the y-direction, we have to consider two things. The first is the upward buoyant force caused by the displacement of water. Since both balls have an identical volume, both of them displace the same amount of water. Consequently, both balls will experience the same upward buoyant force. However, we must also consider the downward force caused by the weight of the ball itself. In this scenario, the downward weight of Ball B is greater than that of Ball A.

We can write the net force acting on the ball in the y-direction as the difference between the upward buoyant force and the downward weight as follows:

Due to the fact that Ball A has a smaller weight (a smaller  component in the above equation), the result is that the net upward force is greater than that of Ball B. Thus, we would expect Ball A to have a greater acceleration.

Consider all the forces on the gold marble:

Recall the equation for buoyant force:

Substitute:

Find the mass of the marble:

Plug in values:

Note how the buoyant force points up while the gravitational force points down.

Use the equation for buoyant force:

Where is the density of the medium around the object in question

is the gravitational acceleration near the surface of the earth

is the volume of the object in question

Convert  into

Plug in values:

Use the equation for buoyant force:

Where

is the density of the medium

is the acceleration due to gravity

is the volume

Plugging in values:

Determining density:

Volume of a sphere:

Combining equations:

Converting to and plugging in values:

It is denser than water, so it will sink.

According to Archimedes's principle, when the block is placed under water, 3.5L of water are displaced. We can then calculate the buoyant force provided by the water:

Where:

Plugging in our values to the first expression:

Then we can use Newton's 2nd law to determine the acceleration of the block:

There are two forces acting on the block, gravity and buoyancy, and they are in opposite directions. If we designate a downward force as positive, we get:

Substituting in the expression for force due to gravity:

Rearranging for acceleration:

Substituting in values:

We will start with Newton's 2nd law for this problem:

Where there are 4 total forces acting on the scuba diver and ball: gravity and buoyancy acting on both the scuba diver and the ball. However, the diver has the same density as the water, so the gravitational and buoyancy forces acting on the diver will cancel out. If we designate an upward force as being positive, we can say:

 (1)

Where:

Where:

So,

Plugging these expressions back in to expression (1), we get:

Now let's start rearranging for the density of the ball:

We will start with Newton's 2nd law for this problem:

Since the block is traveling at a constant velocity we can say:

There are 3 forces acting on the block: gravitational, buoyant, and drag force. If we denote a downward force being positive, the expression becomes:

Where:

Substituting these in:

Where according to Archimedes's principle:

Plugging this in:

Rearranging for volume:

Plugging in values:

Since the ball is held in place, we know that:

There are three forces acting on the ball: gravitational, buoyant, and tension. If we denote a downward force as positive, we get:

     (1)

And using Archimedes's principal:

Where:

Plugging all of these into expression (1), we get:

Plugging in our values, we get:

We are asked when:

Now we need to develop an expression for the mass in the ball using the rate at which water enters the ball:

Where:

Plugging this into expression (1):

Rearranging for time, we get:

Plugging in our values, we get: