Centrifugal force

Centrifugal force (from Latin centrum, center and fugere, flee ), and centrifugal force is an inertial force that is directed radially outward from the axis of rotation. It is caused by the inertia of the body. The effects of the first Newtonian law are often experienced in everyday life, such as when the whirligig, the seats are pushed to the outside, the laundry in the spin dryer is dry, or the cyclists ' place in the curve " must. The centrifugal force resulting from the centrifugal force by multiplying by the mass.

The centrifugal force can be interpreted with two different concepts. On the one hand it is the apparent power, which must be considered when describing the motion of an object with respect to a translational or rotational motion reference system.

Here, the angular velocity of the reference system. is the mass of the moving object. And the distance between the rotation axis of the reference system and the object.

On the other hand, the centrifugal force is that force, which is the centripetal force at equilibrium.

Here, the local radius of curvature of the trajectory of the object. is the mass of the moving object. And the velocity of the object with respect to the inertial system, in which the web will be described.

• 4.1 related to the centripetal force
• 4.2 centrifugal potential

• 5.1 Rotating liquid
• 5.2 Spin laundry
• 5.3 rollercoaster
• 5.4 Technical Applications
• 5.5 centrifugal force as a substitute for the gravity

History

A qualitative description of the centrifugal force can already be found in the 1644 published Principles of Philosophy by René Descartes. Quantitatively, it was first derived in 1669 in a letter by Christian Huygens to the secretary of the Royal Society Henry Oldenbourg, also mentioned in the Horologium Oscillatorium of 1673 without derivation and detailed in its posthumous writing of 1703 De Vis Centrifuga ( from 1659 ). Isaac Newton described the centrifugal force only after Huygens, but regardless of this.

The forming mold by centrifugal force the liquid surface in a rotating, open water bucket was interpreted by Isaac Newton as proof of the existence of absolute space.

D' Alembert's inertial force

Describes the center of gravity of a body with the mass in an inertial frame a curved path, a centripetal force is necessary for it. According to Newton's second law results in a proportional centripetal acceleration, directed towards the center of curvature of the path:

This fundamental equation of mechanics can on the form:

Be brought.

The negative product of mass and centripetal acceleration is formally regarded as a force and called centrifugal force. A dynamic problem can thus be attributed to a static equilibrium of external forces and inertia force:

For the purposes of the dynamic equilibrium, the centrifugal force is always opposite equal to the centripetal force. The sum of the forces is thus zero if one includes the ( d' Alembert ) inertia force.

The centrifugal force in the d' Alembert sense is always coupled to the centripetal force, in a sense the mirror image. This distinguishes them from the apparent force that must be taken into account only if one formulates the Newtonian equation of motion in an accelerated and rotating reference system. In the special case of a rotating reference system in which the body rests, wherein the origin of the reference system is located in the center of curvature, both definitions are identical.

The amount of centripetal or centrifugal force is calculated from the path velocity and the curvature radius of the path:

As an example of the conversion of a dynamic problem in a static calculation of the skew position of a motorcycle rider is shown in a stationary cornering. If the motorcycle is not to tip over, the resultant force of centrifugal and gravitational force has to go through the tire contact patch. The centripetal force acts at the street level and does not need to be considered in the equilibrium of moments about the wheel. For the skew results

With the gravitational acceleration and the radial acceleration.

Scheinkraft in translation and rotation moving reference system

Image forces must always be taken into account when describing movements in a reference system, which itself is accelerated compared to the inertial system. If one considers, for example, the sparks are separating from a grinding wheel in the inertial frame, so they move in a straight line, as they are force-free. In the rotating reference frame of the grinding wheel, the relative acceleration of the particles is, however, explained by an apparent force.

Notation

To distinguish between the sizes of an object ( eg, location, speed ) in two reference systems, the normal notation is used in the inertial reference system and the nichtinertiale receives the same letter with an apostrophe (English prime ). The latter is referred to as " coated reference system ."

