‘Rotational buoyancy’ – Hydrostatic pressure in solid body rotation

Solid body rotation of a fluid about a vertical axis results in a horizontal pressure variation which provides the centripetal force required to rotate the fluid particles. The pressure gradient is, therefore, given by

dp/dr=ρw2r                                              (1)

Where ρ is the fluid density, w is the angular velocity and r is the distance from the center of rotation.

This is easily demonstrated by spinning a cup of water and showing the paraboloid surface that forms with the low point at the center of pressure. However, the pressure gradient exists regardless of the free surface and, just as with the non-rotating hydrostatic pressure variation, an immersed object will experience a net force in toward the region of lower pressure. This can be demonstrated using a sealed container

Equipment

1. A small cork
2. A marble
3. A Lazy Susan or some other cheap turntable
4. A mason jar
5. Some Velcro strips or something else to secure the Mason jar to the turntable.

Demonstration

1. Place the cork and marble in the jar and fill it with water
2. Seal the jar so that there are no bubbles (or at least no bubbles that are large compared to the size of the cork) and attach it to the turntable so that its long axis is horizontal and it is centered on the turntable (see figure above).
3. Shake the jar until the marble and the cork are near the center of the jar (this is so that when the marble moves it is clearly due to the rotation of the jar).
4. Rapidly spin the turntable. The marble should be pushed to one end of the jar while the cork should remain centered.

Analysis

The horizontal hydrostatic pressure gradient (equation above) means that any submerged object will experience a net pressure force acting toward the center of rotation. For a rectangular object of width s in the radial direction and area A normal to the radial direction located a distance r  from the center of rotation, the net pressure force toward the center of rotation is given by

FpAw2((r+s/2)2-(r-s/2)2)/2                      (2)

See figure below.

Fp=ρAw2rs= ρ∀w2r                                          (3)

where is the volume of the object. Therefore, the net force toward the center of rotation is the angular acceleration multiplied by the mass of water displaced by the object. This is directly analogous to the buoyancy force in a stationary fluid. Therefore, if the object displacing the fluid has a lower density than the fluid then the centripetal pressure force will exceed that needed to maintain the angular velocity of the object and it will be pushed toward the center of rotation (as in the cork). If the object is denser than the fluid the centripetal pressure force will not be enough to maintain rotation and the object will move radially outward (note that this is a somewhat simplified linearized analysis and the integration constant is left out of the pressure equation used in (2) but it gets the appropriate result).

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Video of “Compressibility and incompressibility demonstrated with soda bottles and ketchup”

Here is a video of the “Compressibility and incompressibility demonstrated with soda bottles and ketchup” demonstration. The full video is here.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Compressibility and incompressibility demonstrated with soda bottles and ketchup

Introduction

I teach in a civil engineering department so we pretty much only deal with incompressible flows. However, this is a really simple demonstration to illustrate the compressibility of gasses and the relative incompressibility of liquids that uses stuff you can pick up at a fast food restaurant and recycling bin. I found it in a number of different books on science experiments for kids.

Equipment

1. 2 liter soda bottle with cap
2. a small ketchup (or other condiment) packet that floats (test this before you wedge it in to the bottle).
3. water

Demonstration

1. Stick the ketchup packet into the soda bottle and then fill the bottle with water until there is only a small volume of air below the top of the bottle.
2. Tightly screw on the cap so that the bottle is sealed. The ketchup packet should be floating.
3. Squeeze the bottle firmly with your hand and the ketchup packet should sink.
4. release the bottle and the ketchup packet will float back up to the surface.

Discussion

This is effectively a cheap way to make a Cartesian diver. The demonstration relies on the water being effectively incompressible and the air being compressible. When you squeeze the bottle it is the air pocket at the top of the bottle that is compressed by the change in volume. This increases the pressure in the water but does not compress it so the water density stays the same. However, the ketchup packet has a small air bubble in it which also compresses. This reduces the volume of the bubble enough that the net density of the packet changes from being less than that of water to greater than that of water so it sinks. This process is reversed when you stop squeezing the bottle.

You can also get the packet to sink just by leaving the bottle out in the sun. In this case the water and air both heat up. however, given the finite volume, as the water expands slightly from heating, the air is compressed and the packet sinks.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Buoyancy – Balancing ping-pong balls

Background

I saw this demonstration on Veritasium’s YouTube channel (see Beaker Ball Balance Problem video). It is really simple to run as long as you have a good balance. The result is not obvious and so the reason for the result requires a little analysis. Therefore, I think it would be good to write it up and provide a more detailed explanation of the demonstration complete with free body diagrams.

