Fictitious force - Knowledge

7939: 7207: 7934:{\displaystyle {\begin{aligned}{\frac {d^{2}\mathbf {x} _{A}}{dt^{2}}}&=\mathbf {a} _{AB}+\mathbf {a} _{B}+2\ \sum _{j=1}^{3}v_{j}\ {\frac {d\mathbf {u} _{j}}{dt}}+\ \sum _{j=1}^{3}x_{j}\ {\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}}\\&=\mathbf {\Omega \ \times } \left(\mathbf {\Omega \times X} _{AB}\right)+\mathbf {a} _{B}+2\ \sum _{j=1}^{3}v_{j}\ \mathbf {\Omega \times u_{j}} \ +\ \sum _{j=1}^{3}x_{j}\ {\boldsymbol {\Omega }}\times \left({\boldsymbol {\Omega }}\times \mathbf {u} _{j}\right)\\&=\mathbf {\Omega \ \times } \left(\mathbf {\Omega \times X} _{AB}\right)+\mathbf {a} _{B}+2\ {\boldsymbol {\Omega }}\times \mathbf {v} _{B}\ \ +\ {\boldsymbol {\Omega }}\times \left({\boldsymbol {\Omega }}\times \mathbf {x} _{B}\right)\\&=\mathbf {\Omega \ \times } \left(\mathbf {\Omega \times } (\mathbf {X} _{AB}+\mathbf {x} _{B})\right)+\mathbf {a} _{B}+2\ {\boldsymbol {\Omega }}\times \mathbf {v} _{B}\ \,,\end{aligned}}} 9567: 1346:. For example, consider some persons in rotating chairs holding a weight in their outstretched hands. If they pull their hand inward toward their body, from the perspective of the rotating reference frame, they have done work against the centrifugal force. When the weight is let go, it spontaneously flies outward relative to the rotating reference frame, because the centrifugal force does work on the object, converting its potential energy into kinetic. From an inertial viewpoint, of course, the object flies away from them because it is suddenly allowed to move in a straight line. This illustrates that the work done, like the total potential and kinetic energy of an object, can be different in a non-inertial frame than in an inertial one. 10338: 726: 8887: 5001: 9562:{\displaystyle {\begin{aligned}\mathbf {a} (t)&={\frac {d^{2}R}{dt^{2}}}{\begin{bmatrix}\cos(\omega t+\pi /4)\\\sin(\omega t+\pi /4)\end{bmatrix}}+2{\frac {dR}{dt}}\omega {\begin{bmatrix}-\sin(\omega t+\pi /4)\\\cos(\omega t+\pi /4)\end{bmatrix}}-\omega ^{2}R(t){\begin{bmatrix}\cos(\omega t+\pi /4)\\\sin(\omega t+\pi /4)\end{bmatrix}}\\&=2s\omega {\begin{bmatrix}-\sin(\omega t+\pi /4)\\\cos(\omega t+\pi /4)\end{bmatrix}}-\omega ^{2}R(t){\begin{bmatrix}\cos(\omega t+\pi /4)\\\sin(\omega t+\pi /4)\end{bmatrix}}\\&=2s\ \omega \ \mathbf {u} _{\theta }-\omega ^{2}R(t)\ \mathbf {u} _{R}\,.\end{aligned}}} 4477: 10196:. In the case of the carousel, the kinetic discussion would involve perhaps a study of the walker's shoes and the friction they need to generate against the floor of the carousel, or perhaps the dynamics of skateboarding if the walker switched to travel by skateboard. Whatever the means of travel across the carousel, the forces calculated above must be realized. A very rough analogy is heating your house: you must have a certain temperature to be comfortable, but whether you heat by burning gas or by burning coal is another problem. Kinematics sets the thermostat, kinetics fires the furnace. 6772: 4996:{\displaystyle {\begin{aligned}\mathbf {a} _{\mathrm {A} }&=\mathbf {a} _{\mathrm {B} }+\ 2\sum _{j=1}^{3}v_{j}{\boldsymbol {\Omega }}\times \mathbf {u} _{j}(t)+\sum _{j=1}^{3}x_{j}{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {u} _{j}\ +\sum _{j=1}^{3}x_{j}{\boldsymbol {\Omega }}\times \left\\&=\mathbf {a} _{\mathrm {B} }+2{\boldsymbol {\Omega }}\times \sum _{j=1}^{3}v_{j}\mathbf {u} _{j}(t)+{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \sum _{j=1}^{3}x_{j}\mathbf {u} _{j}+{\boldsymbol {\Omega }}\times \left.\end{aligned}}} 1442: 8854: 739: 8210:. A rider on the carousel walks radially across it at a constant speed, in what appears to the walker to be the straight line path inclined at 45° in Figure 5. To the stationary observer, however, the walker travels a spiral path. The points identified on both paths in Figure 5 correspond to the same times spaced at equal time intervals. We ask how two observers, one on the carousel and one in an inertial frame, formulate what they see using Newton's laws. 5176: 5622: 5370: 6056: 4218: 9774: 8186: 5585: 2513: 8529: 2783: 3039: 11700:(New York, 1965). This could be the first hint of the crisis of the non-relativistic physics: in "non-inertial" frames using non-Euclidean and not flat metrics, fictitious forces transform into force exchanged with "objects" that do not follow the geodesic trajectory (simply with a relative speed respect it). In any case this generalized "Newton's second law" must wait for the 5683: 1199:

plays a role in this piece of movement. It may seem that there must be a force responsible for this movement, but actually, this movement arises because of the inertia of the suitcase, which is (still) a 'free object' within an already accelerating frame of reference. After the suitcase has come into contact with the closed door of the car, the situation with the emergence of

5016: 5215: 3383: 5755: 3972: 4471: 3071:, and two terms related to the rotation of frame B. Consequently, observers in B will see the particle motion as possessing "extra" acceleration, which they will attribute to "forces" acting on the particle, but which observers in A say are "fictitious" forces arising simply because observers in B do not recognize the non-inertial nature of frame B. 2050: 3751: 9615: 5426: 2242: 2796: 6438: 8446: 8849:{\displaystyle {\begin{aligned}\mathbf {v} (t)&={\frac {dR}{dt}}{\begin{bmatrix}\cos(\omega t+\pi /4)\\\sin(\omega t+\pi /4)\end{bmatrix}}+\omega R(t){\begin{bmatrix}-\sin(\omega t+\pi /4)\\\cos(\omega t+\pi /4)\end{bmatrix}}\\&={\frac {dR}{dt}}\mathbf {u} _{R}+\omega R(t)\mathbf {u} _{\theta },\end{aligned}}} 2531: 2214: 2233:. The other is a contribution to velocity due to the fact that different locations in the non-inertial frame have different apparent velocities due to the rotation of the frame; a point seen from a rotating frame has a rotational component of velocity that is greater the further the point is from the origin. 5171:{\displaystyle \mathbf {a} _{\mathrm {A} }=\mathbf {a} _{\mathrm {B} }+2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {B} }+{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {x} _{\mathrm {B} }+{\boldsymbol {\Omega }}\times \left({\boldsymbol {\Omega }}\times \mathbf {x} _{\mathrm {B} }\right)\,.} 9588:, and it represents the acceleration of the walker as the edge of the carousel is neared, and the arc of the circle travelled in a fixed time increases, as can be seen by the increased spacing between points for equal time steps on the spiral in Figure 5 as the outer edge of the carousel is approached. 10167:

in the rotating world, a force exactly opposite in sign to the fictitious force must be applied to reduce the net force on the walker to zero, so Newton's law of inertia will predict a straight line motion, in agreement with what the rotating observer sees. The fictitious forces that must be combated

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force upon the walker. These conclusions do not agree with the inertial observer. To obtain agreement, the rotating observer has to introduce fictitious forces that appear to exist in the rotating world, even though there is no apparent reason for them, no apparent gravitational mass, electric charge

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The rain frame perspective here, rather than being that of a raindrop, is more like that of a trampoline jumper whose trajectory tops out just as the ball reaches the edge of the cliff. The shell frame perspective may be familiar to planet dwellers who rely minute by minute on upward physical forces

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The pseudo force on an object arises as an imaginary influence when the frame of reference used to describe the object's motion is accelerating compared to a non-accelerating frame. The pseudo force "explains," using Newton's second law mechanics, why an object does not follow Newton's second law and

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In this connection, it may be noted that a change in the coordinate system, for example, from Cartesian to polar, if implemented without any change in relative motion, does not cause the appearance of rotational fictitious forces, despite the fact that the form of the laws of motion varies from one

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From the viewpoint of a rotating frame, moving with the car, a fictitious centrifugal force appears to be present pushing the car toward the outside of the road (and pushing the occupants toward the outside of the car). The centrifugal force balances the friction between wheels and the road, making

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Passengers in a vehicle accelerating in the forward direction may perceive they are acted upon by a force moving them into the direction of the backrest of their seats for instance. An example in a rotating reference frame may be the impression that it is a force which seems to move objects outward

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A related problem is that of centrifugal forces for the Earth–Moon–Sun system, where three rotations appear: the daily rotation of the Earth about its axis, the lunar-month rotation of the Earth–Moon system about its centre of mass, and the annual revolution of the Earth–Moon system about the Sun.

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when the reference frame does not move inertially any more but begins to accelerate relative to the free object. In terms of the example of the passenger vehicle, a pseudo force seems to be active just before the body touches the backrest of the seat in the car. A person in the car leaning forward

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Because they differ only by the constant walking velocity, the walker and the rotational observer see the same accelerations. From the walker's perspective, the fictitious force is experienced as real, and combating this force is necessary to stay on a straight line radial path holding a constant

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Applying Newton's laws, multiplying the acceleration by the mass of the walker, the inertial observer concludes that the walker is subject to two forces: the inward radially directed centripetal force and another force perpendicular to the radial direction that is proportional to the speed of the

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A common situation in which noninertial reference frames are useful is when the reference frame is rotating. Because such rotational motion is non-inertial, due to the acceleration present in any rotational motion, a fictitious force can always be invoked by using a rotational frame of reference.

