Centrifugal force

The centrifugal force pulls the passengers of a rotating chain carousel outwards.

The centrifugal force (from the Latin centrum , middle and fugere , to flee), also centrifugal force, is an inertial force that occurs during rotary and circular movements and is directed radially outwards from the axis of rotation . It is caused by the inertia of the body. The effects of centrifugal force can be experienced in many ways in everyday life, for example when the seats are pushed outward in the chain carousel , the water is thrown outward in the salad spinner or a two-wheeler has to " lean into the curve" .

In classical mechanics , centrifugal force describes ...

• ... the resistance that the body, according to the principle of inertia, opposes to the change in its direction of movement when it follows a curved path. The centrifugal force is always the opposite of the centripetal force that causes this change in direction of movement. As d'Alembert's inertial force , the centrifugal force is in dynamic equilibrium with the centripetal force .
• ... a force that must always be taken into account when describing the movement of a body in relation to a rotating frame of reference . This inertial force also occurs in the absence of a centripetal force, but never in an inertial system . The centrifugal force results from the centrifugal acceleration multiplied by the mass.

The centrifugal force is an apparent force and therefore does not satisfy the principle of actio and reaction .

history

A qualitative description of centrifugal force can be found in the Principles of Philosophy by René Descartes , published in 1644 . Quantitatively, it was derived for the first time in 1669 in a letter from Christian Huygens to the Secretary of the Royal Society Henry Oldenbourg , also mentioned without derivation in Huygens' Horologium Oscillatorium of 1673 and in detail in his posthumous pamphlet from 1659 De Vis Centrifuga (published 1703). Huygens obtained the formula from considering how a stone loosened by the slingshot moves away from the continuation of the circular motion in an initially square manner. The concept of centrifugal force is thus even older than that of general mass attraction , which was made known by Isaac Newton in 1686 . Newton also described centrifugal force, but only after Huygens and independently of him. From the observation that the centrifugal force in a rotating bucket deforms the surface of the water , Newton concluded that a rotational movement can be determined absolutely, and therefore an absolute space must exist.

After a long period of uncertainty about the origin of centrifugal force, Daniel Bernoulli recognized in 1746 that it was not an original fact inherent in nature, but that it depends on the choice of the reference system used to describe it. Shortly before, Jean-Baptiste le Rond d'Alembert had formulated the general concept of inertia as the negative of the product of the body's mass and acceleration. Shortly thereafter, Leonhard Euler recognized the general concept of inertial forces in an accelerated frame of reference.

Inertial resistance

D'Alembert inertial force

If the center of gravity of a body with its mass describes a curved path in an inertial system , a force is required which has a component perpendicular to the path curve. This component is called the centripetal force . According to Newton's second law , it causes a centripetal acceleration proportional to it , which is directed towards the (current) center of curvature of the path: ${\displaystyle m}$ ${\displaystyle {\vec {F}}_{\text{Zp}}}$ ${\displaystyle {\vec {a}}_{\text{Zp}}}$

${\displaystyle {\vec {F}}_{\text{Zp}}=m{\vec {a}}_{\text{Zp}}}$

According to d'Alembert , this basic equation of mechanics is written in the form

${\displaystyle {\vec {F}}_{\text{Zp}}-m{\vec {a}}_{\text{Zp}}={\vec {0}}}$

and takes the second term formally as a force. This force is known as centrifugal force . It is an inertial force, more precisely a d'Alembert inertial force . It applies ${\displaystyle F_{\text{Zf}}}$

${\displaystyle {\vec {F}}_{\text{Zp}}+{\vec {F}}_{\text{Zf}}={\vec {0}}}$

and therefore

${\displaystyle {\vec {F}}_{\text{Zf}}=-{\vec {F}}_{\text{Zp}}}$.

The centrifugal force is always the opposite of the centripetal force.

Inertial resistance quantifies a property of inertia, which is expressed by the way in which a body opposes the change in an existing movement through an inertial force (“vis inertiae”).

The centrifugal force in the d'Alembertian sense always presupposes the action of a centripetal force. However, together with the centripetal force, it does not form a force couple in the sense of “ actio and reaction ”, because both forces act on the same body. Nevertheless, the centrifugal force is referred to in some texts as a "counterforce" or "reaction force" to the centripetal force.

