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
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
- A small cork
- A marble
- A Lazy Susan or some other cheap turntable
- A mason jar
- Some Velcro strips or something else to secure the Mason jar to the turntable.
- Place the cork and marble in the jar and fill it with water
- 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).
- 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).
- Rapidly spin the turntable. The marble should be pushed to one end of the jar while the cork should remain centered.
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
See figure below.
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 (firstname.lastname@example.org). I also welcome comments (through the comments section or via email) on improving the demonstrations.