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tied by a cord and spinning around their centre of mass. In this case, the identification of a rotating, non-inertial frame of reference can be based upon the vanishing of fictitious forces. In an inertial frame, fictitious forces are not necessary to explain the tension in the string joining the

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appears. As the car enters a left turn, a suitcase first on the left rear seat slides to the right rear seat and then continues until it comes into contact with the closed door on the right. This motion marks the phase of the fictitious centrifugal force as it is the inertia of the suitcase which

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is the distance of the matter in the tube from the centre of the centrifuge. It is the standard specification of a centrifuge to use the "effective" radius of the centrifuge to estimate its ability to provide centrifugal force. Thus, the first estimate of centrifugal force in a centrifuge can be

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In the inertial frame of reference (upper part of the picture), the black ball moves in a straight line. However, the observer (brown dot) who is standing in the rotating/non-inertial frame of reference (lower part of the picture) sees the object as following a curved path due to the Coriolis or

1067:. The Euler force is typically ignored because the variations in the angular velocity of the rotating surface of the Earth are usually insignificant. Both of the other fictitious forces are weak compared to most typical forces in everyday life, but they can be detected under careful conditions. 1078:

to show that the Coriolis force results from the Earth's rotation. If the Earth were to rotate twenty times faster (making each day only ~72 minutes long), people could easily get the impression that such fictitious forces were pulling on them, as on a spinning carousel. People in temperate and

5365:{\displaystyle \mathbf {a} _{\mathrm {B} }=\mathbf {a} _{\mathrm {A} }-2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {B} }-{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {x} _{\mathrm {B} })-{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {x} _{\mathrm {B} }.} 6051:{\displaystyle {\begin{aligned}{\frac {d^{2}\mathbf {x} _{A}}{dt^{2}}}&=\mathbf {a} _{AB}+\mathbf {a} _{B}+2\ \sum _{j=1}^{3}v_{j}\ {\frac {d\mathbf {u} _{j}}{dt}}+\sum _{j=1}^{3}x_{j}\ {\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}}\\&=\mathbf {a} _{AB}\ +\mathbf {a} _{B}\ ,\end{aligned}}} 3162: 4213:{\displaystyle {\frac {d^{2}\mathbf {u} _{j}(t)}{dt^{2}}}={\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {u} _{j}+{\boldsymbol {\Omega }}\times {\frac {d\mathbf {u} _{j}(t)}{dt}}={\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {u} _{j}+{\boldsymbol {\Omega }}\times \left,} 9600:

The rotating observer sees the walker travel a straight line from the centre of the carousel to the periphery, as shown in Figure 5. Moreover, the rotating observer sees that the walker moves at a constant speed in the same direction, so applying Newton's law of inertia, there is

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it is a natural extension of standard terminology (although there is no standard terminology for this case) to call this term the "centrifugal force". Applying this terminology to the example of a tube in a centrifuge, if the tube is far enough from the center of rotation,

9769:{\displaystyle \mathbf {F} _{\mathrm {fict} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {B} }-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {x} _{\mathrm {B} })-m{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {x} _{\mathrm {B} }.} 5580:{\displaystyle \mathbf {F} _{\mathrm {fict} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {B} }-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {x} _{\mathrm {B} })-m{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {x} _{\mathrm {B} }.} 3519: 2508:{\displaystyle {\frac {d^{2}\mathbf {x} _{\mathrm {A} }}{dt^{2}}}=\mathbf {a} _{\mathrm {AB} }+{\frac {d\mathbf {v} _{\mathrm {B} }}{dt}}+\sum _{j=1}^{3}{\frac {dx_{j}}{dt}}{\frac {d\mathbf {u} _{j}}{dt}}+\sum _{j=1}^{3}x_{j}{\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}}.} 1215:

Suppose a few miles further the car is moving at constant speed travelling a roundabout, again and again, then the occupants will feel as if they are being pushed to the outside of the vehicle by the (reactive) centrifugal force, away from the centre of the turn.

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first moves a bit backward in relation to the already accelerating car, before touching the backrest. The motion in this short period just seems to be the result of a force on the person; i.e., it is a pseudo force. A pseudo force does not arise from any

7192: 926:, or sometimes as a pseudo force. D'Alembert's principle is just another way of formulating Newton's second law of motion. It defines an inertial force as the negative of the product of mass times acceleration, just for the sake of easier calculations. 3508: 8087: 6561: 8108:|, all the matter in the test tube sees the same acceleration (the same centrifugal force). Thus, in this case, the fictitious force is primarily a uniform centrifugal force along the axis of the tube, away from the centre of rotation, with a value | 6286: 7094: 8232: 6271: 3961: 2778:{\displaystyle {\frac {d\mathbf {v} _{\mathrm {B} }}{dt}}=\sum _{j=1}^{3}{\frac {dv_{j}}{dt}}\mathbf {u} _{j}+\sum _{j=1}^{3}v_{j}{\frac {d\mathbf {u} _{j}}{dt}}=\mathbf {a} _{\mathrm {B} }+\sum _{j=1}^{3}v_{j}{\frac {d\mathbf {u} _{j}}{dt}}.} 875:"floats freely" as if weightless. As a frame may accelerate in any arbitrary way, so may pseudo forces also be as arbitrary (but only in direct response to the acceleration of the frame). An example of a pseudo force as defined by Iro is the 10039: 3034:{\displaystyle {\frac {d^{2}\mathbf {x} _{\mathrm {A} }}{dt^{2}}}=\mathbf {a} _{\mathrm {AB} }+\mathbf {a} _{\mathrm {B} }+2\ \sum _{j=1}^{3}v_{j}{\frac {d\mathbf {u} _{j}}{dt}}+\sum _{j=1}^{3}x_{j}{\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}}.} 3782:

To derive expressions for the fictitious forces, derivatives are needed for the apparent time rate of change of vectors that take into account time-variation of the coordinate axes. If the rotation of frame 'B' is represented by a vector

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In the rotating reference frame perceived on the surface of the Earth, a centrifugal force reduces the apparent force of gravity by about one part in a thousand, depending on latitude. This reduction is zero at the poles, maximum at the

9971: 10176:(second term). (These terms are approximate.) By applying forces to counter these two fictitious forces, the rotating observer ends up applying exactly the same forces upon the walker that the inertial observer predicted were needed. 6502: 1104:

cannot detect their own motion; however, observers within an accelerating reference frame can detect that they are in a non-inertial reference frame from the fictitious forces that arise. For example, for straight-line acceleration

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must introduce a fictitious force, this time due to the acceleration from the orbital motion of their entire coordinate frame, that is radially outward away from the centre of rotation of the origin of their coordinate system:

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over non-inertial frames because they do not have physics whose causes are outside of the system, while non-inertial frames do. Fictitious forces, or physics whose cause is outside of the system, are no longer necessary in

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Figure 5: Crossing a rotating carousel walking at a constant speed from the centre of the carousel to its edge, a spiral is traced out in the inertial frame, while a simple straight radial path is seen in the frame of the

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is connected to, i.e. the vehicle in this case. From the viewpoint of the respective accelerating frame, an acceleration of the inert object appears to be present, apparently requiring a "force" for this to have happened.

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As part of the requirement of simplicity, to be an inertial frame, in all other frames that differ only by a uniform rate of translation, the description should be of the same form. However, in the Newtonian system the

3865: 3378:{\displaystyle \mathbf {F} _{\mathrm {A} }=\mathbf {F} _{\mathrm {B} }+m\mathbf {a} _{\mathrm {AB} }+2m\sum _{j=1}^{3}v_{j}{\frac {d\mathbf {u} _{j}}{dt}}+m\sum _{j=1}^{3}x_{j}{\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}}\ .} 3074:

The factor of two in the Coriolis force arises from two equal contributions: (i) the apparent change of an inertially constant velocity with time because rotation makes the direction of the velocity seem to change (a

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becomes current. The centripetal force on the car is now also transferred to the suitcase and the situation of Newton's third law comes into play, with the centripetal force as the action part and with the so-called

6929: 937:'. In terms of the example of the passenger vehicle above, a contact force emerges when the body of the passenger touches the backrest of the seat in the car. It is present for as long as the car is accelerated.) 7004: 4466:{\displaystyle {\frac {d^{2}\mathbf {x} _{\mathrm {A} }}{dt^{2}}}=\mathbf {a} _{\mathrm {B} }+2\sum _{j=1}^{3}v_{j}\ {\frac {d\mathbf {u} _{j}}{dt}}+\sum _{j=1}^{3}x_{j}{\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}},} 2045:{\displaystyle {\frac {d\mathbf {x} _{\mathrm {A} }}{dt}}={\frac {d\mathbf {X} _{\mathrm {AB} }}{dt}}+\sum _{j=1}^{3}{\frac {dx_{j}}{dt}}\mathbf {u} _{j}+\sum _{j=1}^{3}x_{j}{\frac {d\mathbf {u} _{j}}{dt}}\,.} 9906: 6724: 1152:. The physical explanation of motions in an inertial frame is the simplest possible, requiring no fictitious forces: fictitious forces are zero, providing a means to distinguish inertial frames from others. 7947: 9832: 6563:

it is a natural extension of standard terminology (although there is no standard terminology for this case) to call this term a "centrifugal force". Whatever terminology is adopted, the observers in frame

3746:{\displaystyle \mathbf {F} _{\mathrm {fictitious} }=-m\mathbf {a} _{\mathrm {AB} }-2m\sum _{j=1}^{3}v_{j}{\frac {d\mathbf {u} _{j}}{dt}}-m\sum _{j=1}^{3}x_{j}{\frac {d^{2}\mathbf {u} _{j}}{dt^{2}}}\,.} 80: 8892: 8534: 8033: 7212: 6824:

points along the axis of the tube toward its opening at its top. It also resembles the Earth–Moon system, where the Moon always presents the same face to the Earth. In this example, unit vector

6507: 5760: 4482: 1318:). Neglecting air resistance, an object dropped from a 50-meter-high tower at the equator will fall 7.7 millimetres eastward of the spot below where it is dropped because of the Coriolis force. 1314:, which is observed in rotational frames, is ordinarily visible only in very large-scale motion like the projectile motion of long-range guns or the circulation of the Earth's atmosphere (see 10445: 3144: 1182: 6729:

This "centrifugal force" has differences from the case of a rotating frame. In the rotating frame the centrifugal force is related to the distance of the object from the origin of frame

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seen by observers in the rotational frame B, but has several additional geometric acceleration terms associated with the rotation of B. As seen in the rotational frame, the acceleration

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When moving along the equator in a ship heading in an easterly direction, objects appear to be slightly lighter than on the way back. This phenomenon has been observed and is called the

10050: 4227:. These derivative formulas now are applied to the relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with time-varying angular velocity ω( 1714: 7099: 3064:, but in addition, there are three acceleration terms related to the movement of the frame-B coordinate axes: one term related to the acceleration of the origin of frame B, namely 6433:{\displaystyle \mathbf {a} _{AB}={\frac {d^{2}}{dt^{2}}}\mathbf {X} _{AB}=\mathbf {\Omega \ \times } \left(\mathbf {\Omega \times X} _{AB}\right)=-\omega ^{2}\mathbf {X} _{AB}\,.} 3418: 11678: 8441:{\displaystyle \mathbf {r} (t)=R(t)\mathbf {u} _{R}={\begin{bmatrix}x(t)\\y(t)\end{bmatrix}}={\begin{bmatrix}R(t)\cos(\omega t+\pi /4)\\R(t)\sin(\omega t+\pi /4)\end{bmatrix}},} 6446: 1418: 1223:

From the viewpoint of an inertial reference frame stationary with respect to the road, the car is accelerating toward the centre of the circle. It is accelerating, because the

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To agree with the inertial observer, the forces applied to the walker must be exactly those found above. They can be related to the general formulas already derived, namely:

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Java physlet by Brian Fiedler illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and as seen from a non-rotating point of view.

6181: 3876: 2209:{\displaystyle {\frac {d\mathbf {x} _{\mathrm {A} }}{dt}}=\mathbf {v} _{\mathrm {AB} }+\mathbf {v} _{\mathrm {B} }+\sum _{j=1}^{3}x_{j}{\frac {d\mathbf {u} _{j}}{dt}}.} 8514: 9982: 2219:

The interpretation of this equation is that the velocity of the particle seen by observers in frame A consists of what observers in frame B call the velocity, namely

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The interpretation of this equation is as follows: the acceleration of the particle in frame A consists of what observers in frame B call the particle acceleration

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Java physlet by Brian Fiedler illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and from a non-rotating point of view.

