Problem no. 7 "Merry-go-round"

Description of the experiment:

We filled a glass with water, with r = 8 cm and h = 20 cm. The ball was made of tin foil filled with a small amount of air and a very small piece of a magnetic material. With a big magnet we could hold the ball down on the bottom of the vessel. In our case the rotating disks were turntables.

angular velocity = 0 angular velocity = pi s-1 angular velocity = 1,5*pi s-1
Ball 14 s15 s30 s
Ball 28 s30 s60 s

Here we have the time the ball needs to reach the surface at different angular velocities. We can see that the velocity of the ball on the rotating disk is nearly 10 times slower than on the stationary disk.

Theory (without rotation):

First we considered all the acting forces:
Acting forces without rotation

The forces which act on the ball are the gravitational force, the buoyancy force and the friction force. There are two possibilities for the friction force: the flow resistance force or Stoke's friction force.
Trying to find out which one we have to use we calculated the Reynolds's number. The calculated Reynolds number was approximately 150. So the flow pattern is definitely laminar. In a laminar flow the flow resistance is characterised by Stoke's law.
By using Newton's law we summed up all the acting forces and set them equal to mass times acceleration.
We reworded the formula so that the velocity is now shown explicit.

formula for the velocity of the ball

You can see that for a very large, nearly infinite time t, the velocity becomes constant because the exponential function converges to 0.
So the formula for this limit speed is
formula for the limit speed

In order to keep the limit speed small we had three possibilities:

Theory (with rotation):

When the liquid is rotating the same forces act. There is, however, one new force acting which is the centripetal force. This force makes the ball moving towards the axis of rotation.

froces on the ball (with rotation)

Pressure distribution:

When the liquid is rotating the former plane isobarics become parabolic.

pressure distribution

The mentioned parabolic isobarics are the explanation for the observed phenomenon that the ball is slower when the liquid is rotating.
We want to explain you our final solution:

final explanation

The red line in the centre represents the diameter d of the ball. The green lines symbolise the isobarics in the non-rotating liquid. In the non-rotating case we have a pressure difference delta p on the height difference d which is the diameter of the ball. When the liquid is rotating and the isobarics become parabolic the vertical distance between the isobarics increases everywhere except in the axis of rotation.
So we have a pressure difference lower than delta p on the vertical distance d. With a lower pressure difference in vertical direction we get a lower buoyancy force in vertical direction which causes a lower upward speed.

In order to prove this theory we made the following experiment: We put a glass on a disk and put a ball on a rope inside. We led the rope across 2 rollers and fixed a weight on the other side which was standing on electronic scales. Then we started to rotate the disk with the glass on it. By increasing the angular velocity the weight shown by the electronic scales decreased, because the buoyancy force on the ball decreased, as predicted by our theory. So the ball "lifted" the weight on the scales.

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Sketch of the experimental setup
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Photo of the experimental setup
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