Re: Centrifugation
- From: "Jim Black" <tramspap@xxxxxxxxx>
- Date: 9 Feb 2007 23:38:35 -0800
On Feb 9, 9:29 pm, "qpwoeiruty" <dharring...@xxxxxxxxxxxx> wrote:
Synopsis: I don't understand the physics of centrifuges.
A few details:
Can someone spell out the details of how centrifugal force directed
toward the axis can pellet a particle AWAY from the axis?
Which direction does the "buoyant force" point, and why?
If a dense particle in water solution moves away from the axis of
rotation, does that increase or decrease the potential energy?
The wording of your question seems confused. A centrifugal force
(take the word apart -- the "centri" means "center" and the "fugal"
meens "fleeing") is a fictitious force *away* from the axis that must
be introduced in a rotating frame of reference.
Ignoring the "centrifugal," it sounds like you want an explanation in
terms of a non-rotating frame of reference, in which the sides of the
centrifuge exert an inward force on the liquid. Remember that no
force is needed to make the moving particle travel away from the axis;
it would go there of its own inertia. However, if you want the
particle to stay at the same distance from the center, you must exert
an inward force on the particle to make it travel in a circle.
There are forces on both sides of the particle, forces from the liquid
further from the center pushing the particle toward the center, and
forces on the liquid closer to the center pushing the particle away
from the center. (The outward force from liquid closer to the center
is a reaction to the force from the particle on the liquid holding it
in, without which that liquid would not stay in its circle.) By an
analysis similar to the analysis of the forces on a body immersed in a
non-rotating liquid, we can show that the net force on the particle
must be the same as the force needed to make an equal volume of liquid
travel in a circle. This would be your buoyant force, I think, if I'm
not abusing the terminology too badly. If the particle is more dense
than the liquid, the force on the particle is does not accelerate the
particle enough to make it travel in a circle. If the particle is
less dense, the force accelerates the particle too much, the particle
moves in an curve tighter than the circle, and it gets closer to the
center.
|\ More
\ dense
\ particle
\
\
|
|
|
|
Liquid |
-------<------- |
/ \ |
/ \ |
/ \ |
/ Less |\ \|
| dense \ |
| particle \ |
| \|
| |
| |
\ /
\ /
\ /
\ /
---------------
[View diagram in a fixed-width font such as Courier.]
Assuming we're still working in a non-rotating frame of reference,
there is no potential energy change when a dense particle moves to the
outside. But we know the that as the particle moves toward the
outside, there will be friction acting on the particle, and some
energy will be converted to heat. This energy comes from the kinetic
energy of the particle. When the centrifuge rotates, but the
particle's velocity doesn't rotate (enough to keep it in a circle),
the radial part of the particle's kinetic energy increases at the
expense of the component tangent to the circle -- not due to an actual
transfer of energy, but just a change in the way we define those
components. To maintain the circular component of the particle's
kinetic energy, a torque must be applied to the particle, which in the
end will sap energy from whatever is keeping the centrifuge at
constant angular velocity. (Note that there is also an increase in
the speed the particle has to travel at at a greater distance from the
center, which results in even more torque having to be applied to the
particle. But this extra kinetic energy is kept by the particle, and
hence doesn't contribute to the energy dissipated by friction.)
Radial /^
component / |
/ |
Circular \ |
component \ |
of velocity \|
|
|
|
|
|
Liquid |
-------<------- |
/ \ |
/ \ |
/ \
/ ^
| |
| | Initial
| | velocity
| | of particle
| |
\
\ /
\ /
\ /
---------------
Above, the circular component of the particle's velocity decreases
unless a torque on the particle makes up the difference. The
component of the kinetic energy corresponding to the radial component
of the velocity is converted into heat because of friction with the
liquid.
This sort of analysis can get complicated. That's why rotating frames
of reference with imaginary centrifugal and coriolis forces were
invented. Analyzing these sort of problems in a rotating frame of
reference is much easier.
--
Jim E. Black
.
- References:
- Centrifugation
- From: qpwoeiruty
- Centrifugation
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