CENTRIFUGAL FORCE: FORMULAS, HOW IT IS CALCULATED, EXAMPLES, EXERCISES - PHYSICAL - 2025

The centrifugal force tends to push out the rotating bodies taking a curve. It is considered a fictitious force, pseudoforce or inertial force, because it is not caused by interactions between real objects, but is a manifestation of the inertia of bodies. Inertia is the property that makes objects want to retain their state of rest or of uniform rectilinear motion, if they have one.

The term "centrifugal force" was coined by the scientist Christian Huygens (1629-1695). He claimed that the curvilinear motion of the planets would tend to move them away unless the Sun exerts some force to hold them back, and he calculated that this force was proportional to the square of the velocity and inversely proportional to the radius of the described circumference.

Figure 1. When cornering, passengers experience a force that tends to pull them out of it. Source: Libreshot.

For those who travel by car, centrifugal force is not fictional at all. Passengers in a car that turns right feel pushed to the left, and vice versa, when the car turns left, people experience a force to the right, which seems to want to move them away from the center of the curve.

The magnitude of the centrifugal force F _g is calculated by the following expression:

- F _g is the magnitude of the centrifugal force

- m is the mass of the object

- v is the speed

- R is the radius of the curved path.

Force is a vector, therefore bold type is used to distinguish it from its magnitude, which is a scalar.

It must always be borne in mind that F _g appears only when the movement is described using an accelerated reference system.

In the example described at the beginning, the spinning car constitutes an accelerated reference, since it requires centripetal acceleration so that it can turn around.

How is the centrifugal force calculated?

The choice of the reference system is vital for the appreciation of the movement. An accelerated reference frame is also known as a non-inertial frame.

In this type of system, such as a spinning car, fictitious forces such as centrifugal force appear, the origin of which is not a real interaction between objects. A passenger cannot say what pushes him out of the curve, he can only affirm that this is the case.

On the other hand, in an inertial reference system, the interactions occur between real objects, such as the moving body and the Earth, which gives rise to weight, or between the body and the surface on which it moves, which originate friction and normal.

An observer standing on the side of the road and watching the car turn the curve is a good example of an inertial reference system. For this observer, the car turns because a force directed towards the center of the curve acts on it, which forces it not to get out of it. This is the centripetal force produced by friction between the tires and the pavement.

In an inertial reference frame, the centrifugal force does not appear. Therefore the first step in calculating it is to carefully choose the reference system that will be used to describe the movement.

Finally, it should be noted that the inertial reference systems do not necessarily have to be at rest, like the observer watching the vehicle turn the curve. An inertial reference frame, known as a laboratory reference frame, can also be in motion. Of course, with constant speed with respect to an inertial one.

Free-body diagram in an inertial and non-inertial system

In the next figure to the left, an observer O is standing and looking at O ​​', who is on the platform that rotates in the indicated direction. For O, which is an inertial frame, certainly O 'is kept rotating due to the centripetal force F _c produced by the wall of the grid on the back of O'.

Figure 2. A person standing on a turntable is seen from two different reference systems: one fixed and the other that goes with the person. Source: Física de Santillana.

Only in inertial reference frames is it valid to apply Newton's second law, which states that the net force is equal to the product of the mass and the acceleration. And in doing so, with the free-body diagram shown, we get:

Similarly, in the figure on the right there is also a free-body diagram that describes what the observer O 'sees. From his point of view, he is at rest, therefore the forces on him are balanced.

These forces are: the normal F, that the wall exerts on it, in red and directed towards the center and the centrifugal force F _g that pushes it outwards and which does not originate from any interaction, is a non-inertial force that appears in rotating reference systems.

Centrifugal force being fictitious, it is balanced by a real force, the contact or normal force that points towards the center. Thus:

Examples

Although centrifugal force is considered a pseudo force, its effects are quite real, as can be seen in the following examples:

- In any spinning game in an amusement park, centrifugal force is present. She ensures that we “run away from the center” and offers constant resistance if you try to walk into the center of a moving carousel. In the following pendulum you can see the centrifugal force:

- The Coriolis effect arises from the Earth's rotation, which makes the Earth stop being an inertial frame. Then the Coriolis force appears, which is a pseudo-force that deflects objects laterally, as happens with people trying to walk on a turntable.