The centrifugal force, defined as d'Alembert's inertial force for centripetal acceleration, corresponds in direction, strength and point of application to the centrifugal force, as it is calculated as an apparent force in a reference system that rotates around the center of curvature and in which the body rests (see below).

Formulas

For a circular path, the centrifugal force is directed radially outwards from the center point. Its strength can be calculated using the mass , the radius of the circle and the orbital velocity using the same formula as the centripetal force. The following applies (for the derivation see centripetal force # Mathematical derivation ): ${\displaystyle F_{\text{Zf}}}$ ${\displaystyle m}$ ${\displaystyle r}$ ${\displaystyle v}$

${\displaystyle F_{\text{Zf}}=m\,{\frac {v^{2}}{r}}}$

This equation is very general, even if a body traverses an arbitrarily curved path. Here, the radius of curvature of the radius of the circle which conforms to the respective location of the body to its web. ${\displaystyle r}$

The circular movement can also be understood as a rotation around the center of curvature with the angular velocity . The path speed depends on the angular speed and the radius of the circle ${\displaystyle \omega }$ ${\displaystyle v}$

${\displaystyle v=\omega \,r}$.

Therefore, the centrifugal force can also be given as a function of the angular velocity :

${\displaystyle F_{\text{Zf}}=m\,\omega ^{2}\,r}$

Everyday experiences

Playground carousel

• Practical experience teaches that you have to hold on to a carousel in order not to be thrown off. This is usually explained by the centrifugal force. Another equivalent explanation would argue that circular motion requires a centripetal force that can also be felt on the hand.

• The inclination of the chain on a chain carousel is generally attributed to centrifugal force. The resultant of weight and centrifugal force points in the direction of the chain. A not-so-common explanation of the process argues that a centripetal force is required for circular motion. This is caused by the inclined position of the chain and provides the necessary force in the direction of the axis of rotation. Both explanations are equivalent, since centripetal force and centrifugal force are oppositely equal.

• The widespread idea that one would be “carried out” of the curve because the centrifugal force is greater than the centripetal force does not apply. However, the centrifugal force that would occur when driving through the intended curve is greater than the maximum centripetal force that can be transmitted from the roadway to the vehicle. The actually acting centripetal force is then not sufficient to bring about the change in the direction of movement along a circular path at the given speed . Example: The static friction of the car tires is insufficient.

• A driver with a mass of 70 kg ( 700 N) drives at 15 m / s (54 km / h) through a right-hand bend with a radius of 75 m. The centripetal force is then ${\displaystyle mg\approx }$

${\displaystyle F_{\text{Zp}}=70\;\mathrm {kg} {\frac {({15\;\mathrm {m/s} )}^{2}}{75\;\mathrm {m} }}=70\;\mathrm {kg} \cdot 3\;\mathrm {m/s^{2}} =210\,\mathrm {N} .}$

The centripetal force acts on the driver from the left and forces him out of his initially straight-line inertial movement into the curved path, just so that he maintains his position in the car. In this example the force has a strength of approx. 30% of the weight force ( 700 N). It is exerted on the driver from the driver's seat and he feels it because he feels himself pressed into the seat by a force. This centrifugal force has the same magnitude, but the opposite direction as the centripetal force. ${\displaystyle mg\approx }$

However, since the centrifugal force is a volume force as a result of acceleration, the points of application of centripetal force and centrifugal force are different. Because of the different character of the forces, they are also perceived differently . In common parlance, the term centrifugal force is therefore used more often when the effect on the entire body is to be described. For fighter pilots, staying in human centrifuges is part of their training.

Reference system dependent apparent forces

Generally accelerated frame of reference

The sparks from an angle grinder fly in a straight line. For an observer who is on the disk and rotates with it, the sparks would follow a helical path, as shown in the picture below.

A "released object" in the rotating frame of reference describes an involute of a circle.

Apparent forces must always be taken into account when setting up an equation of motion in a reference system that is itself accelerated compared to the inertial system. If one considers z. B. the sparks that come loose from a grinding wheel in an inertial system, they move in a straight line because they are force-free. In the rotating reference system of the grinding wheel, however, the relative acceleration of the particles is explained by an apparent force.

