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Archimedes' Principle (2012)

Page history last edited by Joe Redish 11 years, 5 months ago

Working content >MacroModels > Fluids > Pressure

 

Prerequisites

 

In our initial discussion of the concept of pressure we explicitly ignored gravity. In this webpage, we'll consider the effects and see that gravity has a significant impact on the behavior of fluids and of objects immersed in them.

 

Most of you have had lots of personal experiences immersing yourself in a fluid. We're almost always immersed in air, but air is a fluid with density is so low -- about 1 kg/m3 -- that we hardly notice it (unless we are moving at a significant velocity relative to the air). When we immerse ourselves in water, which has a density of about 1000 kg/m3 -- comparable to our own -- that we personally can see effects of gravity on fluids.

 

Think about going swimming in a pool or lake. Do you float? Sink? Most people float, at least somewhat. What if you take a beachball filled with air and try to hold it under water? Does the ball sink? Float? Fly up out of the water? Get pushed down to the bottom? If you don't know what happens, find a ping-pong or tennis ball and hold it under water to see.

 

Although most of us know what happens, it's a bit difficult to reconcile this with the physics we are learning. We know that gravity pulls everything down. If I hold the beachball under water there is water on top of it. Gravity is pulling that water down. Why doesn't that push the ball down to add to the downward pull of gravity on the ball? Why does the ball fly up in the air?

 

Warning! Notational inconsistency -- In chemistry and biology, density is often represented by the symbol "d".  In physics, the standard notation for density is "ρ" (Greek letter "rho"). (Though this is also often used for charge density in physics as well as for mass density.) We will use the physics notation here since what matters to this analysis is the depth -- and we want to use "d" to represent that.

 

Pressure under gravity

The way to understand what's happening to objects that are immersed in water is to start with the simplest possible example. We can choose any object we we want. Let's consider a cylinder of still water with a cross sectional area A as shown at the right (the whole thing). But let's choose as the object we want to look at as the top disk of water, a thickness d  (the depth), shown in lighter blue.  This is really no different from any of the other water -- this is just the part we are considering.

 

Now let's use Newton's theoretical framework to analyze the forces on this top disk of water . Let's assume that the water is open on the top to the air. From the Newtonian framework we know we can isolate a part of an object and consider the forces acting on it. 

 

There are three forces acting on the disk:

 

  • the force of the air pushing down on top, p0A, where p0 is the air pressure at the top;
  • the weight of the disk due to gravity, W = mg = (ρV)gρgdA; 
    just density x volume x g.
  • the force of the water beneath the disc pushing up on it, pA, where p is the pressure of the water at the bottom of the disk.
 

We know that the water on top will not spontaneously flow, so the net force on the water object must be zero.  Thus the up forces must balance against the down forces.   We get

 

pA = p0A + ρgdA;

 

In words: the upward force from the water underneath the disk balances the weight of the disk plus the force of the air pushing down on the top. We see that there is an A in each term so we can cancel it to get

 

p = p0 + ρgd.

 

The pressure p as you recall is measured at depth d in the water.  This equation shows that pressure in the water increases with the depth d. This tells us:

 

As you descend into a fluid, the pressure in the fluid increases as a result of gravity.

 

This makes good sense since the farther down you go, the more water that's above you has to be held up. The water "trying" to fall squeezes the water beneath it and increases the pressure.

 

Archimedes Principle

Well that seems interesting, but what does it have to do with the beach ball?  The implication is actually quite striking. 

Suppose we  immerse a small cylinder of height h and area a in the water at a depth d as shown in the figure at the right.  The water on the top exerts a pressure downwards on the object and the water on the bottom exerts a pressure upwards on the object.

 

But since the bottom is deeper than the top, the pressure at the bottom is greater than the pressure at the top. (The sideways forces all cancel.) The result is a net upward force! The result can be expressed in an interesting way.  Taking up as positive, if we write the balance of forces, we get

 

There's a lot of cancellation, but the result is the density of the water times the volume of the object times the gravitational field, g.  The density of water times the volume of the object is the mass of the water that would have been in the place where the object is. This gives the result is that the magnitude of the upward force the water exerts on the small cylinder is exactly equal to the weight of an equal sized cylinder made out of water!

 

Archimedes' Principle: When an object is imbedded in a fluid, it feels an upward (Buoyant) force that is equal to the weight of the displaced fluid.

 

This seems a bit strange. Why should the displaced fluid matter?  It isn't there anymore?  Did it just come out of the math this way as an accident?  Or can we make sense of it?

 

Consider an arbitrary object (a rock) that we are going to place under water as shown in the figure at the right below.

Archimedes principle tells us that the upward force on the rock exactly balances the weight of the fluid displaced by the rock. To see whether that makes sense lets replace the rock that you see in the image on the right with a plastic bag filled with water.  In that case the buoyancy force on the plastic bag filled with water would exactly balance the weight of the water in the bag - the water filled bag will simply stay in place and neither sink nor float.  (Assuming, of course that we are ignoring the mass of the plastic bag -- or it could be an imaginary bag.) Does it make sense that water can hold up other water? 

 

Since the water outside the plastic bag doesn't "know" what's inside, all it can do it exert the same forces whether there is water in the bag, or a rock, or air. The result is an upward force equal to "the weight of the water displaced"! If the object is lighter than the water displaced, it will be pushed upward. If it is heaver, it will fall, but it's apparent weight will be less.

 

Follow-on:

 

Joe Redish 10/23/11; revised 10/27/12

Wolfgang Losert 10/27/2012

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