Fluids: Archimedes' Principle

According to what we have learned in the last topic, we know that the depth of the object has a significant impact on the pressure experienced by the object. Then, you may be asking, what about the pressures on the top and bottom parts of the object? Don't they have different depths? Then surely they must have different pressures. In retrospect, yes, they have different pressures, and because of this, we need to study Archimedes' Principle and buoyant forces. Before we begin, it is imperative that we must first understand what buoyancy is.

Well, what is buoyancy?

This. Is. Buoyancy.

Okay, joking aside, buoyancy is simply the upward force at the lowest point of an object. Suppose we have Kanye's head submerged underneath the ocean, as we go deeper, or as depth h increases, the pressure experienced by Kanye's head increases. Hence, in this situation, the lowermost part of the head has a higher pressure compared to the topmost part of the head.

What's even more interesting and crucial to note is that even with different shapes (whether it is a cone, cube, pyramid, cylinder, prism, or a sphere), the lowermost part of the submerged object will always have a higher pressure compared to the topmost part.

The buoyant force of an object is calculated by this formula,

How was the formula derived then?

Suppose we're in a situation where we have a cube submerged under the ocean.

To get its buoyant force, we can simply find the difference of pressures between the lowermost part and the topmost part of the object.

Hence,

The subscripts b and t signify the bottom and top part of the object respectively.

If you're wondering why P_bA_b is subtracted by P_tA_t, it's because we know that the bottom part has a higher pressure compared to the top part.

To find the area of a cube, we know its formula is A = s²

And since we have h₂as a side, it becomes,

Then by isolating the area, we will have,

Since we are aware that the formula for pressure

is P = ρgh, then we can just plug it into our equation.

So it becomes,

Then by distribution and cancellation,

Simplify,

Here's the fun part, h³ is the formula for Volume right?

So, we can just replace it with V which in hindsight, denotes Volume.

And here we got our formula.

Now, let's move on to Archimedes' Principle.

Archimedes' Principle states that:

The buoyant force of an object is equal to the weight of the fluid displaced by the object.

In mathematical terms, this is,

Surprising right? Archimedes' Principle is simply Newton's second law of motion,

which is F = mass * acceleration. Moreover, they discovered it way before Newton even existed.

Moving back to the topic, to illustrate Archimedes' Principle, let's again make use of Kanye's head. Suppose two containers with the same amount of fluid, one of which contains Kanye's head. In the illustration below, we can observe that the fluid in the container which contains Kanye's head seemingly increased. However, that's not what is happening here, that seemingly "excess" fluid is being displaced by the presence of Kanye's head.

Going back to Archimedes' Principle, the upward buoyant force of Kanye's head is equal to the Weight of the "excess" or displaced fluid. So if we were to say that the upward buoyant force of Kanye's head is 100N, then in retrospect, the Weight of the displaced fluid must also equal that of 100N.

To better understand this concept, let's do a simple exercise.

Kanye East's head is submerged under a container. We measured the volume of water in the container before and after Kanye's head was submerged and concluded that the displaced fluid has a volume of 0.02m³. We know that the standard density for water is 1000kg/m³ . Find the mass of the displaced fluid.

First, let's bring up the equation in accordance with Archimedes' Principle.

We know that F_buoyant = ρgV and Wfluid = mg

Hence,