Equation of motion

In order to differentiate between the sizes of an object (location, speed, acceleration) in two reference systems, the normal notation in the inertial system is used and the non-inertial reference system is given the same letter with an apostrophe ( prime ). The latter is then also referred to as the “deleted reference system”.

	meaning
${\displaystyle {\vec {r}}}$	Position of the object in S (inertial system).
${\displaystyle {\vec {r}}\,'}$	Relative position of the object in S '(non-inertial system).
${\displaystyle {\vec {v}}={\dot {\vec {r}}}}$	Speed ​​of the object in S
${\displaystyle {\vec {v}}\,'}$	Relative speed of the object in S '
${\displaystyle {\vec {a}}={\dot {\vec {v}}}}$	Acceleration of the object in S
${\displaystyle {\vec {a}}\,'}$	Relative acceleration of the object in S '
${\displaystyle {\vec {r}}_{B}}$	Position of the origin of S 'in S
${\displaystyle {\vec {v}}_{B}={\dot {\vec {r}}}_{B}}$	Velocity of the origin of S 'in S
${\displaystyle {\vec {a}}_{B}={\dot {\vec {v}}}_{B}}$	Accelerating the origin of S 'in S
${\displaystyle {\vec {\omega }}}$	Angular velocity of the system S 'in S
${\displaystyle {\vec {\alpha }}={\dot {\vec {\omega }}}}$	Angular acceleration of the system S 'in S

In its original form, Newton's second law is only valid in the inertial system. In this reference system, the change in momentum is proportional to the external force : ${\displaystyle {\vec {F}}}$

${\displaystyle m\,{\vec {a}}={\vec {F}}}$

If you want to set up an analog equation of motion in a reference system that is not an inertial system, apparent forces must be taken into account. With the help of kinematic relationships , the acceleration in the inertial system is expressed by quantities that are given in an accelerated reference system:

${\displaystyle {\vec {a}}={\frac {d{\vec {v}}}{dt}}={\vec {a}}_{B}+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}{\;'})+{\dot {\vec {\omega }}}\times {\vec {r}}{\;'}+2\,{\vec {\omega }}\times {\vec {v}}{\;'}+{\vec {a}}{\;'}}$

Insertion into Newton's equation of motion and conversion according to the term with the relative acceleration results in:

${\displaystyle m\,{\vec {a}}{\;'}={\vec {F}}-m\,{\vec {a}}_{B}\underbrace {-m\,{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}{\;'})} _{{\vec {F}}_{\text{zentrifugal}}}\underbrace {-m\,{\dot {\vec {\omega }}}\times {\vec {r}}{\;'}} _{{\vec {F}}_{\text{Euler}}}\underbrace {-2m\,{\vec {\omega }}\times {\vec {v}}{\;'}} _{{\vec {F}}_{\text{Coriolis}}}}$

The product of mass and relative acceleration corresponds to the sum of the forces acting in this reference system. These are composed of the external forces and the apparent forces. ${\displaystyle m}$ ${\displaystyle {\vec {a}}{\;'}}$

The term is the centrifugal force that must be taken into account when applying the law of momentum in the accelerated frame of reference. This force is independent of whether there is a centripetal force or not. The centrifugal force is directed radially outwards perpendicular to the angular velocity in the reference system. The centrifugal force is zero on an axis that goes through the origin of the frame of reference and points in the direction of the angular velocity, even if the origin of the frame of reference executes a circular motion. The other apparent forces are (sometimes called the Einstein force), the Euler force and the Coriolis force . ${\displaystyle -m\,{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}{\;'})}$ ${\displaystyle {\vec {\omega }}}$ ${\displaystyle -m\,{\vec {a}}_{B}}$

If the radius vector and the angular velocity are perpendicular to one another, the result for the amount of centrifugal force is: ${\displaystyle r=\left|{\vec {r}}{\;'}\right|}$

${\displaystyle F_{\text{Zf}}=m\omega ^{2}r}$

Rotating frame of reference

Rotations are often described in a reference system in which the origin lies in the stationary or instantaneous center of curvature and which rotates at an angular velocity around this center of curvature. The origin of the reference system is not accelerated. Assuming that only the centripetal force acts as the external force, the equation of motion in the rotating frame of reference is generally: ${\displaystyle {\vec {\omega }}}$

