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Centrifuges


old man emu
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Least you think you have dragged me to the Dark Side as a believer in a thing called centrifugal force, you haven't. But that does not prevent me from calling a machine that spins at very high RPM a "centrifuge". That is because the word "centrifuge" is more or less a trade name, or else a name made up by the inventor of something that has never existed before. Why is a throwing stick that can return to its starting point called a "boomerang"?  A young Veronese girl expressed it well, "A rose, by any other name would smell as sweet".

 

In 1659 the Dutch mathematician and scientist Christiaan Huygens created the term “centrifugal force” in his work “De vi centrifuga”, and that is when the rot started.

https://www.princeton.edu/~hos/mike/texts/huygens/centriforce/huyforce.htm  Since then this term centrifugal force has been misinterpreted. Huygens was too early on the scene to apply vector analysis which was not discovered until the 1880s by Josiah Willard Gibbs, of Yale University. Gibbs's lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into a textbook, Vector Analysis, published in 1901.

 

However he was able to associate circular velocity to weight, and I use weight as F=ma. Here is his idea:

PROPOSITION VI

Given the height that a moving body traverses in a certain time, say a second, in falling perpendicularly from rest, to find the circle in the circumference of which a body moving around horizontally and completing its revolution also in a second has a centrifugal force equal to its weight.

 

Now go way back in this thread to where I was talking about the relationship between centripetal force and inertia of the object. If you put an object down on a surface, Newton tells us that it has inertia, something that has to be overcome by a force to change the velocity of the object. As it sits on the surface, the object has an amount of inertia equal to its mass multiplied by the acceleration due to gravity, ma. Now, tie a string to the object and pull on the string towards you. The string will become straight and the pull is the centripetal force which we call "Tension" in this case. The object will not move until the Tension is greater than the inertia of the object. 

 

From our experience we know that as we start to swing the weight in a circle we also lift it off the surface. Now the Tension has two jobs - to keep the object on a circular path, and also to counter the weight force of the object. This can be represented in this vector diagram:

image.thumb.png.0eaf4b3767588dee16a6b0bfca0c4fdb.png

Where

T = Total centripetal force vector

Tx = Horizontal component vector of T - which has a direction towards the centre of the circle

Ty = Vertical component vector of T - which has a direction at 90 degrees to Tx

mg = Total weight force vector

 

In order to keep the object circling at the same height above the surface, Ty  must equal mg. Tx  equals the centripetal force. Therefore, to keep the object circling, the total tension required to keep the object circling must be greater that the amount of tension to simply overcome the object's inertia when it is no moving plus the tension to move the object. The total tension is the Pythagorean sum of the horizontal and vertical vectors (think wind slide on your whizz wheel) This is all summarised here

image.thumb.png.bc5159562ce3f32566508cf08316bb8f.png

 

T=  {\displaystyle F_{c}=ma_{c}={\frac {mv^{2}}{r}}} which means that if the mass of an object and the radius of the circle remain constant, then the magnitude of Tx is dependant only on the velocity of the object. Therefore, as the velocity increases, the ratio of the vectors Tand Ty changes, so that the ratio TyTx approaches 0:1. That means that Tx approximates ma and we can ignore the contribution of Ty. However, if the velocity and radius remain constant, the magnitude of Tx depends on the mass of the object. What's this to do with centrifuge machines? It's the physics behind the way they work.


Let's have a solution of various objects. Blood is a good example. It is a solution of various types of cells in a medium called plasma. Each cell has a different weight. When the centrifuge machine is spun up it reaches a velocity RPM) set by the operator of the machine. That takes care of the v2  in the equation. The blood is put into vials which have a length. That takes care of the r in the equation. When the centrifuge spins, the centripetal force, which acts towards the centre of rotation, that is applied to the components of the blood, depends only on their masses. Light objects have a smaller force, so remain closer to the centre, and so on with the heaviest objects being located at the greatest extent of possible radii.

 

 

So the whole centripetal - centrifugal argument can be sheeted home to Huygens who didn't know anything of vector analysis.

 

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