Does Bernoulli's Principle Apply to Aerodynamics at Different Heights?

[SamJ96]

Homework Statement

Hi everyone. I have a ping pong levitation system with a ping pong ball in a tube being pushed by a dc motor fan. Can anyone describe the force effect of a slit on the side of the tube on the pin pong ball? I know there is a drag force applied by the fan going against the balls weight, but I don't know how the slit effects the motion. Is it a decrease in pressure? Can anyone quantify it within a force diagram? The setup and FBD i have currently are attached. Cheers

Relevant Equations

F=m*a[ball]=-m*g+0.5*Cd*p*A*(v[air]-v[ball])-F[slit]
What is F[slit]?

 

[Baluncore]

Welcome to PF.

Can anyone describe the force effect of a slit on the side of the tube on the pin pong ball? I know there is a drag force applied by the fan going against the balls weight, but I don't know how the slit effects the motion. Is it a decrease in pressure?

The pressure under the ball, and the flow past the ball, support the ball. For a fixed size ball in a fixed size tube, that pressure, flow and force, should be a constant no matter what height the ball 'floats' in the tube.

As the ball rises, more of the slot is uncovered, so more air can escape through the slot, before reaching the ball.

The height of the ball is therefore a function of the volume of air being blown into the tube by the fan.

 

[SamJ96]

Hi Baluncore,
Thank you very much for your answer. Just wondering, doesn't the static pressure decrease based on Bernoulli's principle as the the potential energy increases with height? Also, is the function that relates the balls height with the air volume blown into the tube different from the force equation that I showed, or is my force equation completely wrong? Thank you once again.

 

[Baluncore]

Just wondering, doesn't the static pressure decrease based on Bernoulli's principle as the the potential energy increases with height?

The air is not being significantly compressed, and can be taken as the same temperature throughout the experiment. PE of hydrostatic head is not significant when the fluid has buoyancy equal to the surrounding atmosphere.