${\displaystyle m\,{\vec {a}}{\;'}={\vec {F}}_{\text{Zp}}\underbrace {-m\,{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}{\;'})} _{{\vec {F}}_{\text{zentrifugal}}}\underbrace {-m\,{\dot {\vec {\omega }}}\times {\vec {r}}{\;'}} _{{\vec {F}}_{\text{Euler}}}\underbrace {-2m\,{\vec {\omega }}\times {\vec {v}}{\;'}} _{{\vec {F}}_{\text{Coriolis}}}={\vec {F}}_{\text{Zp}}+{\vec {F}}_{\text{Zf}}+{\vec {F}}_{\text{E}}+{\vec {F}}_{\text{C}}}$

Special cases

First, the special case is considered where the body rests in the rotating reference system and remains constant. Then the relative acceleration, the Coriolis force and the Euler force are zero. It stays: ${\displaystyle {\vec {\omega }}}$

${\displaystyle {\vec {F}}_{\text{Zp}}+{\vec {F}}_{\text{Zf}}={\vec {0}}}$

The same relationship results as with dynamic equilibrium. Nevertheless, the result is linked to different conditions. While the d'Alembert inertial force is the negative product of mass and absolute acceleration in the inertial system, a special reference system is assumed here.

In this reference system, the outward centrifugal force and the inward centripetal force compensate each other . To put it clearly: If an object is to "stop" on a rotating disk, something has to hold the object in place. The centrifugal force and the centripetal force add up to zero, so that the body remains “at rest”, ie at the same point in the rotating reference system. ${\displaystyle {\vec {F}}_{\text{Zf}}}$ ${\displaystyle {\vec {F}}_{\text{Zp}}}$

A ball rotating around a post held in place by a spring (simple model of a thread). Force (1) is centrifugal force. All other forces are either the centripetal force or its reaction, since they lie on a line of action.

Examples:

• If an occupant is held in a car , for example by a seat belt, by static friction on the seat, by contact forces, etc. , then the force in the rotating reference system that is opposite to the centrifugal force exerts an equal force on him. This force acts precisely as a centripetal force to keep the occupant on the same curved path that the car travels. In this sense , centrifugal force and centripetal force are opposing forces of equal magnitude.
• In the case of an astronaut orbiting the earth in a satellite , the gravitational acceleration for the space capsule and himself is the same and, as the centripetal acceleration, ensures that both traverse the same orbit around the earth. When describing this circular path in a satellite system with its origin in the center of the earth, two forces act on the astronaut: gravitational force and centrifugal force. The centrifugal force just cancels the force of gravity.
• The thread, which holds a body on a circular path, is tensioned by the centripetal force (force (4) in the picture opposite) and the associated reaction force (force (3)). This can e.g. B. can also be measured with a spring balance independently of the reference system

In this perspective, the spring exerts a centripetal force on the ball, so that it is forced onto a circular path, and vice versa, the ball also pulls on the spring.

If you transfer the picture opposite to a person rotating around a post (the spring symbolizes the arm, the ball the body), then it corresponds to everyday language that you feel an outwardly pulling centrifugal force and you compensate for this by holding on to the post got to. The feeling of being pulled outwards, however, is not the result of the "acting centrifugal force" - the stretching in the arm is caused by the centripetal force (4) and its reaction (3) and thus proves to be independent of the reference system.

In the next important special case, the external force is missing (e.g. with the sparks that come off):

${\displaystyle m\,{\vec {a}}{\;'}=\underbrace {-m\,{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}{\;'})} _{{\vec {F}}_{\text{zentrifugal}}}\underbrace {-m\,{\dot {\vec {\omega }}}\times {\vec {r}}{\;'}} _{{\vec {F}}_{\text{Euler}}}\underbrace {-2m\,{\vec {\omega }}\times {\vec {v}}{\;'}} _{{\vec {F}}_{\text{Coriolis}}}}$

With this definition, the centrifugal force depends on the choice of the reference system, but regardless of whether an external force is present or not. In this case, however, the relative movement can only be interpreted by a combination of several apparent forces. B. the Coriolis force can also have a radial direction. An example of this would be the movement of a body at rest in the inertial system, which is described in a rotating reference system.