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term) and (ii) an apparent change in the velocity of an object when its position changes, putting it nearer to or further from the axis of rotation (the change in

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The observer at rest describes the path followed by the walker as a spiral. Adopting the coordinate system shown in Figure 5, the trajectory is described by

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Many problems require use of noninertial reference frames, for example, those involving satellites and particle accelerators. Figure 2 shows a particle with

10811: 1384:). Developing this insight, Einstein formulated a theory with gravity as a fictitious force, and attributed the apparent acceleration due to gravity to the 11505: 1616: 6075: 11267: 10241: 6575: 3797: 5392:. If their observations are to result in the correct force on the object when using Newton's laws, they must consider that the additional force 1039:: "The field of all possible space-time null geodesics or photon paths unifies the absolute local non-rotation standard throughout space-time.". 11825: 11348: 2226:, plus two extra terms related to the rate of change of the frame-B coordinate axes. One of these is simply the velocity of the moving origin 1824: } cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vector 6851: 770: 10610: 3756:

Thus, problems may be solved in frame B by assuming that Newton's second law holds (with respect to quantities in that frame) and treating

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is not perpendicular to the path. Thus, the designation "centripetal acceleration" is an approximate use of this term. See, for example,

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With some browsers, hitting will freeze the motion for more detailed analysis. However, the page may have to be reloaded to restart.

478: 11799: 6733:, while in the case of an orbiting frame, the centrifugal force is independent of the distance of the object from the origin of frame 1239:

between the wheels and the road. The car is accelerating, due to the unbalanced force, which causes it to move in a circle. (See also

10922: 9785: 1293:. After release, the number of forces is zero. For someone in the spinning frame the object moves in a complicated way that needs a 8881:

with the radial vector is zero) and pointing in the direction of travel. The acceleration is the first derivative of the velocity:

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For Newton, the appearance of acceleration always indicates the existence of absolute motion – absolute motion of matter where

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of the suitcase. Now however the inertia appears in the form of a manifesting resistance to a change in its state of motion.

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Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems: Texts in applied mathematics, 17

10875: 5745:, but maintains its coordinate axes fixed in orientation, as in Figure 3. The acceleration of an observed body is now (see 11154: 11039: 10863:

The gravitational mass and the inertial mass are found experimentally to be equal to each other within experimental error.

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Figure 5 shows another example comparing the observations of an inertial observer with those of an observer on a rotating

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tropical latitudes would, in fact, need to hold on, in order to avoid being launched into orbit by the centrifugal force.

11261: 433: 8023:{\displaystyle \mathbf {\Omega \ \times } \left(\mathbf {\Omega \times } (\mathbf {X} _{AB}+\mathbf {x} _{B})\right)\,,} 3766:

Below are a number of examples applying this result for fictitious forces. More examples can be found in the article on

11849: 11208: 10536: 9584:. The first term is perpendicular to the radial direction, and pointing in the direction of travel. Its magnitude is 2 1421:

Rain and shell frame perspectives of physical (red) and fictitious (blue) forces for an object that rolls off a cliff.

1121:, the motion of a mechanical system takes place as if the coordinate system were inertial, but on every point of mass 11809: 11788: 11767: 11748: 11693: 11469: 11449: 11376: 11331: 11304: 11245: 11218: 11164: 11137: 11079: 11049: 11022: 10995: 10965: 10932: 10909: 10885: 10848: 10794: 10710: 10648: 10546: 10505: 10413: 5669: 5643: 1285:

Map and spin frame perspectives of physical (red) and fictitious (blue) forces for an object released from a carousel

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is necessary to explain observations. In an inertial frame outside the Earth, no such fictitious force is necessary.

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the speed of walking. According to simple kinematics, the velocity is then the first derivative of the trajectory:

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always points toward the centre of rotation. This example might apply to a test tube in a centrifuge, where vector

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arising from the physical interaction between two objects, which is the subject of Newton's third law – 'action is

10337: 10153:{\displaystyle \mathbf {F} _{\mathrm {fict} }=-2m\omega s\mathbf {u} _{\theta }+m\omega ^{2}R(t)\mathbf {u} _{R}.} 8180: 4231:). From the previous section, where subscript A refers to the inertial frame and B to the rotating frame, setting 3092: 1256:

spheres. In a rotating frame, Coriolis and centrifugal forces must be introduced to predict the observed tension.

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discussion does not delve into the mechanism by which the required forces are generated. That is the subject of

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where the angular acceleration term is zero for the constant rate of rotation. Because the first term, which is

1805:{\displaystyle \mathbf {x} _{\mathrm {A} }=\mathbf {X} _{\mathrm {AB} }+\sum _{j=1}^{3}x_{j}\mathbf {u} _{j}\,.} 10216: 5647: 5598: 1364:. All fictitious forces are proportional to the mass of the object upon which they act, which is also true for 1195: 354: 94: 7187:{\displaystyle \ {\frac {d}{dt}}\mathbf {u} _{2}=\mathbf {\Omega \times u_{2}} =-\omega \mathbf {u} _{1}\ \ .} 800: 11740: 10638: 10595: 6061:

where the summations are zero inasmuch as the unit vectors have no time dependence. The origin of the system

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to maintain the circular motion. This force is exerted by the ground upon the wheels, in this case, from the

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is a unit vector in the radial direction pointing from the centre of the carousel to the walker at the time

8082:{\displaystyle {\boldsymbol {\Omega }}\times \left({\boldsymbol {\Omega }}\times \mathbf {x} _{B}\right)\,,} 6556:{\displaystyle {\boldsymbol {\Omega }}\times \left({\boldsymbol {\Omega }}\times \mathbf {x} _{B}\right)\,,} 11870: 10307: 10246: 1704:

is the vector displacement of the particle as expressed in terms of the coordinates in frame B at the time

1361: 921: 845: 792: 756: 743: 504: 427: 349: 192: 11368:

A Handbook for Wireless/ RF, EMC, and High-Speed Electronics, Part of the EDN Series for Design Engineers

11041:

Problems and Solutions on Mechanics: Major American Universities Ph.D. Qualifying Questions and Solutions

10364: 7089:{\displaystyle {\frac {d}{dt}}\mathbf {u} _{1}=\mathbf {\Omega \times u_{1}} =\omega \mathbf {u} _{2}\ ;} 1095: 423: 224: 11644: 10769:

and it could exist even without friction, for example, a sledge running down a curve of a bobsled track.

8155:

drops to zero as the matter comes to an equilibrium distribution. For more details, see the articles on

202: 11865: 11502:

type of curvilinear coordinate system to another, depending from the (purely spatial) delta-curvature:

10302: 10262: 1227:

of the velocity is changing, despite the car having constant speed. This inward acceleration is called

1205: 1194:. Observed from a non-inertial frame of reference attached to the car, the fictitious force called the 895: 643: 532: 458: 318: 251: 8451:

where the added π/4 sets the path angle at 45° to start with (just an arbitrary choice of direction),

1289:

For someone in the map perspective only one force is sufficient to explain the motion: the red arrow:

8124:

based upon the distance of the tubes from the centre of rotation, and corrections applied if needed.

6266:{\displaystyle \mathbf {v} _{AB}={\frac {d}{dt}}\mathbf {X} _{AB}=\mathbf {\Omega \times X} _{AB}\ ,} 3956:{\displaystyle {\frac {d\mathbf {u} _{j}(t)}{dt}}={\boldsymbol {\Omega }}\times \mathbf {u} _{j}(t),} 1831:

simply locates the origin of frame B relative to frame A, and so cannot include rotation of frame B.

985: 397: 3779:

Despite this complication, the use of fictitious forces often simplifies the calculations involved.

11709: 11705: 10317: 10287: 5632: 1228: 1055:

problems exactly in an Earthbound reference frame, three fictitious forces must be introduced: the

1048: 796: 673: 517: 11488:

of the walker's trajectory, so "centrifugal" and "Coriolis" are approximate uses for these terms.

11294: 11190: 11112: 8148:, that is, against the wall of the tube. If the tube is spun for a long enough time, the velocity 10727: 10327: 10272: 5636: 1428:

from their environment, to protect them from the geometric acceleration due to curved spacetime.

975: 623: 387: 10957: 10460: 10034:{\displaystyle {\boldsymbol {\Omega }}\times \mathbf {u} _{\theta }=-\omega \mathbf {u} _{R}\,,} 663: 10731: 10312: 1397: 1155:

An example of the detection of a non-inertial, rotating reference frame is the precession of a

856:

Such an additional force due to nonuniform relative motion of two reference frames is called a

668: 485: 16:

Force on objects moving within a reference frame that rotates with respect to an inertial frame

11838: 10786: 10780: 9844:

a unit vector in the radial direction. The position of the walker as seen on the carousel is:

8478: 4238:= 0 to remove any translational acceleration, and focusing on only rotational properties (see 10764: 10497: 1381: 678: 653: 339: 157: 11069: 9966:{\displaystyle {\boldsymbol {\Omega }}\times \mathbf {u} _{R}=\omega \mathbf {u} _{\theta }} 6786:, j = 1, 2, 3 rotate to face the rotational axis, while the origin of the coordinate system 940:

Four fictitious forces have been defined for frames accelerated in commonly occurring ways:

883:

would also be a fictitious force (pseudo force) in a field model in which particles distort

11716:

density tensor that is derived from the covariant divergence of the energy-momentum tensor.

11617: 10679: 10257: 10206: 10164: 6771: 4224: 698: 658: 566: 562: 554: 544: 334: 327: 83: 10734:

connects them. The two transformations agree for speeds of translation much less than the

6497:{\displaystyle \mathbf {\Omega \ \times } \left(\mathbf {\Omega \times X} _{AB}\right)\,,} 3156:

To put matters in terms of forces, the accelerations are multiplied by the particle mass:

1572:

A. Consider a non-inertial frame B whose origin relative to the inertial one is given by

693: 8: 11681: 10231: 5182: 1377: 1052: 1015: 880: 473: 414: 392: 137: 132: 127: 27: 11184: 11106: 10683: 10582: 11701: 10950: 10669: 10297: 10267: 10252: 10193: 10180:

speed. It is like battling a crosswind while being thrown to the edge of the carousel.

1594:). What is the force on the particle as expressed in the coordinate system of frame B? 1555: 1393: 1355: 1083: 1028: 1019: 934: 888: 603: 344: 219: 187: 147: 944:

one caused by any acceleration relative to the origin in a straight line (rectilinear

11805: 11784: 11778: 11763: 11744: 11689: 11485: 11465: 11445: 11423: 11392: 11372: 11327: 11300: 11273: 11241: 11214: 11160: 11133: 11075: 11045: 11018: 10991: 10961: 10928: 10905: 10881: 10844: 10817: 10790: 10706: 10644: 10542: 10501: 10490: 10409: 10277: 10221: 10211: 10173: 9577: 8127:

Also, the test tube confines motion to the direction down the length of the tube, so

3767: 1597:

To answer this question, let the coordinate axis in B be represented by unit vectors

1552: 1294: 1290: 1232: 1172: 1156: 1148:

Other accelerations also give rise to fictitious forces, as described mathematically

1075: 1060: 952: 613: 570: 527: 522: 463: 239: 229: 122: 11607:{\displaystyle {\ddot {x}}^{j}+\Gamma ^{j}{}_{lk}{\dot {x}}^{l}{\dot {x}}^{k}=F^{j}} 10474: 10430: 10388: 5701:

rotate, but maintain a fixed orientation, while the origin of the coordinate system

10563: 10242:

Curvilinear coordinates § Fictitious forces in general curvilinear coordinates

1687:{\displaystyle \mathbf {x} _{\mathrm {B} }=\sum _{j=1}^{3}x_{j}\mathbf {u} _{j}\,.} 1339: 1252: 837: 708: 688: 633: 628: 574: 549: 262: 207: 182: 11830: 10523: 6161:{\displaystyle \mathbf {X} _{AB}=R\left(\cos(\omega t),\ \sin(\omega t)\right)\ ,} 1376:

observer in a closed box would not be able to detect the force of gravity; hence,

1372:

to wonder whether gravity could be modeled as a fictitious force. He noted that a

1071: 11734: 11235: 11180: 11012: 10700: 10226: 9581: 8207: 3788: 1369: 1327: 1191: 1106: 1023: 979: 703: 648: 598: 593: 512: 8198:. The carousel rotates at a constant angular velocity represented by the vector 3870:

then the time derivative of any of the three unit vectors describing frame B is

1251:

A classic example of a fictitious force in circular motion is the experiment of

11875: 10735: 10343: 10236: 10169: 6640:{\displaystyle \mathbf {F} _{\mathrm {fict} }=m\omega ^{2}\mathbf {X} _{AB}\,,} 5592: 1610:

any of { 1, 2, 3 } for the three coordinate axes. Then

1569: 1343: 1311: 1219:

The situation can be viewed from inertial as well as from non-inertial frames.