Exercises

Exercise 1

A car turning with acceleration A to the right has a stuffed toy hanging from the inside rearview mirror. Draw and compare the free-body diagrams of the toy seen from:

a) The inertial frame of reference of an observer standing on the road.

b) A passenger traveling in the car.

Solution to

An observer standing on the road notices that the toy is moving rapidly, with acceleration A to the right.

Figure 3. Free-body diagram for exercise 1a. Source: F. Zapata.

There are two forces acting on the toy: on the one hand the tension in the string T and the vertical weight down W. The weight is balanced with the vertical component of the tension Tcosθ, therefore:

The horizontal component of stress: T. sinθ is the unbalanced force responsible for the acceleration to the right, therefore the centripetal force is:

Solution b

For a passenger in the car, the toy hangs in equilibrium and the diagram is as follows:

Figure 4. Free-body diagram for exercise 1b. Source: F. Zapata.

As in the previous case, the weight and the vertical component of the tension are compensated. But the horizontal component is balanced by the fictitious force F _g = mA, such that:

Exercise 2

A coin is on the edge of an old vinyl record player, the radius of which is 15 cm and it is rotating at 33 revolutions / minute. Find the minimum coefficient of static friction necessary for the coin to stay in place, using the frame of reference solidarity with the coin.

Solution

In the figure is the free-body diagram for an observer moving with the coin. The normal N that the turntable exerts vertically up is balanced by the weight W, while the centrifugal force F _g is compensated by the static _friction F _friction.

Figure 5. Free body diagram for exercise 2. Source: F. Zapata.

The magnitude of the centrifugal force is mv ² / R, as said at the beginning, then:

On the other hand, the static friction force is given by:

Where μ _s is the coefficient of static friction, a dimensionless quantity whose value depends on how the surfaces are in contact. Substituting this equation is:

The magnitude of the normal remains to be determined, which is related to weight according to N = mg. Substituting again:

Back to the statement, it reports that the coin rotates at a rate of 33 revolutions / minute, which is the angular velocity or angular frequency ω, related to the linear speed v:

The results of this exercise would have been the same if an inertial reference frame had been selected. In such a case, the only force capable of causing acceleration towards the center is static friction.

Applications

As we have been saying, the centrifugal force is a fictitious force, which does not appear in inertial frames, which are the only ones in which Newton's laws are valid. In them, the centripetal force is responsible for providing the body with the necessary acceleration towards the center.

The centripetal force is not a different force from those already known. On the contrary, it is precisely these that play the role of centripetal forces when appropriate. For example, the gravity that makes the Moon orbit around the Earth, the tension in a rope by which a stone is rotated, static friction and electrostatic force.

However, as accelerated reference frames abound in practice, fictitious forces have very real effects. For example, here are three important applications where they have tangible effects:

Centrifuges

Centrifuges are instruments widely used in the laboratory. The idea is to make a mixture of substances rotate at high speed and those substances with greater mass experience a greater centrifugal force, according to the equation described at the beginning.

Then the most massive particles will tend to move away from the axis of rotation, thus being separated from the lighter ones, which will remain closer to the center.

Washing machines

Automatic washers have different spin cycles. In them, the clothes are centrifuged to eliminate the remaining water. The higher the revolutions of the cycle, the less wet the clothes will be at the end of the wash.

The cant of curves

Cars are better at cornering on roads, because the track slopes slightly towards the center of the curve, known as cant. In this way, the car does not depend exclusively on the static friction between the tires and the road to complete the turn without leaving the curve.

References

1. Acosta, Victor. Construction of a didactic guide on centrifugal force for students in cycle V grade 10. Retrieved from: bdigital.unal.edu.co.
2. Toppr. Laws of Motion: Circular Motion. Recovered from: toppr.com.
3. Resnick, R. (1999). Physical. Vol. 1. 3rd Ed. In Spanish. Compañía Editorial Continental SA de CV
4. Autonomous University of the State of Hidalgo. Centrifugal force. Recovered from: uaeh.edu.mx
5. Wikipedia. Centrifuges. Recovered from: es.wikipedia.org.