Examples:

• If there is an apple on the front passenger seat, the driver can see how the apple is accelerated to the side in every curve. Here the acceleration of the apple is explained by an apparent force that is not opposed by an equally large centripetal force.

• If the centripetal force suddenly disappears from a body on a circular path, it describes an involute of a circle as a flight path in the co-rotating reference system , while in the non-rotating reference system it flies in a straight line in the direction of the tangent . The involute of a circle points exactly away from the axis of rotation only in its first part.

Centrifugal potential

The centrifugal force in the rotating reference system results from a pure conversion of coordinates between a rotating and an inertial reference system. It can therefore also be spoken of without a certain body describing a curved path in an inertial system or without a certain body being considered at all. This means that the centrifugal force is as present in a rotating reference system as an additional force field . This force field is conservative , so it can also be expressed by a potential :

${\displaystyle V_{\mathrm {Zf} }({\vec {r}})=-{\frac {({\vec {\omega }}\times {\vec {r}})^{2}}{2}}}$

It is called centrifugal potential, its gradient , taken negatively, is centrifugal acceleration. The direct physical meaning (after multiplication with the mass) is the (negatively counted) kinetic energy of the body resting in the rotating reference system, which it has due to its orbital speed : ${\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}}$

${\displaystyle m\;V_{\mathrm {Zf} }({\vec {r}})=-m\;{\frac {({\vec {\omega }}\times {\vec {r}})^{2}}{2}}=-m\;{\frac {v^{2}}{2}}=-E_{\mathrm {kin} }}$

This centrifugal potential can be calculated in the same way as with the usual potential energies in the inertial system, which belong to the force fields caused by interactions (e.g. gravitation, Coulomb force) in the rotating reference system . In contrast to these, the field of centrifugal force is not related to another body that would be the source of this field. Instead, the centrifugal force is related to the axis of rotation of the reference system, and it does not become weaker but stronger as the distance increases. The centrifugal potential can be combined to form the effective potential with another central potential .

Application example

For example, you work against the centrifugal force when you bring a body closer to the axis of rotation using only internal forces (e.g. pulling the tether). This work can be found quantitatively in the increase of the potential energy in the centrifugal potential. For the calculation it must be taken into account that the angular velocity increases when approaching, because the angular momentum remains constant (simplified formula for coordinate origin in the plane of motion). The angular velocity and the potential centrifugal energy thus vary along the path ${\displaystyle L=\omega mr^{2}}$

${\displaystyle \omega ={\frac {L}{mr^{2}}}\quad ,\quad m\;V_{\mathrm {Zf} }(r)=-{\frac {L^{2}}{2mr^{2}}}}$

The same change in potential energy results when the centrifugal force is expressed accordingly

${\displaystyle {\vec {F}}(r)=m\omega ^{2}r={\frac {L^{2}}{2mr^{3}}}}$

and the work is determined by integration over the radius. When the rope is subsequently loosened, the centrifugal force completely returns this work.

Useful examples

Surface of water in a rotating vessel

Stir in a water glass

Rotating liquid

In the case of a cylindrical vessel filled with water that rotates around its vertical axis, the surface of the water assumes a curved shape, the water level being higher on the outside than in the middle. The water particles are forced onto a circular path by a centripetal force. In the stationary state, the vector sum of centrifugal force and weight must be perpendicular to the surface at every point. Describes the acceleration due to gravity , the distance from the axis of rotation ( ) and the angle of the water surface with respect to the horizontal, the following applies: ${\displaystyle g}$ ${\displaystyle r}$ ${\displaystyle 0\leq r\leq R}$ ${\displaystyle \alpha }$

${\displaystyle \tan {\alpha }={\frac {{\omega }^{2}\;r}{g}}}$

The tangent of the angle is the slope of the water surface:

${\displaystyle {\frac {dz}{dr}}=\tan {\alpha }}$

Since the centrifugal force is proportional to the distance from the axis, the surface has the shape of a paraboloid of revolution and its cross-section by integration has the equation:

${\displaystyle z={\frac {{\omega }^{2}}{2\;g}}\;r^{2}+z_{0}}$

The constant volume results in the rise in the water level at the edge of the glass:

