# Centrifugal force

The centrifugal force is a particular case of fictitious force which appears in Physique in the context of the motion study of the objects in noninertial reference frame S .

## Description

### Fictitious forces

In Mechanical Newtonian, the equation of the movement $\ vec \left\{F\right\} = Mr. \ vec \left\{has\right\}$ applies only in a inertial Référentiel. It is sometimes useful or simpler to deal with problem in reference frame which is noninertial.

When this choice is made, one can disregard noninertial character of the reference frame on the condition of adding additional force S in the problem. One then uses this equation but while including in the term of force of the additional forces which one consequently calls of the fictitious forces.

The effects of these fictitious forces are perfectly perceptible since the noninertial reference frame in the direction where they are precisely added so that the perception which an observer has of the movement of the objects since this reference frame is coherent with the Loi of Newton. Nevertheless, they should be distinguished from the other fundamental forces which are they independent of the reference frame.

### Particular case of the centrifugal force

The centrifugal force is a particular case of inertia of drive, which appears in reference frames in uniform rotation compared to a reference frame galiléen. It will be stressed that the object to which the reference frame is attached undergoes a “centripetal” acceleration (see Composition of the movements).

If one studies the movement of an object in a revolving reference frame, one can consequently use the equation F = m.a on the condition of adding, in particular, a centrifugal force like acting on the object.

If, moreover, since the turning reference frame, the object is perceived as with balance (= 0 have), then the centrifugal force is the only fictitious force which it is necessary to add. It is for example the case for reference frames attached to objects in rotation since if the reference frame is attached to the object, the object is perceived there in balance, since it is not perceived there moving. In the contrary case, it is advisable to add another fictitious force, the Force of Coriolis.

The expression of the centrifugal force to add is: $F = m.v^2/R$ where

m is the mass of the studied object;
v is the speed of the revolving reference frame;
R is the radius of curvature of the trajectory of the reference frame;
all measured since a single not-inertial reference frame.

## Demonstration of the expression of the centrifugal force

### Parameters

Let us consider a material point P of mass M turning around a presumedly fixed axis in a reference frame galiléen G. Its circular motion is made at constant speed ω.

As it is about a point it will not be necessary to seek the influence of the moments of force on the possible clean rotation of this point.

The study of such a movement is mathematically simplified, when one expresses the whole of the vectors (positions, speeds, accelerations and forces) in a reference mark turning with the point (what does not want to say that one is located in this reference frame). The reference mark thus chosen is such as:

• $\ vec \left\{COp\right\} = R. \ vec \left\{X\right\}$
• (O, $\ vec \left\{Z\right\}$) is the axis of rotation.

Under these conditions let us recall that: $\ frac \left\{D \ vec \left\{X\right\}\right\} \left\{dt\right\} = \ Omega \ vec \left\{there\right\}$; $\ frac \left\{D \ vec \left\{there\right\}\right\} \left\{dt\right\} = - \ Omega. \ vec \left\{X\right\}$ and $\ frac \left\{D \ vec \left\{Z\right\}\right\} \left\{dt\right\} = \ vec \left\{0\right\}$

Since the law of movement is given to us (position of P in the reference frame G) it is possible to calculate, by derivative successive, the speed then acceleration of the point for each position:

One obtains then: $\ vec \left\{V\right\} \left(P/G\right) =R \ Omega \ vec \left\{there\right\}$ and $\ vec \left\{has\right\} \left(P/G\right) =-R \ omega^2 \ vec \left\{X\right\}$

What can be also written: $\ vec \left\{has\right\} \left(P/G\right) = \left\{\ left|\ vec \left\{V\right\} \left(P/G\right) \ right|^2 \ over R\right\} \ vec \left\{X\right\}$

### Basic principle of dynamics

The basic principle of dynamics (“  law of Newton  ”) gives us, for any body moving in a Référentiel galiléen, a relation between acceleration and the external forces undergone: $\ vec \left\{F\right\} _ \left\{ext. \ to P\right\} = M \ vec \left\{has\right\} \left(P/G\right)$

Maybe after projection on the axis $\ displaystyle \left(O, \ vec \left\{X\right\}\right)$ :
(1) $\ displaystyle Fx_ \left\{ext. \ to P\right\} = - MR. R \ omega^2$

The study of each situation then makes it possible to quantify the term $\ displaystyle Fx_ \left\{ext. \ to P\right\}$ so that such a movement is possible. It will be noted that this term is negative on $\ vec \left\{X\right\}$; it is about the centripetal Force without which the movement would not be: Too much strong the turn is closed again, too weak it opens, null the mass by in straight line.

Note: $\ displaystyle Fx_ \left\{ext. \ to P\right\}$ gathers the whole of the external efforts without giving the detail of it. Moreover, this study is independent of the direction of the Gravitation. If the Poids intervenes in the expression $\ displaystyle Fx_ \left\{ext. \ to P\right\}$ (case of a horizontal axis for example) that means that there is at least another force which comes into play to ensure the balance (for example the tension then variable of the wire which would hold P).

### Centrifugal force

Now let us modify the equation on the left while placing all the terms of the sign equality, one obtains:
• (1) $Fx_ \left\{ext->P\right\} + MR. R \ omega^2 = 0$

Small reminder : in a reference frame galiléen, there is balance of a system (i.e. absence of movement), if the vectorial sum of the external forces which apply to him is null. The 3 conditions (reference frame, balance, null sum) are indissociable.

Let us return to our equation: the second term (no one) is connected in the condition summons forces null. Does it represent a balance? yes, that of P in the revolving reference mark (O, X, there, Z). In this case the expression $MR \ omega^2 \ vec \left\{X\right\}$ must be then regarded as an external force; it has of it already dimension. This quantity is called “centrifugal force” because its sign directs it towards outside, and that it contributes to the cancellation of the forces in the equation.

Thus one restores in an artificial way the triplet (reference frame, balance, equation), which should not in no case to be posted as a manifestation of the principle of dynamics since at least of the 3 conditions is not respected. This is why the centrifugal force is described as fictitious force.

## Examples

There exist cases where the centrifugal effect can be required, for example during the drying of the linen in a drum of machine; Inevitable for the systems in rotation, it can constitute a nuisance, for the passengers of a vehicle negotiating a change of management, one then has recourse to artifices to cancel, or rather to compensate for this effect: G-suit of the pilots of Fighter plan, pendular system of certain trains, corners banked of the roads.

### regulating for purpose centrifuges (on old steam engines)

The diagram given reproduced the principle of the regulator of James Watt. Actuated, via the belt, by the machine, the rotor sees its runners deviating. A linkage orders a valve then. The action on the valve has an opposite effect on the power provided to the machine: it is the principle of the Asservissement. Too much quickly one closes the vapor, too slow one opens, the system ending up finding right balance, and consequently a controlled mode.
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