1176: 1160: 1056: 996:

forces are concerned; absolute motion of the reference system, where so-called

956: 876: 730: 638: 539: 256: 10840:

Introduction to Special and General Relativity - geodesic deviation p. 104-105

5375:

The net force upon the object according to observers in the rotating frame is

1497:= 1, 2, 3. Using these axes, the coordinates of the object according to frame 11859: 11730: 11179: 8160: 8156: 1315: 1200: 930: 833: 618: 445: 11292: 10920: 3860:{\displaystyle |{\boldsymbol {\Omega }}|={\frac {d\theta }{dt}}=\omega (t),} 1441: 945: 683: 608: 297: 177: 11132:. Cambridge UK: Cambridge University Press. pp. Eq. (17.16), p. 475. 10956:. Reston VA: American Institute of Aeronautics, and Astronautics. p. 8878: 5604: 1240: 1181: 1064: 962: 799:. Fictitious forces are invoked to maintain the validity and thus use of 10674: 1208:

as the reaction part. The reactive centrifugal force is also due to the

11713: 10292: 10189: 6924:{\displaystyle \mathbf {u} _{1}=(-\cos \omega t,\ -\sin \omega t)\ ;\ } 1331: 824: 468: 142: 11743:. Vol. 1 (3rd ed.). Butterworth-Heinenan. pp. 128–130. 3787:

pointed along the axis of rotation with the orientation given by the

1389: 1385: 1036: 984:

The role of fictitious forces in Newtonian mechanics is described by

884: 490: 9572:

The last term in the acceleration is radially inward of magnitude ω

6999:{\displaystyle \mathbf {u} _{2}=(\sin \omega t,\ -\cos \omega t)\,.} 5621: 5209:

of the particle is given by rearrangement of the above equation as:

1000:

forces, such as inertial forces or those of Coriolis, are concerned.

8195: 1486:

is determined by the unit vectors along its coordinate directions,

1373: 1236: 1101: 409: 292: 267: 10730:

connects these frames and in the special theory of relativity the

9606:

or what have you, that could account for these fictitious forces.

11801:

Spacecraft Dynamics and Control: A Practical Engineering Approach

11641:

are the contravariant components of the force per unit mass, and

10778: 10322: 10282: 9901:{\displaystyle \mathbf {x} _{\mathrm {B} }=R(t)\mathbf {u} _{R},} 6719:{\displaystyle |\mathbf {F} _{\mathrm {fict} }|=m\omega ^{2}R\,.} 2236:

To find the acceleration, another time differentiation provides:

1365: 1261: 1209: 1149: 828: 382: 235: 152: 11729: 11010: 10524:

NASA notes:(23) Accelerated Frames of Reference: Inertial Forces

8185: 8030:

is of the same form as the normal centrifugal force expression:

7197:

Hence, the acceleration of a moving object is expressed as (see

6504:

is of the same form as the normal centrifugal force expression:

840:

or contact forces. It is just a consequence of the acceleration

791:

that appears to act on a mass whose motion is described using a

11233: 6845:

rotate at the same rate as the origin of coordinates. That is,

6737:, but instead depends upon the distance of the origin of frame 2518:

Using the same formula already used for the time derivative of

2055:

The second term summation is the velocity of the particle, say

1436: 1335: 441: 287: 197: 10583:

The Feynman Lectures on Physics Vol. I Ch. 12-5: Pseudo forces

9827:{\displaystyle \mathbf {v} _{\mathrm {B} }=s\mathbf {u} _{R},} 5737:

rotates with a constant angular speed ω in a circle of radius

5686:

Figure 3: An orbiting but fixed orientation coordinate system

1190:

The effect of a fictitious force also occurs when a car takes

11696:, Section 11.4; or: Adler, R., Bazin, M., & Schiffer, M. 11464:(2nd ed.). Hackensack NJ: World Scientific. p. 37. 10983: 9779:

In this example, the velocity seen in the rotating frame is:

6798:

As a combination example, Figure 4 shows a coordinate system

1417: 1100:

Observers inside a closed box that is moving with a constant

788: 277: 272: 214: 11074:. Sausalito CA: University Science Books. pp. 343–344. 1281: 1089: 810: 11323:

Consolidation of Soils: Testing and Evaluation: a Symposium

8168: 5733:

As a related example, suppose the moving coordinate system

1834:

Taking a time derivative, the velocity of the particle is:

1545: 1360:

The notion of "fictitious force" also arises in Einstein's

965:, caused by a variable rate of rotation, should that occur. 282: 245: 10990:. Cambridge UK: Cambridge University Press. p. xvii. 5682: 1117:

which moves by translation relative to an inertial system

11826:

Q and A from Richard C. Brill, Honolulu Community College

11293:

Bea K Lalmahomed; Sarah Springman; Bhawani Singh (2002).

11037: 10987:

The Principles of Circular Accelerators and Storage Rings

10921:

Alberto Isidori; Lorenzo Marconi; Andrea Serrani (2003).

10843:. Cambridge UK: Cambridge University Press. p. 105. 10698: 10636: 10487: 5195:

on the object is, therefore, not simply the acceleration

1159:. In the non-inertial frame of the Earth, the fictitious 75:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}} 11388:

Physics for Scientists And Engineers With Modern Physics

11152: 10874:

Motz, Lloyd; Weaver, Jefferson Hane (11 November 2013).

10627:

often is limited to this case. See Lanczos, for example.

8181:

Coriolis force § Tossed ball on a rotating carousel

909:, fictitious forces are always proportional to the mass 11418:

is the instantaneous distance from the rotational axis

10640:

The Principles of Electromagnetic Theory and Relativity

1007:

The Principles of Electromagnetic Theory and Relativity

11429:

as seen by the inertial observer, and the unit vector

10924:

Robust Autonomous Guidance: An Internal Model Approach

9395: 9283: 9180: 9068: 8955: 8685: 8583: 8332: 8285: 6790:

moves at constant angular rate ω about the fixed axis

5705:

moves at constant angular rate ω about the fixed axis

3095: 879:, maybe better to be called: the Coriolis effect. The 11831:

NASA's David Stern: Lesson Plans for Teachers #23 on

11647: 11620: 11508: 11213:. New York: Courier Dover Publications. p. 103. 11159:. New York: Courier Dover Publications. p. 233. 10541:. New York: Courier Dover Publications. p. 100. 10053: 9985: 9924: 9853: 9788: 9618: 8890: 8532: 8481: 8235: 8036: 7950: 7210: 7102: 7013: 6937: 6854: 6659: 6578: 6510: 6449: 6289: 6184: 6078: 5758: 5429: 5218: 5019: 4480: 4251: 3975: 3879: 3800: 3522: 3421: 3165: 2799: 2534: 2245: 2071: 1843: 1717: 1619: 1166: 39: 10813:

Fictitional Forces and apparent gravitational fields

10333: 9915:

is zero for uniform angular rotation. Noticing that

11684:of the second kind, see, for instance: David, Kay, 11384: 11296:

Constitutive and Centrifuge Modelling: Two Extremes

11104: 5690:, shown at three different times. The unit vectors 5191:due to what observers in the inertial frame A call 1583:). Let the position of the particle in frame B be 11672: 11633: 11606: 11240:(2nd ed.). NY: Springer-Verlag. p. 251. 11125: 11067: 10949: 10900:Edwin F. Taylor and John Archibald Wheeler (2000) 10489: 10152: 10033: 9965: 9900: 9826: 9768: 9561: 8848: 8508: 8440: 8081: 8022: 7933: 7186: 7088: 6998: 6923: 6718: 6639: 6555: 6496: 6432: 6265: 6160: 6050: 5579: 5364: 5170: 4995: 4465: 4212: 3955: 3859: 3745: 3502: 3377: 3138: 3033: 2777: 2507: 2208: 2044: 1804: 1686: 74: 11260:However, the Earth–Moon system rotates about its 11206: 10534: 10496:. New York: Courier Dover Publications. pp. 6806:as in Figure 3, but the coordinate axes in frame 1380:are equivalent to inertial reference frames (the 929:(A d'Alembert force is not to be confused with a 11857: 11437: 8877:(as can be verified by noticing that the vector 3405:is related to the actual force on the particle, 1349: 1022:in all non-inertial frames. Inertial frames are 11776: 11484:A circle about the axis of rotation is not the 11014:Theoretical Mechanics of Particles and Continua 10719: 6779:similar to Figure 3, but in which unit vectors 11797: 11704:to obtain curvature in spacetime according to 11017:. Courier Dover Publications. pp. 33–39. 10947: 10809: 10694: 10692: 5611: 5006:Collecting terms, the result is the so-called 3773: 1247:the car stationary in this non-inertial frame. 11265: 10836: 10705:. Berlin: Springer. pp. §27 pp. 129 ff. 8472:) increases steadily with time according to: 6171:leading to a velocity of the origin of frame 3139:{\textstyle \sum x_{j}\,d\mathbf {u} _{j}/dt} 1437:Mathematical derivation of fictitious forces 1321: 1031:, since these physics are explained with the 916:The fictitious force that has been called an 816:toward the rim of a centrifuge or carousel. 764: 11044:. Singapore: World Scientific. p. 183. 10779:Daniel Kleppner; Robert J. Kolenkow (1973). 6276:leading to an acceleration of the origin of 5713:passes through the origin of inertial frame 1697:The interpretation of this equation is that 887:due to their mass, such as in the theory of 11804:. Cambridge University Press. Chapter 4.8. 11780:Classical Dynamics of Particles and Systems 11757: 11364: 11319: 11011:Alexander L Fetter; John D Walecka (2003). 10816:. Singapore: World Scientific. p. 40. 10702:Mathematical Methods of Classical Mechanics 10689: 6831:retains a fixed orientation, while vectors 5650:. Unsourced material may be challenged and 4223:as is verified using the properties of the 2525:, the velocity derivative on the right is: 1708:. From frame A the particle is located at: 1392:. This idea underlies Einstein's theory of 11495: 11441:Orbital Mechanics for Engineering Students 11346: 11234:Jerold E. Marsden; Tudor.S. Ratiu (1999). 11129:Classical Mechanics: An Undergraduate Text 10873: 10401: 1326:Fictitious forces can be considered to do 771: 757: 11457: 11414:: There is a subtlety here: the distance 11063: 11061: 10673: 10027: 9551: 8075: 8016: 7923: 6992: 6712: 6633: 6549: 6490: 6426: 5741:about the fixed origin of inertial frame 5670:Learn how and when to remove this message 5164: 3739: 3109: 2038: 1798: 1680: 1330:, provided that they move an object on a 1186:centrifugal forces present in this frame. 1090:Detection of non-inertial reference frame 811:Measureable examples of fictitious forces 11357: 11340: 10984:Philip J. Bryant; Kjell Johnsen (1993). 10830: 10803: 10772: 10630: 10528: 10405:A Modern Approach to Classical Mechanics 8184: 6775:Figure 4: An orbiting coordinate system 6770: 6760: 5681: 1440: 1416: 1280: 1180: 867:A Modern Approach to Classical Mechanics 11254: 11227: 11210:The Variational Principles of Mechanics 11200: 11119: 10538:The Variational Principles of Mechanics 9576:, which is therefore the instantaneous 11858: 11391:. Pearson Prentice-Hall. p. 301. 11183:& William Raymond Longley (1910). 11111:(2nd ed.). McGraw-Hill. pp. 11058: 11004: 10977: 10914: 10857: 10762:The force in this example is known as 10664:Gilson, James G. (September 1, 2004), 10663: 8174: 8141:and the Coriolis force is opposite to 1815:As an aside, the unit vectors { 1125:an additional "inertial force" acted: 11286: 11173: 6745:centre of rotation, resulting in the 1539: 795:, such as a linearly accelerating or 453:Newton's law of universal gravitation 11444:. Butterworth-Heinemann. p. 5. 11146: 11038:Yung-kuo Lim; Yuan-qi Qiang (2001). 10941: 10576: 10561: 10488:Max Born; Günther Leibfried (1962). 9595: 8213: 5648:adding citations to reliable sources 5615: 2790: 11712:and a spacetime form that uses the 11326:. ASTM International. p. 590. 11313: 11153:Georg Joos; Ira M. Freeman (1986). 11031: 10481: 8206:, pointing upward according to the 7955: 7815: 7671: 7464: 6454: 6359: 5008:acceleration transformation formula 803:, in frames of reference which are 434:Mechanics of planar particle motion 42: 13: 11723: 11698:Introduction to General Relativity 11673:{\displaystyle \Gamma ^{j}{}_{lk}} 11649: 11532: 11299:. Taylor and Francis. p. 82. 10699:Vladimir Igorevich Arnold (1989). 10637:Marie-Antoinette Tonnelat (2002). 10395: 10071: 10068: 10065: 10062: 9862: 9797: 9757: 9709: 9670: 9636: 9633: 9630: 9627: 8167:These three motions influence the 6682: 6679: 6676: 6673: 6596: 6593: 6590: 6587: 5725:from the origin of inertial frame 5568: 5520: 5481: 5447: 5444: 5441: 5438: 5353: 5308: 5272: 5244: 5227: 5153: 5115: 5073: 5045: 5028: 4762: 4514: 4493: 4308: 4273: 3584: 3581: 3558: 3555: 3552: 3549: 3546: 3543: 3540: 3537: 3534: 3531: 3491: 3488: 3485: 3482: 3479: 3476: 3473: 3470: 3467: 3464: 3447: 3430: 3214: 3211: 3191: 3174: 2876: 2859: 2856: 2821: 2703: 2549: 2328: 2305: 2302: 2267: 2134: 2117: 2114: 2086: 1895: 1892: 1858: 1746: 1743: 1726: 1628: 1167:Example concerning Circular motion 1140:is the acceleration of the system 14: 11887: 11819: 10443:Harvard University lecture demo, 10408:. World Scientific. p. 180. 6749:centrifugal fictitious force for 6443:Because the first term, which is 5399:is present, so the end result is 2062:as measured in frame B. That is: 11762:(3rd ed.). Addison-Wesley. 11688:(1988) McGraw-Hill Book Company 10877:Example train and gravity, p 101 10596:"Inertia and Newton's third law" 10336: 10137: 10097: 10056: 10017: 9996: 9987: 9953: 9935: 9926: 9885: 9856: 9811: 9791: 9751: 9731: 9703: 9694: 9683: 9664: 9655: 9621: 9541: 9501: 8896: 8829: 8799: 8538: 8267: 8237: 8060: 8051: 8038: 7998: 7980: 7968: 7952: 7910: 7901: 7881: 7858: 7840: 7828: 7812: 7786: 7777: 7764: 7741: 7732: 7712: 7690: 7684: 7668: 7642: 7633: 7620: 7570: 7566: 7559: 7505: 7483: 7477: 7461: 7422: 7346: 7285: 7267: 7230: 7165: 7148: 7144: 7137: 7123: 7070: 7056: 7052: 7045: 7031: 6940: 6857: 6667: 6620: 6581: 6534: 6525: 6512: 6473: 6467: 6451: 6413: 6378: 6372: 6356: 6339: 6292: 6244: 6238: 6220: 6187: 6081: 6028: 6007: 5967: 5894: 5833: 5815: 5778: 5620: 5562: 5542: 5514: 5505: 5494: 5475: 5466: 5432: 5347: 5327: 5302: 5293: 5282: 5266: 5257: 5238: 5221: 5147: 5138: 5125: 5109: 5089: 5067: 5058: 5039: 5022: 4962: 4922: 4909: 4895: 4844: 4815: 4775: 4756: 4720: 4711: 4698: 4650: 4630: 4570: 4561: 4508: 4487: 4432: 4362: 4302: 4267: 4183: 4174: 4161: 4147: 4127: 4087: 4072: 4058: 4038: 3991: 3931: 3922: 3888: 3807: 3711: 3638: 3575: 3525: 3458: 3441: 3424: 3341: 3268: 3205: 3185: 3168: 3115: 3000: 2930: 2870: 2850: 2815: 2751: 2697: 2671: 2619: 2543: 2474: 2404: 2322: 2296: 2261: 2182: 2128: 2108: 2080: 2017: 1965: 1886: 1852: 1788: 1737: 1720: 1670: 1622: 1275:object released from a carousel 1109:presents the following theorem: 738: 737: 724: 57: 11850:Motion over a parabolic surface 11478: 11405: 11097: 11088: 10894: 10866: 10756: 10741: 10657: 10617: 10602: 10587: 10555: 10492:Einstein's Theory of Relativity 10389:"Fictitious force - Britannica" 10365:"What is a "fictitious force"?" 8866:a unit vector perpendicular to 3388:The force observed in frame B, 1445:Figure 2: An object located at 823:might also be referred to as a 819:The fictitious force called a 793:non-inertial frame of reference 11264:, not the Earth's center; see 11195:centrifugal force theoretical. 11105:JL Synge; BA Griffith (1949). 10517: 10467: 10452: 10437: 10422: 10381: 10357: 10183: 10132: 10126: 9880: 9874: 9715: 9690: 9533: 9527: 9463: 9440: 9427: 9404: 9387: 9381: 9354: 9331: 9318: 9295: 9248: 9225: 9212: 9189: 9172: 9166: 9139: 9116: 9103: 9080: 9023: 9000: 8987: 8964: 8906: 8900: 8824: 8818: 8756: 8733: 8720: 8697: 8677: 8671: 8651: 8628: 8615: 8592: 8548: 8542: 8491: 8485: 8424: 8401: 8392: 8386: 8376: 8353: 8344: 8338: 8313: 8307: 8297: 8291: 8262: 8256: 8247: 8241: 8008: 7975: 7868: 7835: 6989: 6953: 6909: 6870: 6689: 6661: 6144: 6135: 6120: 6111: 6065:is located according to frame 5526: 5501: 5314: 5289: 4978: 4972: 4831: 4825: 4736: 4730: 4586: 4580: 4199: 4193: 4103: 4097: 4007: 4001: 3947: 3941: 3904: 3898: 3851: 3845: 3812: 3802: 3791:, and with magnitude given by 1047:The surface of the Earth is a 1: 11741:Course of Theoretical Physics 10904:(Addison Wesley Longman, NY) 10750:"centripetal force, pp 48-49" 10666:Mach's Principle II, p.1, p.9 10475:"Inertial force - Britannica" 10350: 1350:Gravity as a fictitious force 1005:Marie-Antoinette Tonnelat in 969: 836:between two objects, such as 801:Newton's second law of motion 360:Koopman–von Neumann mechanics 11385:Douglas C. Giancoli (2007). 