The Spinning Log "Perpetual Motion" problem, and my attempt at a solution So I was introduced to this "perpetual motion" riddle a few weeks ago. The problem goes like this: we all know perpetual motion machines are not possible, but this riddle seems like it should work as a perpetual motion machine - the problem is to explain why it doesn't work.
Here's the situation:
(1) You take a room, and put a wall down the middle of the room - splitting it in to two equal half-rooms. 
(2) Take a perfectly cylindrical log whose length is the same as the length of the wall we just put in to make two half-sized rooms, and cut out a place halfway up the wall so that the log slips right in, length wise (if you were looking at the middle wall from the inside of one of the rooms, you would not see the ends of the log at all, you would a half-cylinder running the length of the wall - if you looked from the other room, you would see the other side of the cylinders length).
(3) Make sure the space between the log and the wall is air-tight (the log is treated to absorb no water, there is no friction between it and the wall, etc), and fill one of the half-sized rooms all the way to the top with water. Now you have one side of the log exposed to air, and one side totally surrounded by water.
The idea is that, since everyone has experience that logs rise in water, the side exposed to the water should "rise" and the log should spin, creating a perpetual motion machine.
Now, I have two explanations of why it doesn't work. One I came up with myself, and one that is proposed in the solution on the page in which I found the problem.
My Solution:
Water pressure increases as you increase in depth, so the water molecules hitting the log deeper in the water would have more force that the water molecules hitting the log less deep, but at each infinitesimal depth increase, the water is hitting the log at all angles, and the force from the molecules coming on from all directions would cancel out to leave a force pointing toward the center, at every place the log is exposed to water. Since all of the net forces from the water at each point exposed to water is pointing toward the center, there can be no net torque and the log cannot spin. The net force would be a slight force directly upwards (due to the difference in size of the force vectors pointing inwards between the less dense water up top and the more dense water on the bottom), but since the log is held in place, it cant move upwards.
The Proposed Solution
"To understand why this would not work, we need to look at how buoyancy itself works. 
When a lighter than water object (lets say a hollow ball) is submerged it is pushed up by the water apparently in spite of gravity. In fact the opposite is true. It is gravity pulling down on the water that pushes the ball up. The kinetic energy to lift the ball comes from an equal volume of water falling to occupy the space where the ball just was. As the ball moves up, the evacuated area is filled with water from above it; therefore, this water is falling. When the water was above the ball it had potential energy that is exchanged for kinetic as it falls to fill the void left by the ball thus providing energy to lift the ball. 
The opposite is true for a heavier than water object (lets say a brick), but it's still the same principle. As the brick falls through the water, it is filling space that was once filled by water. As the brick falls, it is providing energy to lift the water to fill the space it just evacuated.
Now back to our spinning log example. If the log were to spin, it would not be evacuating any space for the water to fill. Even though a different part of the log would be filling that space, it would still be the same space it was filling before. There would be not downward movement (falling) of water to convert potential energy into kinetic, so there is no energy to cause the log to spin."
Here's my problem
I've never understood this explanation of buoyancy. Why, just because an object is less dense than water, is the water "rushing below it" to fill the space evacuated by the object as it rises? Why doesn't the object just stay where it is? It seems like you're using buoyancy to explain buoyancy - the less dense object floats upward due to buoyancy and the water rushes in below to fill the evacuated space, which causes the buoyant force upwards... what?
Why is the water rushing below less dense objects, and not rushing below more dense objects? Am I missing something in the explanation?
I understand that when you submerge an object in water, it displaces an equal volume of water, and the weight of the water displaced equals the bouyant force upwards, but I just don't understand why the water is "rushing underneath and pushing the object up".
 A: The explanation is using an energy argument.  That for the normal case of a submerged piece of wood, you can assume that if the wood and a parcel of water above it switch places, then the water (which is heavier/more massive) drops in the gravitational field releasing potential energy.  This release is not offset by the rising wood since it is not as massive.  This energy imbalance goes into the kinetic energy to move the log (and the water).
This argument says that in the case of the rotating log, none of the water can move downward to release potential energy.  Since the water stays in the same place, no energy is released, and there is no energy to move the log.

Why doesn't the object just stay where it is?

Because forces will combine to move it to a position with less energy.  You can do a free-body diagram of a ball rolling downhill to see that the net force is down the hill.  But you can also simply say that the ball is going to move in a way that lowers its potential energy (which is downward in the gravitational field).  Using that argument for the cylinder says that there is no position it is free to move to that lowers the potential energy, so no movement will happen.
A: There absolutely is a net force from the water on the log. The fact that it is held still does not mean the water does not exert a force. However what is important is the direction of the force. If you integrate the force field over the half circle submerged in water the net force is diagonally upwards through the center of mass of the circle, which in this case equals the center of rotation of the log. Hence it will not rotate and 
-if nothing is holding the log in place it will be pushed out of the hole
-if it leans on something preventing it from being pushed out but not preventing it from sliding up and down it will slide up if it's mass is low enough, because the net force from the water is pointing slightly up.
Conclusion: There is still a buoyant force. It is pointing slightly up, but it does not create rotation because it is pointing through the center of rotation of the log.
