Back when I was writing my unread (and for good reason) blog “The Bitter Christian”, I used to wonder if my efforts to understand and explain Christianity were a waste of time. I finally concluded that having a solid understanding of what we believe is a part of the Great Commission of the church. If we do not understand the Bible, then how can we explain it? If we find our own explanation unsatisfactory, how can we be confident or enthusiastic when it comes time to share with others? Attempting to understand and explain Christianity, therefore, was an act of service even when it didn’t seem like it was doing any good.
The advantage that I have when it comes to explaining things to others is that I am a very simple guy. If I come across an explanation that is simple enough for me to understand, chances are that other people will find it helpful as well. This coupled with the fact that I won’t claim to understand issues if I don’t understand them means that I have accumulated easy to understand explanations on a variety of topics. Just for fun, I thought I would share a couple of simple examples that have nothing to do with Christianity.
Understanding the Two Coin Paradox
So imagine you have two coins as pictured below and you want to roll the red coin around the black coin such that it is always touching the black coin. How many times does the red coin rotate to get back to its starting position?
When I first saw this problem, I thought, “That is easy. The two coins have the same circumference, therefore one rotation of the red coin should take it completely around the black coin.” This turns out to be incorrect. The correct answer is that it takes 2 rotations for the red coin to make a complete circuit around the black coin. Try it for yourself.
Now this really threw me for a loop and I didn’t understand it at all. As I read other people’s explanations, I became more and more confused. I kept thinking and thinking and nothing made any sense. I finally understood it when I asked a simple question. What happens if you move the red coin along the perimeter of the black coin without rolling it?
Moving the blue dot along the black coin without changing the orientation of the red coin results in the blue dot moving only the circumference of the black coin. It is not, however, at all the same as the case of rolling the coin. If you compare the diagram below with the one above, it is clear that the blue dot has travelled further for the rolling case then for the case where the coin does not change orientation because it ends up further away.
Understanding Countable Infinity
The concept of infinity is very difficult to grasp. This can be illustrated by a simple example. For infinite sets, the number of integers has the same number of elements as the number of integers divisible by 5.
1, 2, 3, 4, 5, 6, 7, 8 . . .
5, 10, 15, 20, 25, 30, 35, 40 . . .
This didn’t make any sense to me until I was looking at multiplication of infinite sets. Imagine multiplying every number in the first set by a factor of 5.
1*5, 2*5, 3*5, 4*5, 5*5, 6*5, 7*5, 8*5 . . .
What is the result?
5, 10, 15, 20, 25, 30, 35, 40 . . .
Writing it that way, it is obvious that every element in the first set corresponds to exactly one element in the second set and therefore that the sets have the same number of elements.
So a while back a friend with a daughter came to me and said he did not know how to explain how buoyancy works to his daughter who was covering the topic in school. At first, my reaction was, “Buoyancy? That is easy. It is just the force that result from the displacement of water. What is the big deal?” His answer ways, “Everybody knows that but how does it work?” As I went over how you could explain how buoyancy works, however, I realized it was not so easy. After thinking about it for a while, I came up with an explanation that I hoped would be helpful.
The key to understanding buoyancy is to understand that pressure comes from the weight of all the stuff above you. For the air, this means that there are 14.7 pounds worth of air above a given square inch of surface at sea level.
PAir = Weight of Air / Surface Area = 14.7 pounds per square inch
Again, the pressure of the water is just the weight of the water divided by the surface area. Using standard relationships between volume, area and density we can show the following:
PWater-ocean = PAir + Weight of Water / Surface Area Water
PWater-ocean = PAir + Volume * Density of Water/ Surface Area Water
Volume = Depth * Surface Area Water
PWater-ocean = PAir + Depth * Density of Water
Now it is a fundamental characteristic of fluids (liquids and gases) that they tend to translate force applied in one direction into all different directions. This has the consequence that the pressure of the water under the boat is the same as the pressure of the water at the same depth in the ocean.
PWater-ocean = PWater-boat
The same is not true of solids. If the ocean was made of concrete, for example, the pressure under the boat would not be the same as the pressure further away. You can see how fluids transmit force by filling up your cheeks with air and popping them with your hands. As you apply pressure to the air in 1 direction, the air applies force in another direction. This is the principle by which hydraulics work.
So the pressure pushing up on the bottom side of the boat is the pressure of the air combined with the pressure of the water at the same depth. The pressure of the boat is just due to the weight of the air and the weight of the boat.
PBoat = PAir + Weight of Boat / Surface Area Boat
If the pressure applied by the boat is greater than the pressure applied by the water, the boat sinks until the pressure increases enough to support the boat. At this point, the two pressures are equal.
PWater-ocean = PBoat
PAir + Weight of Boat / Surface Area Boat = PAir + Depth * Density of Water
Weight of Boat = Depth * Surface Area Boat * Density of Water
Weight of Boat = Volume of Boat in water * Density of Water
Weight of Boat = Weight of Water displaced by boat
To understand buoyancy, therefore, we have to understand that it is a phenomena that is caused by pressure and the way liquids transmit force. As shown above, the force transmitted to the bottom of the boat is the same as the weight of the displaced water.
What can be calculated for a simple rectangular boat can be generalized to a large number of tiny rectangular boat slices and integrated together to yield a result for a boat of arbitrary shape.