10782:An Introduction to Mechanics 10747:Science of everyday things, 10308:Non-inertial reference frame 6767:Clohessy–Wiltshire equations 5602:, and the third term is the 5590:Here, the first term is the 1411:ball that rolls off a cliff 1404: 1378:freefalling reference frames 1362:general theory of relativity 1268: 846:non-inertial reference frame 428:Non-inertial reference frame 7: 11733:and E. M. Lifshitz (1976). 11363:For a similar example, see 11350:The Popular Science Monthly 11272:. Read Books. p. 307. 11126:R. Douglas Gregory (2006). 11068:John Robert Taylor (2004). 10199: 9911:and the time derivative of 6802:that orbits inertial frame 5612:Orbiting coordinate systems 3774:Rotating coordinate systems 3047: 1482:. The orientation of frame 1096:Inertial frame of reference 1042: 1014:Fictitious forces arise in 844:of the physical object the 827:. It is due to an object's 355:Appell's equation of motion 225:Inertial frame of reference 10: 11892: 11844:Motion over a flat surface 11489: 11427:of the walker's trajectory 11371:. Newnes. pp. 60–61. 11207:Cornelius Lanczos (1986). 10611:"about Newton's third law" 10535:Cornelius Lanczos (1986). 8178: 7198: 6764: 6753:objects observed in frame 5746: 4239: 1353: 1322:Fictitious forces and work 1206:reactive centrifugal force 1170: 1093: 1063:(described below) and the 973: 11438:Howard D. Curtis (2005). 11422:. However, it is not the 10643:. Springer. p. 113. 5717:, so the origin of frame 5596:, the second term is the 961:and a fourth, called the 920:is also referred to as a 11777:Jerry B. Marion (1970). 11710:Einstein field equations 10972:orbit coordinate system. 10927:. Springer. p. 61. 10608:Physics stack exchange, 10318:Rotating reference frame 10288:Inertial reference frame 9578:centripetal acceleration 8509:{\displaystyle R(t)=st,} 3763:as an additional force. 1229:centripetal acceleration 1049:rotating reference frame 951:two involving rotation: 797:rotating reference frame 518:Rotating reference frame 350:Hamilton–Jacobi equation 11798:Marcel J. Sidi (1997). 11189:. Boston: Gin. p. 11108:Principles of Mechanics 10948:Shuh-Jing Ying (1997). 10810:Fritz Rohrlich (2007). 10785:. McGraw-Hill. p. 10728:Galilean transformation 10328:Uniform circular motion 10303:Newton's laws of motion 10273:Generalized coordinates 8464:. The radial distance 1456:is located at location 1113:In a coordinate system 976:Absolute space and time 459:Newton's laws of motion 319:Newton's laws of motion 11674: 11635: 11608: 11266:Simon Newcomb (2007). 10837:Hans Stephani (2004). 10732:Lorentz transformation 10313:Orthogonal coordinates 10263:Frenet–Serret formulas 10247:d'Alembert's principle 10154: 10035: 9967: 9902: 9828: 9770: 9563: 8850: 8510: 8442: 8191: 8083: 8024: 7935: 7605: 7544: 7393: 7324: 7188: 7090: 7000: 6925: 6795: 6720: 6641: 6557: 6498: 6434: 6267: 6162: 6052: 5938: 5872: 5730: 5581: 5366: 5172: 4997: 4949: 4882: 4802: 4686: 4612: 4549: 4467: 4406: 4340: 4214: 3957: 3861: 3747: 3685: 3619: 3504: 3379: 3315: 3249: 3140: 3035: 2974: 2911: 2779: 2732: 2652: 2589: 2509: 2448: 2368: 2210: 2163: 2046: 1998: 1935: 1806: 1775: 1688: 1657: 1536: 1467:. The origin of frame 1463:in accelerating frame 1422: 1286: 1187: 1146: 1012: 872: 486:Simple harmonic motion 399:Euler's laws of motion 193:D'Alembert's principle 76: 11675: 11636: 11634:{\displaystyle F^{j}} 11609: 11186:Theoretical Mechanics 10902:Exploring black holes 10172:(first term) and the 10155: 10036: 9968: 9903: 9829: 9771: 9564: 8851: 8511: 8443: 8188: 8084: 8025: 7936: 7585: 7524: 7373: 7304: 7189: 7091: 7001: 6926: 6774: 6761:Orbiting and rotating 6721: 6642: 6558: 6499: 6435: 6268: 6163: 6053: 5918: 5852: 5685: 5582: 5367: 5183:physical acceleration 5173: 4998: 4929: 4862: 4782: 4666: 4592: 4529: 4468: 4386: 4320: 4215: 3958: 3862: 3748: 3665: 3599: 3505: 3380: 3295: 3229: 3141: 3036: 2954: 2891: 2780: 2712: 2632: 2569: 2510: 2428: 2348: 2211: 2143: 2047: 1978: 1915: 1807: 1755: 1689: 1637: 1444: 1420: 1382:equivalence principle 1284: 1184: 1111: 990: 854: 340:Hamiltonian mechanics 158:Statistical mechanics 77: 11758:Keith Symon (1971). 11706:stress–energy tensor 11645: 11618: 11506: 11365:Ron Schmitt (2002). 11320:Raymond Nen (1986). 11094:Kleppner pages 62–63 10562:Seligman, Courtney. 10258:Free motion equation 10207:Analytical mechanics 10165:straight-line motion 10051: 9983: 9922: 9851: 9786: 9616: 8888: 8530: 8479: 8233: 8034: 7948: 7208: 7100: 7011: 6935: 6852: 6810:turn so unit vector 6657: 6576: 6508: 6447: 6287: 6182: 6076: 5756: 5721:is a fixed distance 5644:improve this section 5427: 5216: 5193:real external forces 5017: 4478: 4249: 4225:vector cross product 3973: 3877: 3798: 3520: 3419: 3163: 3093: 2797: 2532: 2243: 2069: 1841: 1715: 1617: 834:physical interaction 563:Angular acceleration 555:Rotational frequency 335:Lagrangian mechanics 328:Analytical mechanics 84:Second law of motion 37: 11871:Classical mechanics 11682:Christoffel symbols 11461:Accelerator physics 11424:radius of curvature 11347:D Appleton (1877). 11156:Theoretical Physics 11071:Classical Mechanics 10684:2004physics...9010G 10564:"Fictitious Forces" 10458:ThoughtCo website, 10402:Harald Iro (2002). 10369:Scientific American 10232:Classical mechanics 8175:Crossing a carousel 1053:classical mechanics 1016:classical mechanics 896:Newton's second law 881:gravitational force 415:Harmonic oscillator 393:Equations of motion 28:Classical mechanics 22:Part of a series on 11783:. Academic Press. 11702:general relativity 11670: 11631: 11604: 11458:S. Y. Lee (2004). 10298:Kinetics (physics) 10268:General relativity 10253:Dynamics (physics) 10249:of inertial forces 10150: 10031: 9963: 9898: 9824: 9766: 9559: 9557: 9468: 9359: 9253: 9144: 9028: 8846: 8844: 8761: 8656: 8506: 8438: 8429: 8318: 8192: 8079: 8020: 7931: 7929: 7184: 7086: 6996: 6921: 6796: 6716: 6650:and of magnitude: 6637: 6553: 6494: 6430: 6263: 6158: 6048: 6046: 5731: 5577: 5362: 5168: 4993: 4991: 4463: 4210: 3953: 3857: 3743: 3500: 3375: 3136: 3031: 2775: 2505: 2206: 2042: 1802: 1684: 1568:) in a particular 1540:General derivation 1537: 1452:in inertial frame 1423: 1394:general relativity 1356:General relativity 1287: 1188: 1029:general relativity 1020:special relativity 889:general relativity 852:As stated by Iro: 731:Physics portal 345:Routhian mechanics 220:Frame of reference 72: 11866:Fictitious forces 11582: 11563: 11519: 11486:osculating circle 11398:978-0-13-149508-1 11279:978-1-4067-4574-0 11269:Popular Astronomy 11103:See for example, 10952:Advanced Dynamics 10872:Motz and Weaver, 10823:978-981-270-004-9 10461:"Coriolis effect" 10278:Generalized force 10222:Centripetal force 10217:Centrifugal force 10212:Applied mechanics 10188:Notice that this 10174:centrifugal force 9744: 9596:Rotating observer 9538: 9498: 9492: 9058: 8948: 8795: 8576: 8214:Inertial observer 7957: 7922: 7899: 7817: 7762: 7756: 7753: 7730: 7673: 7618: 7584: 7578: 7557: 7523: 7466: 7448: 7406: 7372: 7365: 7337: 7303: 7256: 7180: 7177: 7119: 7105: 7082: 7027: 6973: 6920: 6914: 6893: 6456: 6361: 6335: 6259: 6216: 6154: 6128: 6040: 6022: 5993: 5951: 5913: 5885: 5851: 5804: 5697:, j = 1, 2, 3 do 5680: 5679: 5672: 5599:centrifugal force 5555: 5340: 5102: 4857: 4662: 4643: 4525: 4458: 4381: 4353: 4295: 4140: 4115: 4051: 4026: 3916: 3837: 3768:centrifugal force 3737: 3657: 3371: 3367: 3287: 3146:due to change in 3055: 3054: 3026: 2949: 2890: 2843: 2770: 2690: 2615: 2564: 2500: 2423: 2394: 2343: 2289: 2201: 2101: 2036: 1961: 1910: 1873: 1432: 1431: 1398:Eötvös experiment 1334:that changes its 1306: 1305: 1297:: the blue arrow. 1295:centrifugal force 1291:centripetal force 1233:centripetal force 1196:centrifugal force 1173:Centrifugal force 1157:Foucault pendulum 1076:Foucault pendulum 1061:centrifugal force 953:centrifugal force 781: 780: 528:Centrifugal force 523:Centripetal force 479:Euler's equations 464:Relative velocity 240:Moment of inertia 70: 44: 11883: 11815: 11794: 11773: 11754: 11717: 11679: 11677: 11676: 11671: 11669: 11668: 11660: 11657: 11656: 11640: 11638: 11637: 11632: 11630: 11629: 11613: 11611: 11610: 11605: 11603: 11602: 11590: 11589: 11584: 11583: 11575: 11571: 11570: 11565: 11564: 11556: 11552: 11551: 11543: 11540: 11539: 11527: 11526: 11521: 11520: 11512: 11499: 11493: 11482: 11476: 11475: 11455: 11409: 11403: 11402: 11382: 11361: 11355: 11354: 11344: 11338: 11337: 11317: 11311: 11310: 11290: 11284: 11283: 11258: 11252: 11251: 11231: 11225: 11224: 11204: 11198: 11197: 11177: 11171: 11170: 11150: 11144: 11143: 11123: 11117: 11116: 11101: 11095: 11092: 11086: 11085: 11065: 11056: 11055: 11035: 11029: 11028: 11008: 11002: 11001: 10981: 10975: 10974: 10955: 10945: 10939: 10938: 10918: 10912: 10898: 10892: 10891: 10870: 10864: 10861: 10855: 10854: 10834: 10828: 10827: 10807: 10801: 10800: 10776: 10770: 10760: 10754: 10753: 10745: 10739: 10723: 10717: 10716: 10696: 10687: 10686: 10677: 10661: 10655: 10654: 10634: 10628: 10625:d'Alembert force 10621: 10615: 10614: 10606: 10600: 10599: 10591: 10585: 10580: 10574: 10573: 10571: 10570: 10559: 10553: 10552: 10532: 10526: 10521: 10515: 10514: 10512:inertial forces. 