‘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.

Photo Apr 13, 3 04 44 PM

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.

rotation

Expanding leads to

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 “Buoyancy: Throwing rocks from boats (and weighing coins)”

Here is the video of the “Buoyancy: Throwing rocks from boats (and weighing coins)” demonstration. The two GIFs show the dropping of the coins into the cup and then the dumping of them into the water. The full video is linked here.

Dropping the coins into the cup showing that 9 quarters displaces 50 ml of water.

drop

Dumping the coins out showing that, when submerged, they displace a lot less than 50 ml.

dump


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: Throwing rocks from boats (and weighing coins)

This is a simple demonstration of a classic buoyancy question about throwing a rock out of a boat (the full scale problem in discussed by Physics Girl here). The question is ‘if you are sitting in a boat holding a rock and throw the rock into the water will the water level go up or down’. The answer is that, when the rock is in the (floating) boat it displaces its weight but when it is thrown in the water (and sinks) it displaces its volume. Therefore, the water level drops when it is thrown out of the boat.

This is a simple scaled down version that clearly demonstrates the puzzle result but can also be used to weigh the heavy object (in this case coins).

Equipment

  1. Small plastic cup
  2. measuring cup
  3. syringe (for small adjustments in the volume of water in the measuring cup
  4. a hand full of identical coins (I used 9 US quarters).

Photo May 10, 1 53 34 PM

Demonstration

  1. fill the measuring cup so that the water level is one line below the top volume measurement when the empty cup is floating in the water
  2. drop the coins in one by one until the water level rises up to the top line (record the number of coins needed)
  3. pour the coins into the water and put the plastic cup back on the water.
  4. note that the water level is below what it was when the coins were floating in the cup.

Analysis

The puzzle answer is fairly straightforward. However,the demonstration can also be used to estimate the weight of the individual coins (hence the need to use all the same coins). The change in volume recorded as the coins are added is equal to the volume displaced by the coins (see figure below). Therefore, the volume change multiplied by the density of water will equal the mass of the coins added. I measured a 50 ml change in volume when I added 9 quarters. The density of water is approximately 1 g/ml so the 9 coins have a mass of 50 g. The weight of the individual quarters is, therefore 50/9=5.56 g (which is very close to the actual standard mass of a quarter of 5.67 g). It is also possible to work out the volume of the coins by measuring the volume they displace when they sink (see figure). My measuring cup did not have enough resolution to get a good volume. The total volume of the 9 coins would be 7.5 ml based on US Mint dimensions.

boat

Figure showing (from left to right) the cup displacing the coins weight, the empty cup, and the coins displacing their mass with the water level lower than when the coins are floating in the cup.

The other great thing about the demonstration is that it is easy to ask a lot of questions around the demonstration. Obviously you can pose the original question about the water level going up or down. There are also simple calculations that can be asked such as “what is the mass (or weight) of the coins?”, “what is the approximate density of the coins?”. The US mint website has information about size and metal content that can be used to calculate the actual density for comparison.

Ii is also really easy to have this as an activity in class. I used a version of it on the last day of class one semester using pennies.I used small and large plastic cups and gave the students the dimensions of the small cup and the equation for the cup volume (here). In this case you count how many pennies it takes to sink the cup so that the mass of pennies matches the density of water multiplied by the cup volume.  You will also need a roll of paper towel for the clean up. I did not explain the demonstration I just gave them the equipment and told the to calculate the weight of a penny. It was well received by the students though I had to give some groups a hint or two as to how it worked.


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.

Spin up, boundary layers, and tracking tea leaves

Background:

Boundary layers play an important role in many fluid mechanics applications including drag, lift, and flow in conduits. This demonstration illustrates the role of boundary layers as part of the classic spin-up problem. The demonstration is a cheap and easy version of one written up by Nicholas Rott in the book ‘Experiments in fluid mechanics’ (out of print but worth getting a second hand copy of). The demonstration uses tea leaves to visualize the secondary vortex that forms during spin-up. See here for more on tea cup fluid dynamics.

Equipment:

  • Turntable (Lazy Susan)
  • Bag of tea
  • Scissors
  • Half-filled glass of water
  • Tape (to secure the glass to the turntable)

Procedure:

  1. Place the tea bag in a glass of hot water to wet the tea leaves.
  2. Fasten the half-filled glass of water to the turntable with tape.
  3. Using the scissors, cut open the used tea bag and dump roughly half of the tea leaves into the glass of water fastened to the turntable.
  4. Spin the turntable quickly, so that the tea leaves move to the outer edge of the glass. Keep spinning until the water is fully spun up (at least thirty seconds for the glass we used you will need to test this out prior to using the demonstration).
  5. After the elapsed thirty seconds, stop the turntable abruptly.
  6. The tea leaves should move from the outer edge and settle in a heap in the center of the bottom of the glass.
  7. Alternatively, if you do not allow the water in the cup to fully spin up, when you stop it the tea leaves will form a circle at the edge of the secondary vortex (see analysis below).

CAUTION: If not attached well, the glass of water can slide off of the turntable when rotated.

The images below show the tea leaves location when, from left to right, the cup is being spun up, the cup is stopped having been fully spun up, and the cup has been stopped after partial spin up.

spinupspindownfullspindownpart

Analysis (qualitative)

When, starting from rest, the cup is initially spun, a boundary layer forms along the base of the cup. This drives the fluid in a circumferential direction. However, in the absence of any force to balance the resulting normal acceleration, the water in the boundary layer is driven radially outward. This drives the tea leaves to the edge of the cup. The radial outflow is then forced up the side of the cup, though the tea leaves stay in the corner at the base as they are denser than the water.

The vertical flow then turns back in toward the cup center and then down when it reaches the water surface. This creates a cylindrical vortex around the edge of the cup (see figure below). Inside the cylindrical vortex is a non-rotating core with a flat water surface.

partialup

Over time, the cylindrical vortex grows toward the center of the cup until there is no longer a non-rotating core and the water surface is curved all the way across (see figure below). At this point the flow is fully spun up and the tea leaves should still be at the corner of the cup.

fullup

When the cup is abruptly stopped, the water in contact with the base also stops moving. There is, therefore, no longer anything driving the flow radially outward. Instead, there is a hydrostatic pressure gradient toward the center of the cup due to the curved water surface (the water surface remains curved as all the fluid outside the boundary layer does not know the cup has stopped and is still rotating). Therefore, the flow in the bottom boundary layer reverses and the tea leaves are driven into the center of the cup (see figure below).

spindown

In the event that the cup is not fully spun up (step 7 in the procedure section), the hydrostatic pressure gradient only extends from the side of the cup to the edge of the cylindrical vortex (recall that the water surface in the non-rotating core is horizontal). Therefore, the lower boundary layer only flows radially inward to the edge of the cylindrical vortex. The tea leaves thus accumulate at the inner edge of the cylindrical vortex (see figure below).

partialdown

This is a remarkably robust experiment. It is almost impossible for it not to work (provided that the cup is secured to the center of the turntable). Thanks to Alex, and Meredith for putting together this write up and demonstration. Videos to follow soon.

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.

Videos of “Buoyancy – Balancing ping-pong balls”

Here are the videos of the “Buoyancy – Balancing ping-pong balls” demonstration. The full video is here. The GIF titles link to shorter videos. Obviously the YouTube videos from @veritasium here, here, and here have slightly higher production standards.

Setup 1

balance1 00_00_00-00_00_16~1

 

Setup 2

balance2 00_00_00-00_00_27~1

The payoff

balance3 00_00_01-00_00_06~1

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.