Equipment

1. Balance
2. Three identical cups
3. Water
4. Two ping pong balls, one filled with sand of something to weigh it down (actually any two balls with the same diameter as long as at only one of them floats)
5. Tape
6. String
7. A stand

Procedure

1. Place identical volumes of water in two cups but leave enough space for at the top of the cups so that it will not overflow when you place the balls in the cups.’
2. lock the balance so that it is level for steps 3-7.
3. Take the empty ping-pong ball and tape some string to it and then tape the string to the base of the empty cup. Use as little string as possible. When you pour the water in, the ball has to be fully submerged.
4. Pour the water from one of the cups into the one with the taped ball.
5. Place the two cups with water (one with a ball attached) on either side of the balance.
6. Suspend the heavy ball from the stand into the second cup such that it is fully submerged.
7. Poll your students to see which way they think the balance will tip when it is released.
8. Unlock the balance and observe which way the balance tips (it should go up on the side with the ball taped to the bottom of the cup)

Analysis

We start by looking at the setup (see figure below). The two cups have identical volumes of water in them and both have a submerged ball with identical volumes which, therefore, displace identical volumes of water. Therefore, the depth of water in each cup is the same. For the purposes of this explanation light ball is denoted as ball (1) and the heavy suspended ball is ball (2).

We now examine the forces acting each ball (see figure below). Ball (1) has weight down (W1), buoyancy force up (FB), and the tension force in the string acting down (T1). The forces sum to zero as the system is in equilibrium. Ball (2) has the same set of forces acting on it. However, for ball (2) the tension force (T2) is acting up as the ball is denser than water such that the buoyancy force is less than the weight. Hence the ball must be suspended from above. The only important thing here is that the balls displace the same volume of water such that the water levels in each cup are identical.

We now turn our attention to the forces acting on the cups (see figure below).

For cup (1) there is the hydrostatic pressure force acting down, PA, that depends only on the cup geometry and the depth of water in the cup. There is also the weight of the cup (Wcup), the tension on the string acting up (T1) and the force due to the scale Fscale(1). Therefore,

Fscale(1) = PA+Wcup-T                                                                                                (1)

The forces acting on cup (2) are the hydrostatic pressure force acting down, PA, the weight of the cup (Wcup), and the force due to the scale Fscale(1). Therefore,

Fscale(2) = PA+Wcup                                                                                                        (2)

As the cups are identical and the water depth in is the same in each then both PA and Wcup are the same in equations (1) and (2). Therefore, subtracting (1) from (2) gives

Fscale(2) – Fscale(1)= T1

or

Fscale(2) > Fscale(1)

and the balance goes down on the side with the suspended ball.

In fact, you could run this test with any two sized balls as long as:

1. One ball floats and is tethered to the base of a cup
2. One ball does not float and is suspended from above
3. The water level in the two identical cups is the same.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Take home lab experiment – Density of oil

This is another early semester take home lab. When I give this to my students they have already been down to the lab and used a U-tube manometer to measure the specific gravity of oil (See post https://teachingfluids.wordpress.com/2014/01/13/measuring-specific-gravity-of-oil-with-a-u-tube-manometer/). This goes a step further and requires them to find 2 more ways to measure the density of common cooking oil. They have to look at the course material covered so far (typically we just finished hydrostatics when I hand this out) and work out what topics will enable them to measure the density of cooking oil. The two main goals are to (1) have them review the course to find possible measurement techniques and (2) to start doing some error quantification. They are required to give 3 different values of density based on three different measurement tecuniques and explain any possible differences. At this stage in the semester I typically only ask for estimates of their direct measurement uncertainty but not their uncertainty in their final calculated density.

In general the students are able to come up with three different methods for doing the measurement. They also, typically, measure densities that are slightly less than that of water. This puts them in the right ball park for the actual density. However, the students sometimes end up measuring some very small quantities (such as the height differences in the U-tube manometer. This in turn leads to large measurement percentage errors and even larger density percentage errors. The analysis of this error propagation is left to later take home labs though I do discuss error propagation in class around the time when I hand back the graded lab reports.

As with all the take home labs I will not publish methods for conducting the tests as I still use them in class and want my students to figure it out on their own.

Introduction

Fluid density is needed for many fluid mechanics calculations (hydrostatic pressure, forces on submerged structures, buoyancy, conservation of mass calculations). You have already measured the specific gravity of cooking oil in the lab. You are now required to measure its density.

1. Use three different methods to measure the density of cooking oil (you can repeat the approach you used in the lab if you like).
2. Write a brief report that
1. Is 3 pages max including photos of you running your experiments
2. Details how you made the measurements
3. Details how you used the measurements to calculate the cooking oil density
4. Has clear diagrams showing your setup
5. Compares the three different measured values of density.
6. Quantifies potential sources of error in your measurements including a table of what you measured directly, typical values, and estimated uncertainty.

Rules

1. You can use tubing from the fluids lab and a scale from the materials lab if you can get permission and access. Otherwise you are limited to household implements.
2. You can take the density of water to be 1,000 kg/m3.
3. If you need to go to a store to buy something please come and see me first. I may be able to lend you something or I will buy it and then lend it to you. You will need to explain why you need it and it should be cheap.

Due in 2 weeks

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.