# Gravitational force, centrifuge?

So after going through 5 pages on google on why and how a centrifuge separates out particles of different densities, I still don't quite get this.

If Galileos statement is right that force can be split into two or more directions and they don't have any effect on each other like a bullet fired in the horizontal direction will arrive on the floor at the same time as dropping it from the same height (if we don't consider other factors). So how is it that the higher the RPM, the quicker the sedimentation, if (in this case) the centrifugal acceleration in the "x" direction has no effect on the gravity in the y-direction?

So after going through 5 pages on google on why and how a centrifuge separates out particles of different densities, I still don't quite get this.

It is because of the centripetal force (aka radial force) in combination with the buoyancy. The centripetal acceleration of a particle is defined as $$a_\text{rad} = \omega^2 r$$, where $$\omega$$ is the angular velocity and $$r$$ is the distance from the center of rotation. Note that all rotating particles in a centrifuge device have the same angular velocity $$\omega$$.

So how is it that the higher the RPM, the quicker the sedimentation

In the context of buoyancy, the centrifugal force in a centrifuge device behaves much like the gravitational force in oceans, but its magnitude is significantly larger. The acceleration due to the centrifugal force can be up to 105 times larger than the acceleration due to the gravitational force $$g$$. For example, for angular velocity of $$\omega = 30000 \text{ rpm}$$ and distance $$r = 10 \text{ cm}$$ the resulting acceleration is approximately $$10^5 g$$. In order to fully understand this, you need to first understand the following principles:

• Uniform circular motion
• Difference between centripetal and centrifugal force
• Buoyancy
• Fluid friction (viscosity)

### Uniform circular motion

When a particle undergoes uniform circular motion, such as in the centrifuge, the net force acting on the particle points towards the center of rotation. This component of the net force is known as the centripetal force or the radial force.

Note that the centrifugal force does not exist in inertial reference frames. It is a pseudo-force introduced to make non-inertial reference frame (e.g. a rotating disc) to satisfy first and second Newton’s laws of motion. Being a pseudo-force, the centrifugal force does not have the reaction pair, which means it does not satisfy the third Newton's law of motion. The centripetal (radial) force itself is also not a force, but merely a component of the net (resultant) force which points in the direction of the center of rotation.

If you are interested in mathematical background of the (non)uniform circular motion, here is one of my earlier posts where I show how to derive expressions for the tangential acceleration and the centripetal acceleration: Angular velocity and banking angle

### Centripetal vs. centrifugal force

Imagine you are on a frictionless turntable which has a vertical wall along the outer rim. If you are initially at rest and stand away of the wall, you will stay at rest as seen from the ground frame even if the turntable starts rotating. This is because there is nothing to provide you with the centripetal force component necessary to rotate (change direction of motion). But if you lean against the wall at the outer rim, the wall will constantly push you towards the center of rotation and provide you the centripetal force component. As a result you will rotate together with the turntable.

Imagine you are leaned on the outer wall and initially you rotate together with the turntable. What happens if the wall suddenly disappears? At that moment the net force acting on you becomes zero and you lose the centripetal force component necessary to keep you rotating. When the net force is zero the object is said to be in equilibrium, and from the first Newton's law of motion it follows that you will keep moving at constant velocity in a straight line. In other words, you leave the turntable, and at the instant when that happens you lose the normal force that was keeping you on the turntable and start falling on the ground due to the gravitational force.

While you are rotating, from the turntable perspective you are actually at rest, i.e. in equilibrium in the context of first Newton's law of motion. Since the wall is pushing you towards the center, an observer on the turntable (non-inertial reference frame) concludes that there must be some force pushing you away from the center - this virtual force is called the centrifugal force, and the normal force by the wall is actually the centripetal force. From this it is clear that the centrifugal force is equal in magnitude and opposite in direction to the centripetal force.

Note that an observer on the ground (inertial reference frame) sees you rotating due to the normal force exerted on by the wall. Hence, the centrifugal force does not exist in inertial reference frame, while centripetal force is just a component of the net force acting on the body in the direction of center of rotation. Although the centrifugal force does not exist in reality, it is not wrong to use it for analysis as long as you are aware of what it really represents.

### Buoyancy

Buoyancy (upthrust) is a force that a fluid (liquid or gas) exerts on a submerged object in a direction opposite to the gravitational force. Buoyancy on the Earth (i.e. outside the centrifuge device) is defined as

$$B = \rho \cdot V \cdot 1g \tag 1$$

where $$\rho$$ is the fluid density, $$V$$ is the volume of the submerged part of the object, and $$1g$$ is the acceleration due to the gravitational force. This means that the object will float if the fluid is more dense and sink if the fluid is less dense than the object.

### Fluid friction (viscosity)

If there were only the gravitational force and the buoyancy, all particles (objects) would either sink or float, as already discussed. Then why submerged particles do not separate naturally? The larger (heavier) particles actually do separate, but smaller (lighter) particles stay trapped within the fluid due to the viscosity - the resistance fluid exerts in order not to change its shape.

With acceleration of only $$1g$$, the net force between gravitational force and buoyancy is not sufficient to overcome fluid viscosity. In the centrifuge the acceleration goes up to $$10^5 g$$ which gives particles enough net force to overcome fluid viscosity. As a result, the submerged particles will either completely sink or completely float.

