Centrifugal Force

Why is it a Pseudo Force?

But then if we are able to see the effects of this phenomenon every day, then why is it called a pseudo force? To understand this we will need to introduce another force - but this one acts towards the center of the circle and is real.

Any physical object that has a mass and is rotating about a point will require a pulling force towards the center of the rotation. Without this force, the object will move in a straight line. In order for an object to move in a circle, it must have a force. This is called the centripetal force requirement. An inward-directed acceleration necessitates the application of an internal push. Without this inward force, an object would continue to move on a straight line parallel to the circle's circumference.

The circular motion would be impossible without this inward or centripetal force. The centrifugal force acts simply as a reaction to this centripetal force. This is why centrifugal force is defined as a sensation that throws objects away from the center of rotation. This can also be attributed to the inertia of an object. In an earlier example, we spoke about how passengers are thrown in the opposite direction when a moving vehicle makes a turn. This is basically the passenger's body resisting a change in their direction of motion. Let us look at this mathematically.

Centrifugal Force Equation

Because centrifugal force is a pseudo force or sensation. we will first need to derive the equation for centripetal force. Remember both these forces are equal in magnitude but opposite in direction.

Imagine a stone tied to a string that is being rotated uniform speed. Let the length of the string be \(r\), which makes it also the radius of the circular path. Now take a picture of this stone that is being rotated. What's interesting to note is that the magnitude of the tangential velocity of the stone will be constant at all points on the circular path. However, the direction of tangential velocity will keep changing. So what is this tangential velocity?

The tangential velocity vector will point towards the tangent of the circular path followed by the stone. As the stone is being rotated this tangential velocity vector is constantly changing its direction.

And what does it mean when the velocity keeps on changing; the stone is accelerating! Now according to Newton's first law of motion, an object will continue to move in a straight line unless an external force acts on it. But what is this force that is making the stone move around in a circular path? You might recall when you spin the stone you're basically pulling the string, creating tension that produces a pulling force on the stone. This is the force that is responsible for accelerating the stone around the circular path. And this force is known as Centripetal force.

The magnitude of a centripetal force or radial force is given by newtons second law of motion: $$\overset\rightharpoonup{F_c}=m\overset\rightharpoonup{a_r},$$

where \(F_c\) is the centripetal force, \(m\) is the mass of the object and \(a_r\) is the radial acceleration.

Every object moving in a circle has radial acceleration. This radial acceleration can be represented as: $$\overset\rightharpoonup{a_r}=\frac{V^2}r,$$

where \(a_r\) is the radial acceleration, \(V\) is the tangential velocity and \(r\) is the radius of the circular path.

combining this with the equation for centripetal force and we get; $$\overset\rightharpoonup{F_c}=\frac{mV^2}r$$

The tangential velocity can also be represented as :$$V=r\omega$$

$$\mathrm{Tangential}\;\mathrm{velocity}\operatorname{= }\mathrm{angular}\;\mathrm{velocity}\times\mathrm{radius}\;\mathrm{of}\;\mathrm{circular}\;\mathrm{path}$$

This gives another equation for centripetal force as: $$\overset\rightharpoonup{F_c}=mr\omega^2$$

But wait, there's more! According to Newton's third law of motion, every action will have an equal and opposite reaction. So then what could possibly act in the opposite direction of centripetal force. This is nothing but centrifugal force. Centrifugal force is called a pseudo force because it only exists due to the action of centripetal force. The centrifugal force will have a magnitude equal to that of the centripetal force in the opposite direction, which means the equation for calculating the centrifugal force is also:

$$\overset\rightharpoonup{F_c}=mr\omega^2$$

where mass is measured in \(\mathrm{kg}\), radius in \(\mathrm{m}\) and \(\omega\) in \(\text{radians}/\text{sec}\). Let's now use these equations in a few examples.

Centrifugal force examples

Here we will go through an example in which we will apply the principles of centrifugal force.

Relative Centrifugal Force Units and Definition

We spoke about how centrifugal force can be used to create artificial gravity. Well, we can also represent the centrifugal force generated by a spinning object relative to the amount of gravity we experience on earth

RCF is expressed as units of gravity, \(\mathrm{G}\). This unit is used in the process of centrifugation instead of just using RPM as it also accounts for the distance from the center of rotation. It is given by the following equation. $$\text{RCF}=11.18\times r\times\left(\frac{\text{RPM}}{1000}\right)$$ $$\text{Relative}\;\text{Centrifugal}\;\text{Force}=11.18\times\mathrm r\times\left(\frac{\text{Revolutions}\;\text{Per}\;\text{Minute}}{1000}\right)^2$$

you might wonder why force is expressed in units of gravity, well as you know the unit of gravity actually measures acceleration. When RCF experienced by an object is \(3\;\mathrm g\) , it means that the force is equivalent to three times the force experienced by an object free falling at a rate of \(g\;=\;9.81\;\mathrm{m/s^2}\).

This brings us to the end of this article. Let's look at what we've learned so far.