Centripetal acceleration

[MDJH]

I was recently thinking about centripetal acceleration (a=vv/r) and I have a few questions about it...

 

 

1. Since gravity's variation with depth has a corresponding air pressure variation with elevation, I was wondering if spinning an enclosed container (with air in it) around in a circle such that the same part of it was always facing the centre (ie. similar to how the same face of the moon is always pointed towards the Earth) then if the stronger forces towards the circle's centre (ie. where the radius is closer) would push air enough to the part of the container further from the centre to create a pressure gradient within the container, in addition to whatever pressure gradient already would exist for height.

 

 

2. I was thinking about how to find centripetal acceleration as a function of frequency of circling, since speed would be difficult to directly measure. So I thought of the following formulae, where Circ means circumference, T means period of oscillation (as in, seconds for the object to go full circle)

 

Circ = 2*pi*r

 

v_circ = Circ / T

 

T = 1 / f

 

Therefore v = 2*pi*r*f

 

... and in turn...

 

a = (2*pi*r*f) * (2*pi*r*f) / r

 

a = 4 * (pi*pi) * r * (f*f)

 

Thus my derived formula suggests that acceleration is proportional to the radius, and proportional to the square of the frequency with which the object is rotated (as in, how many times it is spun full circle per second) with the coefficient being the product of 4 and the square of pi.

 

Are there any problems with my methods and/or results?

[swansont]

The equation is correct. [math]2{\pi}f[/math] is known as the angular frequency (measured in radians per second), so you'll see this written as ^{[math]\omega^2r[/math]}

[MDJH]

The equation is correct. [math]2{\pi}f[/math] is known as the angular frequency (measured in radians per second), so you'll see this written as ^{[math]\omega^2r[/math]}

Ah ok. So it's in units of angle per units of time, rather than in full circles per time... though I suppose "full circle" would qualify as a unit of angle anyway right? As if "full circle" were a unit of angle measure equivalent to 2pi radians?

 

Also, what about my pressure gradient question?

[swansont]

Right. 2 pi radians for a circle.

 

Your gradient question is similar to that of a centrifuge; where the more massive particles will tend to move out and you'll get greater pressure on the outside, but for the earth this is small effect compared to the gravitational force.

 

http://en.wikipedia.org/wiki/Centrifuge

http://en.wikipedia.org/wiki/Gas_centrifuge

[MDJH]

Your gradient question is similar to that of a centrifuge; where the more massive particles will tend to move out and you'll get greater pressure on the outside, but for the earth this is small effect compared to the gravitational force.

 

http://en.wikipedia.org/wiki/Centrifuge

http://en.wikipedia.org/wiki/Gas_centrifuge

Yeah, I've heard of "centrifuges" before; our chemistry labs used them to settle insoluble precipitates. I just wasn't sure if the idea of a centrifuge was applied to gases, and I guess it turns out it is.

 

Also, if someone were spinning some container (let's say a plastic bottle) that had one solvent substances on the "outer" end and a solute gas on the "inner" end (let's say filled with carbon dioxide to dissolve in water) would the centrifuge's pressure gradient force more CO2 to dissolve in the water?

[swansont]

It should, because you will have increased the pressure.

[MDJH]

What if it were a plastic bottle with the cap open, would this pressure gradient suck air in from the surroundings?

[swansont]

What if it were a plastic bottle with the cap open, would this pressure gradient suck air in from the surroundings?

 

A small amount, I expect. It would depend on the speed and radius.

[MDJH]

A small amount, I expect. It would depend on the speed and radius.

But would the average pressure inside the container be greater than the average pressure outside the container?

 

Let's say it was a small radius (ie. the bottle was about 20cm long, and the opening was about 2cm away from the centre of rotation) and a very fast speed (let's say a hundred metres per second) and this therefore created centripetal accelerations ranging from 45454m/s/s to 500000m/s/s; would the pressure in the bottle be significantly higher than external pressure?

[swansont]

Yes. You'd need an equation for pressure to calculate how much.

[MDJH]

... I just realized that if a plastic bottle is spinning around in a circle with one end always pointed at the centre, then that end would be moving at a different speed than the other end... am I correct in assuming this? Also, how do you find the speed at one end from the speed at the other end?

[swansont]

... I just realized that if a plastic bottle is spinning around in a circle with one end always pointed at the centre, then that end would be moving at a different speed than the other end... am I correct in assuming this? Also, how do you find the speed at one end from the speed at the other end?

 

The angular speed is a constant. The relation between linear speed and angular speed is what you figured out in the first post.