10495: 10485: 10479: 10478: 10471: 10465: 10464: 10456: 10450: 10449: 10446:"Coriolis force" 10441: 10435: 10434: 10431:"Coriolis force" 10426: 10420: 10419: 10399: 10393: 10392: 10385: 10379: 10378: 10376: 10375: 10361: 10346: 10341: 10340: 10159: 10157: 10156: 10151: 10146: 10145: 10140: 10122: 10121: 10106: 10105: 10100: 10076: 10075: 10074: 10059: 10040: 10038: 10037: 10032: 10026: 10025: 10020: 10005: 10004: 9999: 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1898: 1889: 1879: 1874: 1872: 1864: 1863: 1862: 1861: 1855: 1845: 1811: 1809: 1808: 1803: 1797: 1796: 1791: 1785: 1784: 1774: 1769: 1751: 1750: 1749: 1740: 1731: 1730: 1729: 1723: 1693: 1691: 1690: 1685: 1679: 1678: 1673: 1667: 1666: 1656: 1651: 1633: 1632: 1631: 1625: 1405: 1269: 1253:rotating spheres 1231:, it requires a 1010: 923:d'Alembert force 870: 838:electromagnetism 785:fictitious force 773: 766: 759: 746: 741: 740: 733: 729: 728: 634:Johann Bernoulli 629:Daniel Bernoulli 550:Tangential speed 454: 430: 405:Fictitious force 400: 252:Mechanical power 242: 183:Angular momentum 81: 79: 78: 73: 71: 69: 61: 60: 51: 46: 45: 19: 18: 11891: 11890: 11886: 11885: 11884: 11882: 11881: 11880: 11856: 11855: 11833:Inertial Forces 11822: 11812: 11791: 11770: 11751: 11726: 11724:Further reading 11721: 11720: 11686:Tensor Calculus 11661: 11659: 11658: 11652: 11648: 11646: 11643: 11642: 11625: 11621: 11619: 11616: 11615: 11598: 11594: 11585: 11574: 11573: 11572: 11566: 11555: 11554: 11553: 11544: 11542: 11541: 11535: 11531: 11522: 11511: 11510: 11509: 11507: 11504: 11503: 11500: 11496: 11483: 11479: 11472: 11452: 11435: 11420:of the carousel 11410: 11406: 11399: 11379: 11362: 11358: 11345: 11341: 11334: 11318: 11314: 11307: 11291: 11287: 11280: 11259: 11255: 11248: 11232: 11228: 11221: 11205: 11201: 11181:Percey F. Smith 11178: 11174: 11167: 11151: 11147: 11140: 11124: 11120: 11102: 11098: 11093: 11089: 11082: 11066: 11059: 11052: 11036: 11032: 11025: 11009: 11005: 10998: 10982: 10978: 10968: 10946: 10942: 10935: 10919: 10915: 10899: 10895: 10888: 10871: 10867: 10862: 10858: 10851: 10835: 10831: 10824: 10808: 10804: 10797: 10777: 10773: 10765:ground reaction 10761: 10757: 10748: 10746: 10742: 10724: 10720: 10713: 10697: 10690: 10675:physics/0409010 10662: 10658: 10651: 10635: 10631: 10622: 10618: 10609: 10607: 10603: 10598:. 3 March 2021. 10594: 10593:Physics Forum, 10592: 10588: 10581: 10577: 10568: 10566: 10560: 10556: 10549: 10533: 10529: 10522: 10518: 10508: 10486: 10482: 10473: 10472: 10468: 10459: 10457: 10453: 10444: 10442: 10438: 10429: 10427: 10423: 10416: 10400: 10396: 10387: 10386: 10382: 10373: 10371: 10363: 10362: 10358: 10353: 10342: 10335: 10332: 10227:Circular motion 10202: 10186: 10141: 10136: 10135: 10117: 10113: 10101: 10096: 10095: 10061: 10060: 10055: 10054: 10052: 10049: 10048: 10021: 10016: 10015: 10000: 9995: 9994: 9986: 9984: 9981: 9980: 9957: 9952: 9951: 9939: 9934: 9933: 9925: 9923: 9920: 9919: 9889: 9884: 9883: 9861: 9860: 9855: 9854: 9852: 9849: 9848: 9843: 9815: 9810: 9809: 9796: 9795: 9790: 9789: 9787: 9784: 9783: 9756: 9755: 9750: 9749: 9736: 9730: 9726: 9724: 9708: 9707: 9702: 9701: 9693: 9682: 9669: 9668: 9663: 9662: 9654: 9626: 9625: 9620: 9619: 9617: 9614: 9613: 9598: 9582:circular motion 9556: 9555: 9545: 9540: 9539: 9518: 9514: 9505: 9500: 9499: 9475: 9474: 9467: 9466: 9455: 9431: 9430: 9419: 9391: 9390: 9372: 9368: 9358: 9357: 9346: 9322: 9321: 9310: 9279: 9278: 9260: 9259: 9252: 9251: 9240: 9216: 9215: 9204: 9176: 9175: 9157: 9153: 9143: 9142: 9131: 9107: 9106: 9095: 9064: 9063: 9050: 9042: 9040: 9027: 9026: 9015: 8991: 8990: 8979: 8951: 8950: 8941: 8937: 8933: 8923: 8919: 8918: 8916: 8909: 8895: 8891: 8889: 8886: 8885: 8872: 8865: 8843: 8842: 8833: 8828: 8827: 8803: 8798: 8797: 8787: 8779: 8777: 8768: 8767: 8760: 8759: 8748: 8724: 8723: 8712: 8681: 8680: 8655: 8654: 8643: 8619: 8618: 8607: 8579: 8578: 8568: 8560: 8558: 8551: 8537: 8533: 8531: 8528: 8527: 8480: 8477: 8476: 8459: 8428: 8427: 8416: 8380: 8379: 8368: 8328: 8327: 8317: 8316: 8301: 8300: 8281: 8280: 8271: 8266: 8265: 8236: 8234: 8231: 8230: 8216: 8208:right-hand rule 8202:with magnitude 8183: 8177: 8154: 8147: 8140: 8134:is opposite to 8133: 8114: 8107: 8096: 8064: 8059: 8058: 8050: 8049: 8045: 8037: 8035: 8032: 8031: 8002: 7997: 7996: 7984: 7979: 7978: 7967: 7966: 7962: 7951: 7949: 7946: 7945: 7928: 7927: 7914: 7909: 7908: 7900: 7885: 7880: 7879: 7862: 7857: 7856: 7844: 7839: 7838: 7827: 7826: 7822: 7811: 7802: 7801: 7790: 7785: 7784: 7776: 7775: 7771: 7763: 7745: 7740: 7739: 7731: 7716: 7711: 7710: 7694: 7683: 7682: 7678: 7667: 7658: 7657: 7646: 7641: 7640: 7632: 7631: 7627: 7619: 7610: 7606: 7600: 7589: 7569: 7565: 7558: 7549: 7545: 7539: 7528: 7509: 7504: 7503: 7487: 7476: 7475: 7471: 7460: 7451: 7450: 7441: 7437: 7433: 7426: 7421: 7420: 7414: 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6104: 6100: 6085: 6080: 6079: 6077: 6074: 6073: 6045: 6044: 6032: 6027: 6026: 6011: 6006: 6005: 5996: 5995: 5986: 5982: 5978: 5971: 5966: 5965: 5959: 5955: 5954: 5952: 5943: 5939: 5933: 5922: 5905: 5898: 5893: 5892: 5888: 5886: 5877: 5873: 5867: 5856: 5837: 5832: 5831: 5819: 5814: 5813: 5806: 5797: 5793: 5789: 5782: 5777: 5776: 5770: 5766: 5765: 5763: 5759: 5757: 5754: 5753: 5696: 5676: 5665: 5659: 5656: 5641: 5625: 5614: 5567: 5566: 5561: 5560: 5547: 5541: 5537: 5535: 5519: 5518: 5513: 5512: 5504: 5493: 5480: 5479: 5474: 5473: 5465: 5437: 5436: 5431: 5430: 5428: 5425: 5424: 5419: 5412: 5405: 5398: 5391: 5381: 5352: 5351: 5346: 5345: 5332: 5326: 5322: 5320: 5307: 5306: 5301: 5300: 5292: 5281: 5271: 5270: 5265: 5264: 5256: 5243: 5242: 5237: 5236: 5226: 5225: 5220: 5219: 5217: 5214: 5213: 5208: 5201: 5190: 5152: 5151: 5146: 5145: 5137: 5136: 5132: 5124: 5114: 5113: 5108: 5107: 5094: 5088: 5084: 5082: 5072: 5071: 5066: 5065: 5057: 5044: 5043: 5038: 5037: 5027: 5026: 5021: 5020: 5018: 5015: 5014: 4990: 4989: 4966: 4961: 4960: 4954: 4950: 4944: 4933: 4921: 4920: 4916: 4908: 4899: 4894: 4893: 4887: 4883: 4877: 4866: 4849: 4843: 4839: 4837: 4819: 4814: 4813: 4807: 4803: 4797: 4786: 4774: 4761: 4760: 4755: 4754: 4745: 4744: 4724: 4719: 4718: 4710: 4709: 4705: 4697: 4691: 4687: 4681: 4670: 4654: 4649: 4648: 4635: 4629: 4625: 4623: 4617: 4613: 4607: 4596: 4574: 4569: 4568: 4560: 4554: 4550: 4544: 4533: 4513: 4512: 4507: 4506: 4499: 4492: 4491: 4486: 4485: 4481: 4479: 4476: 4475: 4451: 4447: 4443: 4436: 4431: 4430: 4424: 4420: 4419: 4417: 4411: 4407: 4401: 4390: 4373: 4366: 4361: 4360: 4356: 4354: 4345: 4341: 4335: 4324: 4307: 4306: 4301: 4300: 4288: 4284: 4280: 4272: 4271: 4266: 4265: 4259: 4255: 4254: 4252: 4250: 4247: 4246: 4237: 4187: 4182: 4181: 4173: 4172: 4168: 4160: 4151: 4146: 4145: 4132: 4126: 4122: 4120: 4107: 4091: 4086: 4085: 4081: 4079: 4071: 4062: 4057: 4056: 4043: 4037: 4033: 4031: 4019: 4015: 4011: 3995: 3990: 3989: 3983: 3979: 3978: 3976: 3974: 3971: 3970: 3935: 3930: 3929: 3921: 3908: 3892: 3887: 3886: 3882: 3880: 3878: 3875: 3874: 3829: 3821: 3819: 3811: 3806: 3801: 3799: 3796: 3795: 3789:right-hand rule 3776: 3762: 3730: 3726: 3722: 3715: 3710: 3709: 3703: 3699: 3698: 3696: 3690: 3686: 3680: 3669: 3649: 3642: 3637: 3636: 3632: 3630: 3624: 3620: 3614: 3603: 3580: 3579: 3574: 3573: 3530: 3529: 3524: 3523: 3521: 3518: 3517: 3463: 3462: 3457: 3456: 3446: 3445: 3440: 3439: 3429: 3428: 3423: 3422: 3420: 3417: 3416: 3411: 3404: 3394: 3360: 3356: 3352: 3345: 3340: 3339: 3333: 3329: 3328: 3326: 3320: 3316: 3310: 3299: 3279: 3272: 3267: 3266: 3262: 3260: 3254: 3250: 3244: 3233: 3210: 3209: 3204: 3203: 3190: 3189: 3184: 3183: 3173: 3172: 3167: 3166: 3164: 3161: 3160: 3151: 3125: 3119: 3114: 3113: 3103: 3099: 3094: 3091: 3090: 3084: 3070: 3063: 3019: 3015: 3011: 3004: 2999: 2998: 2992: 2988: 2987: 2985: 2979: 2975: 2969: 2958: 2941: 2934: 2929: 2928: 2924: 2922: 2916: 2912: 2906: 2895: 2875: 2874: 2869: 2868: 2855: 2854: 2849: 2848: 2836: 2832: 2828: 2820: 2819: 2814: 2813: 2807: 2803: 2802: 2800: 2798: 2795: 2794: 2762: 2755: 2750: 2749: 2745: 2743: 2737: 2733: 2727: 2716: 2702: 2701: 2696: 2695: 2682: 2675: 2670: 2669: 2665: 2663: 2657: 2653: 2647: 2636: 2623: 2618: 2617: 2607: 2600: 2596: 2592: 2590: 2584: 2573: 2556: 2548: 2547: 2542: 2541: 2537: 2535: 2533: 2530: 2529: 2524: 2493: 2489: 2485: 2478: 2473: 2472: 2466: 2462: 2461: 2459: 2453: 2449: 2443: 2432: 2415: 2408: 2403: 2402: 2398: 2396: 2386: 2379: 2375: 2371: 2369: 2363: 2352: 2335: 