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Can you explain to me what causes the buoyant force? Is this a result of a density gradient, or is it like a normal force with solid objects?

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It is the result of a dependence of the pressure with growing depth, due to the gravitational field (i.e. the weight of the water).

You may do an easy calculation with some simple geometrical form, e.g. a cylinder totally submerged in water, to quickly understand how it works. The force due to pressure in each surface element of the curved wall of the cylinder is proportional to the depth of that element, and has the normal direction to the wall, i.e. towards the axis of the cylinder. After an easy integration in polar coordinates, you can see that the resultant force points upwards. That is because the forces in the upper parts are smaller than the ones near the more deeply submerged part of the cylinder.

A surprising conclusion is that a golf ball submerged in a tank of water in the space station, would not go upwards... or that the bubbles in a coke in the hands of an astronaut remain where they origin... I would love to see that.

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    $\begingroup$ @leonardo - if you'd like a more rigorous proof, you can easily show that this is a consequence of the divergence theorem (Gauss's law). $\endgroup$
    – Deep Blue
    Commented Oct 31, 2012 at 3:30
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It's like a teeter-totter.

Some of the fluid is pushed up as the solid thing moves down. There energy cost of doing that is exactly the same as the energy cost of pushing down on one end of a teeter-totter.

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Good answers, but let me try to make it intuitive.

Water weighs 1 gram per cubic centimeter (or 1 kilogram per Liter, a cube 10 centimeters on a side, if you prefer).

So if you have a tube 1 centimeter square, stopped up at the bottom, and you fill it with water to a height of N centimeters, then you know how much pressure there is at the bottom. It is simply the weight of the water in the tube, N grams, right?

Now take the tube, not stopped up at the bottom, and just put it in water to a depth of N centimeters. So the water in it weighs N grams, right? So, since the water stays in the tube (it doesn't run out the bottom) the outside water at the bottom of the tube has to be pushing up with a pressure equal to the weight of the water in the tube - N grams per square centimeter.

Now, with the tube still in the water, N centimeters deep, seal off the bottom of the tube with some kind of membrane, and suck the water out of the tube so it is empty. How much pressure is the outside water pushing up with against the membrane on the bottom of the tube? The same, right? N grams per square centimeter. However, since the tube doesn't have N grams of water inside it, pressing down, the tube itself is being pushed up by the N grams per square centimeter pressure at the bottom, that is not being matched by an equal weight of water in the tube pressing down.

That's bouyancy.

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Causes of buoyant force is the greater upward pressure exerted by water on the bottom of mug because it is greater depth inside the water.

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Let's consider a mug filled with water immersed in a bucket full of water. Water exerts force on sides of the mug as well as on it's top and bottom. the sideways forces exerted on water on the mug are equal and opposite, therefore they cancel out. As the top of the mug is at a lower depth in water it experiences less force downwards. the bottom of the mug is at a greater depth, therefore it experiences more force on the upward direction. Therefore there is a net force on the mug in the upward direction. The net upward force on the mug is equal to the difference in the upward force acting on its bottom and the downward force acting on its top. This net upward force acting on the mug is the buoyant force which reduces the effective weight of the mug and makes it feel lighter.

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One can think of buoyancy as the same force that lets you stand on the ground without sinking into it. When you are standing on the ground, the earth's gravity exerts a downward force equal to your weight, and the ground offers a normal reaction of equal magnitude in the upward direction which balances out your weight. Now, think of the water in a filled tumbler, and visualize the water in upper half as you, and the water in lower half of the glass as the ground. The weight of upper half of the water is supported by the normal reaction, or in this case, the pressure difference created by the rest of the fluid. Using this analogy, it is very simple to understand the Archimedes principle too. The buoyancy force exerted on any immersed body is equal to the weight of the fluid displaced by it, because this weight was initially balanced out by the pressure difference created by the fluid beneath it. Fundamentally, it is the intermolecular forces and the lattice structure which provides solids and fluids a degree of resistance to compression. This resistance is exhibited as the compressional stress in solids and as pressure in fluids.

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