The acceleration due to the centrifugal force is proportional to the angular velocity squared, as in $$a_\text{rad} = \omega^2 r$$. The higher the angular velocity the higher the acceleration and the net difference between centrifugal force and buoyancy is larger. It is exactly this net difference between two forces what separates particles in the fluid

$$F_\text{net} = (\rho - \rho_o) \cdot V_o \cdot a \tag 2$$

where $$\rho$$ is density of the fluid, $$\rho_o$$ and $$V_o$$ are density and volume of the submerged particle (object), and $$a$$ is the acceleration due to the gravitational force in oceans or the centrifugal force in the centrifuge device. When $$\rho > \rho_o$$ the buoyant force dominates and particles float, and when $$\rho < \rho_o$$ the centrifugal force dominates and particles sink.

From $$F_\text{net} = (\rho_o V_o) a_\text{net}$$ and the Eq. (2) it follows that particles with greater $$(1-\rho/\rho_o)$$ term will have greater net acceleration $$a_\text{net}$$ and will tend to sink faster. That is the main reason why you can see layers of different particles in the sediment on the bottom of the test tube.

• @Tomy Yes, particles feel as if pinned to the wall. That is actually the wall pushing them towards the center of rotation. The higher RPM, according to $a_\text{rad} = \omega^2 r$, the higher radial acceleration and the higher the force with which the wall pushes the particles towards the center of rotation. By Newton's third law of motion, the particle is pushing the wall with the same force in the opposite direction. Sorry about the edits, I will do one more.. Feel free to ask more questions! Mar 18, 2022 at 18:59
• Correct, gravitational acceleration by Earth $g$ is very small compared to accelerations provided by the centrifuge which can go up to $10^5 g$. Centripetal force is not in the x direction, but in the direction of the center of rotation - it is different for each particle. The less dense particles group closer towards the center of rotation because more dense particles exert buoyant force towards the center of rotation. Mar 18, 2022 at 19:17
• +1..@Tomy Yes, the wall is responsible for exerting the necessary force for circular motion, but all particles don't feel pinned to the wall because of their relative mass. The thing that is constant here for all particles is the angular acceleration. So for heavier particles to maintain the same angular acceleration as the lighter ones, they feel the need to increase the radius of their circle(as the force is not enough to keep them bound in a tighter circle), and similarly the lighter ones feel the appropriate force at a lower radius, thus rising to the top. Mar 18, 2022 at 19:21
• @Tomy Check my most recent edit. I added more information to clarify some principles. Mar 19, 2022 at 17:59
• @MarkoGulin Thank you so much, really appreciate this!!!
– Tomy
Mar 20, 2022 at 13:47

(Marko's answer covers the physics of the matter very well, so I won't address that. This is just a minor expansion.)

One thing to note is that there are two basic types of lab centrifuge: swinging bucket and fixed angle. In the case of the swinging bucket type the angle of the sample containers being centrifuged aligns with the combined forces acting on it, so the bottom of the test tube is always in line with the resultant 'down' vector. Of course when your centrifugal acceleration is on the order of 10,000g the down portion of the vector is pretty insignificant in comparison, so the combined force vector is very, very close to straight out.

Here's an example of a swinging bucket rotor: Grant Instruments R-6, for LMC-3000 low-speed centrifuge, rated up to 3,000 RPM with an RCF (relative centrifugal force) of 1,700 g. In a lot of cases having a properly aligned sediment direction is a good thing, but there are other situations where a fixed rotor wins for various reasons. They can handle higher centrifugal acceleration which is great when you're trying to separate proteins, DNA or RNA in your samples. And because of the design you can (usually) put more samples through a single run, greatly speeding up bulk processing.

Now for an interesting high capacity fixed angle rotor example: Thermo Scientific's Fiberlite F50L-24 x 1.5 Fixed-Angle Rotor, 100mm radius, rated for 50,000 rpm with an RCF of 280,000 g. Because the sample container in a fixed angle centrifuge doesn't line up with the force vector you end up with some of the sediment against the side of the tube. Here's a visible comparison between fixed rotor (left) and swinging bucket (right) sedimentation patterns.  You can also get some interesting swinging bucket configurations like the microtitre plates that so you can run lots of tiny samples at once. The LMC-3000 (linked above the swinging bucket image) has an option if you want to have a look... I'm not selling these things, I just went down the rabbit hole while looking for images.

(Much of the information and some of the images in this answer came from laboratory-equipment.com, which I am also not affiliated with. There's a button on that page labelled Swinging Bucket vs. Fixed Angle Rotors where you'll find the discussion.)

• specs (RPM, diameter) of the devices in the images would help.
– JEB
Mar 19, 2022 at 20:03
• @JEB Added some numbers and links for the pictured rotors. Hope that helps. Mar 20, 2022 at 0:24

Because a 26,000 RPM bench top centrifuge pulls 50,000 g's, versus Earth's puny 1.

It sounds like you're thinking that the tubes in a centrifuge are oriented vertically, and so you're wondering how the centripetal acceleration (which is a horizontal force) causes sedimentation (which is a vertical motion).

Actually, the tubes in a centrifuge are at an angle, and nearly horizontal. The lighter particles don't end up on top; they end up in the middle (near the center of the centrifuge).

• Yeah thats how I initially imagined it...Half way through Markos answer I realised they can't be vertically oriented....
– Tomy
Mar 20, 2022 at 13:49