The second Newtonian law applicable in its original form only in the inertial frame. The change of momentum is proportional to the external force in this reference system:

If you want to set up an analogous equation of motion in a reference frame that is not inertial, apparent forces must be taken into account. For this purpose the acceleration is expressed by sizes in the inertial system, which are placed in an accelerated reference:

Multiplying the mass and dissolving after the term on the relative acceleration, we obtain:

The product of mass and relative acceleration equal to the sum of the acting in this reference system forces. These are composed of the external forces and the apparent forces.

The term is the centrifugal force, which must be taken into account when the pulse rate is applied in accelerated reference system. This force is independent of whether or not there is a centripetal force. The centrifugal force is directed perpendicular to the angular velocity in the reference frame radially outward.

Rotating frame of reference

Rotations about a fixed axis are often described in a reference system in which the origin is on the rotation axis. With constant angular velocity, the motion equation is simplified:

Rests a body relative to a rotating reference system, the centrifugal force and the inward centripetal force compensating.

In this special case, the apparent force in a rotating reference system and the d' Alembert inertial force are identical.

If there is no external force, such as when the sparks become detached, the definitions differ:

The centrifugal force is coupled according to this definition in the reference system, but regardless of whether an external force is present or not.

In order for a body is held relative to a rotating reference system whose origin is itself unaccelerated, at rest, the centrifugal force and the inward centripetal force must compensate. Clearly formulated: If an object is to " stand " on a rotating disk, the object needs to hold something. The centrifugal and centripetal add up to zero, so that the body " at rest", so stay in the same place the disc.

Describing the object on a rotating disk in an inertial system, so the body under inertia sentence does not want to continue to move in a circular path, but while retaining its speed straight ahead; it acts on a while, but the same "inwardly" directed centripetal force. This is in contrast to the centrifugal force no inertial force, but one to be considered in any reference system external (real) power that causes the body constantly accelerated inward and is thus forced into a circular orbit. It is directed toward the center of rotation and provides the centripetal acceleration that an object moves along a curved path.

The centrifugal force is critical coupled to the reference system. A body which rotates together with a reference system, the centripetal force required to keep it on the curved path, in opposite directions, but by the same amount as the centrifugal force that it senses, in this reference system. The following examples illustrate the differences between the two approaches. They are limited for simplicity to the special case that the origin of the rotating frame of reference is itself unaccelerated.

• If an inmate etc. held for example by a seat belt, by friction on the seat by contact forces in a car, so the exercises from the perspective of a person of the opposite centrifugal force, equal force on him. This force is used just as a centripetal force to keep the occupants on the same curved path that goes through the car. In this sense, centrifugal and centripetal force are opposite to each other, equal forces.

• If, however, in the passenger seat of an apple, so the driver sees in every corner, like the apple in the car is accelerated to the side. Here, the acceleration of the apple is explained with an apparent force which opposes no equal centripetal force.

• When an astronaut orbiting the Earth in a spacecraft is the gravitational acceleration for the space capsule and the same size as him and provides the centripetal acceleration that both go through the same circular path around the Earth. In describing this orbit in a satellite system with the origin at the center of the Earth, two forces act on the astronaut: the gravitational force and the centrifugal force. The centrifugal force just cancels gravity.

Related to the centripetal force

The centrifugal force is described in some texts as " counter-force " or " reaction force " to the centripetal force; Reference is made to Newton's third law. In this view, the spring exerts a centripetal force on the ball out, so that it is forced on a circular path, and vice versa also pulls the ball to the spring. This reaction force of the ball on the cable is sometimes referred to as "reactive centrifugal force ".

However, other authors argue that this power should not be confused with the occurring in rotating frames of reference inertial or fictitious forces and point to a contradiction with the third Newtonian law as centripetal and centrifugal force act on the same body, in contrast, must force pairs, as " action and reaction " are referred to, act on different bodies. The inertial or fictitious forces represent an apparent force equilibrium of a body ( ie the ball ), and depend on the choice of the reference system. In contrast, reaction forces shall constitute the third law is an interaction between two different bodies ( spring and ball ), which occurs independently of the reference system.

The thread that keeps a body on a circular path, is excited by the reaction force to the centripetal force ( force ( 3) in the picture ) and the force ( 4 ) ( centripetal force ). This can for example be measured with a spring balance regardless of the reference system. Only in the special case of a co-rotating with the observed body reference system, the reaction to the centripetal force ( 3) and the centrifugal force ( 1) the magnitude and direction are equal, but not otherwise. Your targets are, however, always different.