${\displaystyle H={\frac {{\omega }^{2}}{4\;g}}\;R^{2}}$

The parabolic shape of a light-reflecting liquid surface is used in the liquid mirrors of astronomical reflecting telescopes , which in the simplest case consist of mercury . Their focal length is:

${\displaystyle f={\frac {g}{2\,\omega ^{2}}}}$

Spin laundry

A washing machine with a drum diameter of 50 cm has a spin speed of 1200 revolutions per minute. The centrifugal acceleration for a piece of laundry rotating with it is given by

${\displaystyle \omega =130\,{\frac {\mathrm {rad} }{\mathrm {s} }},\qquad r=0{,}25\,\mathrm {m} ,\qquad a_{\mathrm {Zf} }=4000\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}.}$

Here is the angular velocity . ${\displaystyle \omega }$

The result is roughly 400 times the acceleration due to gravity . A centrifugal force that is 400 times as great as its weight acts on an item of clothing on the drum wall .

Roller coaster

The centrifugal force is important for the construction of roller coasters , where forces that are unpleasant for the human body should be avoided as much as possible, but those that counteract gravity and thus create a feeling of weightlessness are desirable. For example, in circular loops where weightlessness is created at the highest point, there is an abrupt increase in acceleration at the entry point , so that the body moving with it suddenly becomes five times the weight of inertia. For this reason, roller coaster designer Werner Stengel developed a clothoid shape ( Cornu spiral ) for the trajectory curve for loopings , in which the radius of curvature is inversely proportional to the arc length, which leads to a gentle increase in the inertial forces occurring in the vehicle. The clothoid had previously been used in road construction. ${\displaystyle 5g}$

Technical applications

Technical applications of centrifugal force are the centrifuge , the centrifugal separator , the worm gear , the centrifugal pendulum and the centrifugal governor . With excessive use, it can also come to a centrifugal force .

Centrifugal force as a substitute for gravity

• 

  Agena rocket stage on safety tape, second attempt with Gemini 12 (1966)

• 

  Nautilus-X ( NASA design from 2011)

• 

  Rotating wheel-shaped space station, design by Herman Potočnik (pseudonym: Hermann Noordung, 1929)

For future space stations of different sizes it has been planned to use centrifugal force as a substitute for gravity , because prolonged weightlessness can damage human health. The first attempt to use centrifugal force in a manned spacecraft was made in 1966. This one has the Gemini-11 capsule with the Agena - stage rocket through a 30 meter long guard band connected and both objects with about one rotation every six minutes to the common focus rotate.

As a result of the centrifugal force in a rotating space station, a plumb bob at each location points away from the axis of rotation. Objects that “fall” freely move away from this vertical direction, counter to the direction of rotation of the space station. This deviation can be understood as a consequence of the Coriolis force . The trajectory of a freely falling object has the shape of an involute in the rotating frame of reference of the space station and is independent of the rotational speed of the space station. However, the scale of the involute of a circle depends on the radius of the initial circular path, i.e. H. from the initial distance of the object from the axis of rotation. Viewed from a non-rotating reference system, freely “falling” objects move at constant speed on a straight line that is tangential to their previous circular path.

While many satellites use rotation for stabilization, this effect was deliberately used to simulate gravity on the German EuCROPIS mission (2018/19). With a view to future lunar missions, plant growth should be studied under reduced gravity. However, due to a technical error, the greenhouse could not be operated as planned.

Web links

Wiktionary: Centrifugal force  - explanations of meanings, word origins, synonyms, translations

Wiktionary: Centrifugal force  - explanations of meanings, word origins, synonyms, translations

• Centrifugal force at student level at LEIFI
• Video: Coriolis and centrifugal force in the rotating frame of reference . Institute for Scientific Film (IWF) 2007, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-13095 .

