Centripetal Force & Gas Centrifuges

[Jimmy87]

Hi,

I am studying circular motion at the moment and we were asked to go away and find an example of a device which has a very large centripetal acceleration. Initially I found a washing machine has up to about 300g but then I was looking at nuclear enrichment gas centrifuges which have huge accelerations. I can;t find a source that mentions the acceleration but I have found sources that tell you the information you need:

Radius of Centrifuge = 4 inches
Frequency of Rotation = 2000Hz

a = v²/r

v = 2πf

So, a = (2πf)²/r = (2π x 2000)²/0.1016 = 1.55 x 10⁹ ms^-2

So I come to my main question - what have I done wrong as this is way too high.

I also have a second question. When an object is undergoing uniform circular motion how do you interpret the acceleration. What I mean is that it is really easy to understand a car accelerating in a straight line because it tells you how many m/s his speed is increasing every second. For example, an acceleration for a car of 3m/s² that is not changing direction means that every second the speed is increasing by 3m/s. What does 3m/s² mean when the object is going round in a circular at constant speed though. How can 3m/s every second relate to a change in direction?

Thanks for any help,

 

[Orodruin]

v = 2πf

This is obviously false as the units do not match. The left-hand side has units of length/time and the right-hand side has units of 1/time.

What does 3m/s2 mean when the object is going round in a circular at constant speed though. How can 3m/s every second relate to a change in direction?

Acceleration is not the change in speed, which is a scalar. It is the change in velocity, which is a vector.

 

[Jimmy87]

This is obviously false as the units do not match. The left-hand side has units of length/time and the right-hand side has units of 1/time.

Acceleration is not the change in speed, which is a scalar. It is the change in velocity, which is a vector.

Thanks - how silly of me. So would it be:

So, a = (2πfr)²/r = (2π x 2000 x 0.1016)²/0.1016 = 1.6 x 10⁷ ms^-2

That still seems very high - is that plausible for a gas centrifuge?

About my second question - I think I wasn't clear with what I was asking. I know velocity is a vector and that the speed can remain constant whilst the object is accelerating (e.g. circular motion). However, how do you interpret the value of the acceleration if the speed is remaining constant? If the acceleration is 3m/s² for example and the speed is not changing then what does 3m/s per second actually mean? That the direction is changing by 3m/s per second? What does that even mean?

 

[Orodruin]

I am sorry, but I cannot understand how you can say this

I know velocity is a vector and that the speed can remain constant whilst the object is accelerating (e.g. circular motion).

and yet not be ok with this

However, how do you interpret the value of the acceleration if the speed is remaining constant?

Acceleration is change of velocity with time, if there is a non-zero acceleration, then the velocity will be different at a later time. If the difference in time is small, then
$$
\vec v_2 = \vec v_1 + \vec a (t_2-t_1).
$$
All that is being said is how the velocity changes with time for small times. Note that acceleration is also a vector, it has a direction - the direction in which the velocity changes.

 

[A.T.]

That the direction is changing by 3m/s per second? What does that even mean?

https://physics.stackexchange.com/a/193623

 

[Jimmy87]

https://physics.stackexchange.com/a/193623

Thanks. The second answer with the diagram is what I was looking for - thank you.

 

[Jimmy87]

I am sorry, but I cannot understand how you can say this

and yet not be ok with this

Acceleration is change of velocity with time, if there is a non-zero acceleration, then the velocity will be different at a later time. If the difference in time is small, then
$$
\vec v_2 = \vec v_1 + \vec a (t_2-t_1).
$$
All that is being said is how the velocity changes with time for small times. Note that acceleration is also a vector, it has a direction - the direction in which the velocity changes.

I am perfectly fine with the fact that an object can accelerate even though speed is constant. Changing direction requires a resultant force and F=ma tells us there must then be an acceleration. It is not this I am not ok with it is just interpreting the value of the acceleration in terms of changing direction only. It makes perfect sense to say that an acceleration of 3m/s² means that the speed increases by 3m/s every second. However, if the speed is constant then we are now saying that the direction changes by 3m/s every second. I can see it mathematically but can't picture what that means. Would that equation you gave work for acceleration at constant speed? The magnitude of your v₂ will be different to v₁ and I thought the magnitude is the speed and won't change in this example?

 

[jbriggs444]

The magnitude of your $v_2$ will be different to $v_1$

In uniform circular motion, the magnitude of $v_2$ and $v_1$ will be identical. But, because their directions are different, their vector difference will be non-zero. If the difference in angle between $v_1$ and $v_2$ is tiny, the vector difference can be visualized as the base of a tall, thin isoceles triangle with $v_1$ and $v_2$ as the equal sides and the vector difference as the remaining side.

 

[mfb]

That still seems very high - is that plausible for a gas centrifuge?

It is the right order of magnitude