# ‘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.