Transferring picture on the left to a man that rotates around a pole (the spring symbolizes the arm, the ball hits the body ), so it corresponds to the ordinary language, that one feels a pulling outward centrifugal force and to compensate for this by clinging to the post needs. In the event that you force ( 1) and (3) does not differ and ignores any possible contradictions to the interaction principle, such a statement is also possible. On closer inspection, is the feeling of being pulled outward, no power, but you can feel a stretch in the arm. This is due to the centripetal force (4) and the reactio (3) produced.

Centrifugal potential

Since the centrifugal force, like the force of gravity is proportional to the mass of the body, the centrifugal acceleration due to gravity can be similar to the point as a local factor which indicates the acceleration at a given location, which might know a body due to the centrifugal force at this location.

The energy in the centrifugal potential is equal to the kinetic energy:

Practical examples

Rotating fluid

In a water-filled cylindrical vessel which rotates about its vertical axis, the surface assumes a curved shape, wherein the water level is higher than the outside in the center. The water particles are forced by a centripetal force on a circular path. In the stationary state, the vector sum of the centrifugal force and weight must be at any point of the surface on this perpendicular. It is the same formula that has already been derived during the motorcyclist, wherein r denotes the distance from the axis of rotation. is the angle of the surface with respect to the horizontal.

Since the speed of the angular velocity of the fluid can be calculated, the result is:

The tangent of the angle of the slope of the water surface. Since the centrifugal force is proportional to the radius, the surface is in the form of a paraboloid of revolution, and whose cross-section by integrating the equation:

The parabolic shape of a light reflective surface of the liquid is applied to the liquid mirror astronomical reflecting telescopes, which consist in the simplest case of mercury.

Spin laundry

A washing machine with a drum diameter of 50 cm makes the spin 1200 revolutions per minute. The centrifugal acceleration for a co-rotating piece of laundry is given by

Here, the angular velocity. is one revolution per minute.

The result corresponds to about 400 times the acceleration of gravity. A sock to the drum wall thus a centrifugal force that is 400 times as large as its weight.

Roller coaster

The centrifugal force is for the construction of roller coasters of importance in which the human body unpleasant forces should be avoided whenever possible, but such that counteract the force of gravity, thus creating a feeling of weightlessness, are desired. For example, results from circular loops, which just weightlessness is generated at the highest point, at the entry point, an abrupt increase in acceleration over so suddenly five times the force of gravity occurs for the body as a co-moving inertial force. Therefore, a clothoid form ( Cornu spiral) is designed of the trajectory, in which the radius of curvature is inversely proportional to the arc length, resulting in a gentle increase in the inertial forces occurring in the vehicle by the roller coaster design engineer for Werner stem loops. The clothoid had previously been already used in road construction.

Technical Applications

Technical application of the centrifugal force, the centrifuge, the centrifugal separator, the centrifugal force pendulum and the governor.

Centrifugal force as a substitute for the gravity

For future space stations of different sizes you have planned to use the centrifugal force as a substitute for gravity, since prolonged weightlessness on human health can be harmful. The first relatively tentative attempt at a manned spacecraft controlled centrifugal force to produce, took place in 1966. This one has the Gemini 11 capsule connected with the Agena rocket stage through a 30 -meter safety belt and allow both objects rotate every six minutes around the common center of gravity with about a revolution. In a rotating space station, a plumb line at each location would show from the rotational axis away, but free-falling objects would move away from the perpendicular direction more and more in a direction opposite the direction of rotation of the space station -looking. This deviation can be interpreted as a consequence of the Coriolis force. The shape of this curve case, an involute is completely independent of the rotational speed of the space station, while their size scale depends on the radius of the initial orbit. From a non-rotating reference frame of view freely falling objects at a constant speed in a straight line would move tangential to its previous orbit. In a horizontal throw in the space station, opposite to the rotation of the space station and with the rotation speed of the space station, the thrown object would constantly fly horizontally on as long as one can neglect air resistance. From a non-rotating reference system, this object would simply stand still, while the space station continues to rotate.