We spend quite a bit of time in class building mental models of different things for our students, but one thing we rarely do is break those models. We may tell our students under what circumstances the models are ok, but we almost never show them what it looks like when a model fails. For instance, what would the position vs. time graph of a pendulum look like if we calculated using the small angle approximation but looked at the graph for a complete 360° swing.
My plan is create simple models for various systems and then push them to the limit until they break. To the students reading this, I hope this gives you some confidence to build models of your own and push them to the limits. Not all models are good ones, but they can be a lot of fun.
What is a liquid? Anything that isn’t a plasma, solid, or gas? The textbook definition is that a liquid is any substance that takes the shape of the container it is in and it has well defined surfaces or volume. I should point out that these definitions of states of matter are somewhat arbitrary; there are many materials that fit in the cracks between the states of matter.
Assume the liquid is comprised of a series of tiny cubes, stacked in an orderly fashion on top of one another.
It has well defined surfaces along the edges of the outermost blocks, but if we were to stick our stack of cubes inside a cardboard box the cubes wouldn’t flow to fit the shape of the box. Strike one for the model.
Anyone who has ever gone diving in a swimming pool knows that the pressure increases as you dive deeper. Does our model include this behavior? It does and in fact I think that is the one thing this model does well. Imagine sticking your hand between the top block and next block down in a single stack. If each blocks weighs W kg and has a surface area of A m2 the pressure on your hand is:
If you stuck your hand between the top block and next block down for all three stacks the pressure on your hand is
The pressure is the same. So what happens if we go down a level? Does the pressure increase accordingly?
Stick you hand between the ground and all three blocks in a single stack. The pressure on your hand is:
and if you balance all nine blocks on your hand the pressure is:
So based on this simple model we can say that the pressure between any two layers of blocks increases linearly with depth and only depends on how deep we go and not how large an area we use to measure the pressure. Victory for our model.
One thing to note is that the blocks in the stack support the weight of your hand, which wouldn’t happen in a fluid. Strike 2. The fluid below would provide some upward force, equal to the weight of the fluid directly overhead. This is why, when you are underwater, you don’t feel like you have to actually lift all the fluid over you; the fluid under you is pushing up and supporting that fluid. You can think of the squeeze you feel as you go deeper due to the liquid above you pushing down at the same time the liquid below you pushes up with equal force. As we will see in our later post, the fluid to either side pushes in with the same force to support the liquid.
- Model demonstrates how fluid pressure increases with depth
- Model does not explain why liquids take the shape of the container they are in.
- Model does not explain why liquids exert sideways forces (these two are actually the same)