Buoyancy

Many of us have gotten used to seeing boats carry items or passengers from place to place and not pay mind to it. However, to a curious thinker, this may be quite weird to look at in terms of the forces on the boat. A boat has weight so it should be accelerating downwards due to Earth's gravitational field, right? This is correct but this doesn't tell the whole story. It's quite clear that the boat doesn't vertically accelerate(for the most part) so that means its weight must be counteracted by the force the water exerts on it upward. This would also be true but doesn't tell the whole story. Looking at it like this would imply the water is applying a normal force that doesn't allow the boat to penetrate into the ocean. However, because it's obviously possible to drop something into the water and watch it sink, the water's support force isn't the same as the normal force because otherwise, you wouldn't be able to penetrate the water. So, why is it such that some objects on water float above the water while others sink? It can't be solely because of weight or else we wouldn't see a tennis ball sink in water while a cruise ship doesn't when the cruise ship has a lot more weight than the tennis ball.

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The answer to that is simple but requires a decent bit of math and intuition. First, let's define this mysterious force on the objects floating. This force is universally known as the buoyant force on the object(the force exerted that buoys the object up above a fluid). The first thing to know about the buoyant force is that it always acts upwards on an object whether the object floats at the top or sinks to the bottom. The way to prove why is very simple with an application of hydrostatic pressure. If you look at the image below, the force exerted by the liquid molecules on the bottom is greater than the force exerted by the liquid molecules on the top.

The pressure at the bottom of the object is at a greater depth than the pressure at the top so for a given area, the force at the bottom will exceed the force at the top, causing the buoyant force to be upwards at all times. Thus, the net difference between the forces is upward, which gives the buoyant force of the liquid on the object inside. You may be asking why the object doesn't horizontally accelerate. The reason for that is because the force of the liquid on the left cancels out the force of the liquid on the right, making a net 0 total horizontal force. You also may be asking if this works with objects of any shape and the answer is yes. Remember that the net difference in total force exerted from the bottom minus the total force exerted from the top must be upward. You will see that this pretty much works for any object because if the object isn't uniformly structured but we divide up the horizontal and vertical components from the forces of the molecules of liquid everywhere on the object, the total upward components will still exceed the total downward components in magnitude.

Learning conceptually how the buoyant force works is great but it'd be also great to know how to apply the buoyant force mathematically. Thanks to Archimedes, approaching this mathematically through the application of Archimedes' Principle. Archimedes' Principle states that the upward buoyant force on an object in a fluid equals the weight of the fluid displaced, which will make sense in a moment. However, before we can internalize Archimedes' Principle, we need to analyze the math behind buoyancy, shown below with the given equation.

The formula for the buoyant force proves Archimedes' principle correct. The density of a fluid is the quotient of its mass and volume so if you multiply the fluid's density by its displaced volume, you get the fluid's mass. Multiplying by gravitational acceleration gives the weight of the displaced fluid as stated in the principle. This equation has some serious implications for how we understand buoyancy in static fluids.

First off, the depth of the object the buoyant force acts on doesn't matter because height is not in the equation. Conceptually, this makes sense because buoyancy is less about the depth an object is at and more about the difference in its bottom's depth and the top's depth, which should be constant because the object itself shouldn't be changing dimensionally.

Second, we can regard g as constant for simplicity's sake, mainly because the object in the liquid would ideally not be that close to Earth's center of mass. Next, the volume in this equation is the object's volume and not the fluids because it is the height difference of the object multiplied by its cross-sectional area. However, if and only if the object is submerged, the volume term in this equation equals the volume of liquid displaced which would make sense because if the object takes up a certain volume in the fluid, the volume must be displaced elsewhere(the top) in the fluid.

This equation above may not look useful at first glance but it can help us understand many things about how objects float, and if they even do float. The density of water is a good benchmark to use because it can be well approximated to 1000 kg/m3.

Let's say we put an object with a density of 560 kg/m3 into a pool(filled with water). Since the density of the object is less than the density of water, assuming volume is constant between the fluid the object displaces and the object itself, the density of water will cause the buoyant force to exceed the object's force. This means that the object will float because it is being buoyed, or pushed, up.

However, another interesting aspect of this is that this specific gravity is also the proportions between the volumes as shown in the derivation above. This means that if the specific gravity is 0.56, then 56% of the object's volume is submerged. However, if the specific gravity exceeds 1, you can't have more than 100% of the object's volume submerge so the object simply sinks. If the object has a density greater than that of water, the object will sink because its weight will exceed the buoyant force. This also means that the specific gravity of the object in water is greater than 1.

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Simply put,

If the specific gravity of an object in a fluid is:

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Greater than 1, the object sinks

Less than 1, the object floats with (specific gravity x 100)% of its volume submerged

Equals 1, the object partially floats while fully submerged in the water(it's submerged but doesn't sink)

Note that specific gravity is dimensionless since the density units cancel out.