Individual evidence

1. ↑ Hans J. Paus : Physics in experiments and examples . 3rd, updated edition. Hanser, Munich 2007, ISBN 3-446-41142-9 , pp.  33–35 ( limited preview in Google Book search).
2. ↑ ^{a b} Bruno Assmann, Peter Selke: Kinematik und Kinetik (=  Technical Mechanics . Volume 3 ). 15th, revised edition. Oldenbourg, Munich 2011, ISBN 978-3-486-59751-6 , pp. 252 ( limited preview in Google Book search). "The centrifugal force is the reaction force of the centripetal force, which forces the curved path."
3. ↑ ^{a b c} Martin Mayr: Technical mechanics: statics, kinematics - kinetics - vibrations, strength theory . 6th revised edition. Hanser, 2008, ISBN 978-3-446-41690-1 ( limited preview in Google book search). “When moving on a curved path, normal or centripetal acceleration also occurs. We call the associated inertial force centrifugal force. "
4. ↑ René Descartes: The principles of philosophy, translated by Artur Buchenau . 7th edition. Felix Meiner Verlag, Hamburg 1965, p. 86 ff .
5. ^ John Herivel : The Background of Newton's Principia, and John Herivel: Newton's Discovery of the law of Centrifugal Force. In: The Isis. Volume 51, 1960, p. 546.
6. ^ Domenico Bertoloni Meli: The Relativization of Centrifugal Force . In: Isis . tape 81 , no. 1 , 1990, p. 23–43 ( online [PDF; accessed July 15, 2019]).
7. ↑ Dietmar Gross, Werner Hauger, Jarg Schrader, Wolfgang A. Wall: Technical Mechanics. Volume 3: Kinetics . 10th edition. Gabler Wissenschaftsverlage, 2008, ISBN 978-3-540-68422-0 , p. 191 ( limited preview in Google Book search). "We are writing now and take the negative product of the mass and acceleration formally as a force to which we [...] D'Alembert inertial force call: . This force is not a force in the Newtonian sense, since there is no counterforce to it (it violates the axiom actio = reactio!), We therefore call it an apparent force. " ${\displaystyle F-ma=0}$ ${\displaystyle m}$ ${\displaystyle a}$ ${\displaystyle F_{T}}$ ${\displaystyle F_{T}=-ma}$
8. ↑ ^{a b} Cornelius Lanczos: The Variational Principles of Mechanics . Courier Dover Publications, New York 1986, ISBN 0-486-65067-7 , pp. 88–110 ( limited preview in Google Book search). P. 88: "We now define a vector I by the equation I = -m A. This vector I can be considered as a force created by the motion. We call it the “force of inertia”. With this concept the equation of Newton can be formulated as follows: F + I = 0. "
9. ^ Mahnken: Textbook of technical mechanics. Dynamism . Springer, 2012, ISBN 978-3-642-19837-3 ( limited preview in Google book search). "We also note that the centrifugal force is in equilibrium with the centripetal force, which is directed towards the center."
10. ^ Alfred Böge, Wolfgang Böge, Klaus-Dieter Arnd a. a .: Handbook of Mechanical Engineering: Fundamentals and Applications of Mechanical Engineering, Hardcover - 22nd edition . Springer Verlag, 2014, ISBN 978-3-658-06597-3 , pp. B 14-B 15 ( preview ).
11. ↑ Ludwig Bergmann , Clemens Schaefer : Mechanics, Relativity, Warmth . Ed .: Thomas Dorfmüller (=  Textbook Der Experimentalphysik . Volume  1 ). 11th, completely revised edition. de Gruyter, Berlin 1998, ISBN 3-11-012870-5 , p. 240 ff . ( limited preview in Google Book search).
12. ↑ Notation mainly based on Karl Schilcher: Theoretical Physics compact for teaching. P. 89.
13. ↑ Ekbert Hering, Rolf Martin, Martin Stohrer: Physics for engineers . 11th edition. Springer, 2012, ISBN 978-3-642-22568-0 , pp. 51–52 ( limited preview in Google Book search).
14. ↑ Walter Greiner: Classical Mechanics I . Harri Deutsch, 2007, ISBN 978-3-8171-1815-1 ( limited preview in Google book search).
15. ↑ Peter R. Hook Esch: Lecture notes fluid mechanics. P. 51 ff. (PDF; 7.19 MB).
16. ↑ Verena Heintz, Ann-Marie Martensson-Pendrill, Anette Schmitt, Klaus Wendt: Riding a rollercoaster in physics lessons. In: Physics in Our Time. 2009, issue 2.
17. ↑ Compact satellite in polar orbit - Farewell to Mission Eu: CROPIS. DLR, January 13, 2020, accessed on January 19, 2020 .