2327: 2326: 2321: 2320: 2316: 2314: 2301: 2300: 2295: 2294: 2282: 2278: 2274: 2266: 2265: 2260: 2259: 2253: 2249: 2248: 2246: 2244: 2241: 2240: 2232: 2225: 2193: 2186: 2181: 2180: 2176: 2174: 2168: 2164: 2158: 2147: 2133: 2132: 2127: 2126: 2113: 2112: 2107: 2106: 2093: 2085: 2084: 2079: 2078: 2074: 2072: 2070: 2067: 2066: 2061: 2028: 2021: 2016: 2015: 2011: 2009: 2003: 1999: 1993: 1982: 1969: 1964: 1963: 1953: 1946: 1942: 1938: 1936: 1930: 1919: 1902: 1891: 1890: 1885: 1884: 1880: 1878: 1865: 1857: 1856: 1851: 1850: 1846: 1844: 1842: 1839: 1838: 1830: 1823: 1792: 1787: 1786: 1780: 1776: 1770: 1759: 1742: 1741: 1736: 1735: 1725: 1724: 1719: 1718: 1716: 1713: 1712: 1703: 1674: 1669: 1668: 1662: 1658: 1652: 1641: 1627: 1626: 1621: 1620: 1618: 1615: 1614: 1605: 1589: 1578: 1563: 1542: 1534: 1525: 1516: 1507: 1492: 1477: 1462: 1451: 1439: 1370:Albert Einstein 1358: 1352: 1324: 1310:The fictitious 1179: 1169: 1107:Vladimir Arnold 1098: 1092: 1045: 1011: 1004: 982: 980:Preferred frame 972: 957:Coriolis effect 871: 864: 813: 777: 736: 723: 722: 715: 714: 713: 588: 580: 579: 559: 513:Circular motion 507: 497: 496: 495: 452: 422: 419: 398: 377: 369: 368: 365: 364: 322: 312: 304: 303: 302: 261: 257:Mechanical work 250: 234: 172: 164: 163: 162: 117: 109: 86: 62: 56: 52: 50: 41: 40: 38: 35: 34: 17: 12: 11: 5: 11889: 11879: 11878: 11873: 11868: 11854: 11853: 11847: 11841: 11839:Coriolis Force 11836: 11828: 11821: 11820:External links 11818: 11817: 11816: 11810: 11795: 11789: 11774: 11768: 11755: 11749: 11725: 11722: 11719: 11718: 11667: 11664: 11655: 11651: 11628: 11624: 11601: 11597: 11593: 11588: 11581: 11578: 11569: 11562: 11559: 11550: 11547: 11538: 11534: 11530: 11525: 11518: 11515: 11494: 11477: 11470: 11450: 11433: 11404: 11397: 11377: 11356: 11353:. p. 276. 11339: 11332: 11312: 11305: 11285: 11278: 11253: 11246: 11226: 11219: 11199: 11172: 11165: 11145: 11138: 11118: 11096: 11087: 11080: 11057: 11050: 11030: 11023: 11003: 10996: 10976: 10966: 10940: 10933: 10913: 10893: 10886: 10865: 10856: 10849: 10829: 10822: 10802: 10795: 10771: 10755: 10740: 10736:speed of light 10718: 10711: 10688: 10656: 10649: 10629: 10616: 10601: 10586: 10575: 10554: 10547: 10527: 10516: 10506: 10480: 10466: 10451: 10436: 10421: 10414: 10394: 10380: 10355: 10354: 10352: 10349: 10348: 10347: 10344:Physics portal 10331: 10330: 10325: 10320: 10315: 10310: 10305: 10300: 10295: 10290: 10285: 10280: 10275: 10270: 10265: 10260: 10255: 10250: 10244: 10239: 10237:Coriolis force 10234: 10229: 10224: 10219: 10214: 10209: 10203: 10201: 10198: 10185: 10182: 10170:Coriolis force 10161: 10160: 10149: 10144: 10139: 10134: 10131: 10128: 10125: 10120: 10116: 10112: 10109: 10104: 10099: 10094: 10091: 10088: 10085: 10082: 10079: 10073: 10070: 10067: 10064: 10058: 10042: 10041: 10030: 10024: 10019: 10014: 10011: 10008: 10003: 9998: 9993: 9989: 9974: 9973: 9960: 9955: 9950: 9947: 9942: 9937: 9932: 9928: 9909: 9908: 9897: 9892: 9887: 9882: 9879: 9876: 9873: 9870: 9864: 9858: 9841: 9835: 9834: 9823: 9818: 9813: 9808: 9805: 9799: 9793: 9777: 9776: 9765: 9759: 9753: 9748: 9742: 9739: 9733: 9729: 9723: 9720: 9717: 9711: 9705: 9700: 9696: 9692: 9689: 9685: 9681: 9678: 9672: 9666: 9661: 9657: 9653: 9650: 9647: 9644: 9638: 9635: 9632: 9629: 9623: 9597: 9594: 9570: 9569: 9554: 9548: 9543: 9535: 9532: 9529: 9526: 9521: 9517: 9513: 9508: 9503: 9495: 9489: 9486: 9483: 9480: 9478: 9476: 9471: 9465: 9462: 9458: 9454: 9451: 9448: 9445: 9442: 9439: 9436: 9433: 9432: 9429: 9426: 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7300: 7297: 7292: 7287: 7282: 7277: 7274: 7269: 7264: 7261: 7259: 7252: 7248: 7244: 7237: 7232: 7225: 7221: 7214: 7213: 7195: 7194: 7183: 7172: 7167: 7162: 7159: 7156: 7150: 7146: 7142: 7139: 7135: 7130: 7125: 7117: 7114: 7110: 7085: 7077: 7072: 7067: 7064: 7058: 7054: 7050: 7047: 7043: 7038: 7033: 7025: 7022: 7018: 7006: 6995: 6991: 6988: 6985: 6982: 6979: 6976: 6970: 6967: 6964: 6961: 6958: 6955: 6952: 6947: 6942: 6917: 6911: 6908: 6905: 6902: 6899: 6896: 6890: 6887: 6884: 6881: 6878: 6875: 6872: 6869: 6864: 6859: 6842: 6835: 6828: 6821: 6814: 6783: 6762: 6759: 6727: 6726: 6715: 6711: 6706: 6702: 6698: 6695: 6691: 6684: 6681: 6678: 6675: 6669: 6663: 6648: 6647: 6636: 6630: 6627: 6622: 6615: 6611: 6607: 6604: 6598: 6595: 6592: 6589: 6583: 6552: 6547: 6541: 6536: 6531: 6527: 6522: 6518: 6514: 6493: 6488: 6483: 6480: 6475: 6472: 6469: 6464: 6459: 6453: 6441: 6440: 6429: 6423: 6420: 6415: 6408: 6404: 6400: 6397: 6393: 6388: 6385: 6380: 6377: 6374: 6369: 6364: 6358: 6354: 6349: 6346: 6341: 6331: 6327: 6323: 6317: 6313: 6307: 6302: 6299: 6294: 6274: 6273: 6262: 6254: 6251: 6246: 6243: 6240: 6235: 6230: 6227: 6222: 6214: 6211: 6207: 6202: 6197: 6194: 6189: 6169: 6168: 6157: 6150: 6146: 6143: 6140: 6137: 6134: 6131: 6125: 6122: 6119: 6116: 6113: 6110: 6107: 6103: 6099: 6096: 6091: 6088: 6083: 6059: 6058: 6043: 6035: 6030: 6025: 6017: 6014: 6009: 6004: 6001: 5999: 5997: 5989: 5985: 5981: 5974: 5969: 5962: 5958: 5946: 5942: 5936: 5931: 5928: 5925: 5921: 5917: 5911: 5908: 5901: 5896: 5891: 5880: 5876: 5870: 5865: 5862: 5859: 5855: 5848: 5845: 5840: 5835: 5830: 5825: 5822: 5817: 5812: 5809: 5807: 5800: 5796: 5792: 5785: 5780: 5773: 5769: 5762: 5761: 5694: 5678: 5677: 5628: 5626: 5619: 5613: 5610: 5593:Coriolis force 5588: 5587: 5576: 5570: 5564: 5559: 5553: 5550: 5544: 5540: 5534: 5531: 5528: 5522: 5516: 5511: 5507: 5503: 5500: 5496: 5492: 5489: 5483: 5477: 5472: 5468: 5464: 5461: 5458: 5455: 5449: 5446: 5443: 5440: 5434: 5417: 5410: 5403: 5396: 5389: 5379: 5373: 5372: 5361: 5355: 5349: 5344: 5338: 5335: 5329: 5325: 5319: 5316: 5310: 5304: 5299: 5295: 5291: 5288: 5284: 5280: 5274: 5268: 5263: 5259: 5255: 5252: 5246: 5240: 5235: 5229: 5223: 5206: 5199: 5188: 5179: 5178: 5167: 5162: 5155: 5149: 5144: 5140: 5135: 5131: 5127: 5123: 5117: 5111: 5106: 5100: 5097: 5091: 5087: 5081: 5075: 5069: 5064: 5060: 5056: 5053: 5047: 5041: 5036: 5030: 5024: 5004: 5003: 4988: 4984: 4980: 4977: 4974: 4969: 4964: 4957: 4953: 4947: 4942: 4939: 4936: 4932: 4928: 4924: 4919: 4915: 4911: 4907: 4902: 4897: 4890: 4886: 4880: 4875: 4872: 4869: 4865: 4861: 4855: 4852: 4846: 4842: 4836: 4833: 4830: 4827: 4822: 4817: 4810: 4806: 4800: 4795: 4792: 4789: 4785: 4781: 4777: 4773: 4770: 4764: 4758: 4753: 4750: 4748: 4746: 4742: 4738: 4735: 4732: 4727: 4722: 4717: 4713: 4708: 4704: 4700: 4694: 4690: 4684: 4679: 4676: 4673: 4669: 4665: 4657: 4652: 4647: 4641: 4638: 4632: 4628: 4620: 4616: 4610: 4605: 4602: 4599: 4595: 4591: 4588: 4585: 4582: 4577: 4572: 4567: 4563: 4557: 4553: 4547: 4542: 4539: 4536: 4532: 4528: 4522: 4516: 4510: 4505: 4502: 4500: 4495: 4489: 4484: 4483: 4473: 4462: 4454: 4450: 4446: 4439: 4434: 4427: 4423: 4414: 4410: 4404: 4399: 4396: 4393: 4389: 4385: 4379: 4376: 4369: 4364: 4359: 4348: 4344: 4338: 4333: 4330: 4327: 4323: 4319: 4316: 4310: 4304: 4299: 4291: 4287: 4283: 4275: 4269: 4262: 4258: 4235: 4221: 4220: 4209: 4205: 4201: 4198: 4195: 4190: 4185: 4180: 4176: 4171: 4167: 4163: 4159: 4154: 4149: 4144: 4138: 4135: 4129: 4125: 4119: 4113: 4110: 4105: 4102: 4099: 4094: 4089: 4084: 4078: 4074: 4070: 4065: 4060: 4055: 4049: 4046: 4040: 4036: 4030: 4022: 4018: 4014: 4009: 4006: 4003: 3998: 3993: 3986: 3982: 3964: 3963: 3952: 3949: 3946: 3943: 3938: 3933: 3928: 3924: 3920: 3914: 3911: 3906: 3903: 3900: 3895: 3890: 3885: 3868: 3867: 3856: 3853: 3850: 3847: 3844: 3841: 3835: 3832: 3827: 3824: 3818: 3814: 3809: 3804: 3775: 3772: 3760: 3754: 3753: 3742: 3733: 3729: 3725: 3718: 3713: 3706: 3702: 3693: 3689: 3683: 3678: 3675: 3672: 3668: 3664: 3661: 3655: 3652: 3645: 3640: 3635: 3627: 3623: 3617: 3612: 3609: 3606: 3602: 3598: 3595: 3592: 3586: 3583: 3577: 3572: 3569: 3566: 3560: 3557: 3554: 3551: 3548: 3545: 3542: 3539: 3536: 3533: 3527: 3511: 3510: 3499: 3493: 3490: 3487: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3460: 3455: 3449: 3443: 3438: 3432: 3426: 3409: 3402: 3392: 3386: 3385: 3374: 3363: 3359: 3355: 3348: 3343: 3336: 3332: 3323: 3319: 3313: 3308: 3305: 3302: 3298: 3294: 3291: 3285: 3282: 3275: 3270: 3265: 3257: 3253: 3247: 3242: 3239: 3236: 3232: 3228: 3225: 3222: 3216: 3213: 3207: 3202: 3199: 3193: 3187: 3182: 3176: 3170: 3149: 3135: 3132: 3128: 3122: 3117: 3112: 3106: 3102: 3098: 3082: 3068: 3061: 3053: 3052: 3043: 3041: 3030: 3022: 3018: 3014: 3007: 3002: 2995: 2991: 2982: 2978: 2972: 2967: 2964: 2961: 2957: 2953: 2947: 2944: 2937: 2932: 2927: 2919: 2915: 2909: 2904: 2901: 2898: 2894: 2887: 2884: 2878: 2872: 2867: 2861: 2858: 2852: 2847: 2839: 2835: 2831: 2823: 2817: 2810: 2806: 2788:Consequently, 2786: 2785: 2774: 2768: 2765: 2758: 2753: 2748: 2740: 2736: 2730: 2725: 2722: 2719: 2715: 2711: 2705: 2699: 2694: 2688: 2685: 2678: 2673: 2668: 2660: 2656: 2650: 2645: 2642: 2639: 2635: 2631: 2626: 2621: 2613: 2610: 2603: 2599: 2595: 2587: 2582: 2579: 2576: 2572: 2568: 2562: 2559: 2551: 2545: 2540: 2522: 2516: 2515: 2504: 2496: 2492: 2488: 2481: 2476: 2469: 2465: 2456: 2452: 2446: 2441: 2438: 2435: 2431: 2427: 2421: 2418: 2411: 2406: 2401: 2392: 2389: 2382: 2378: 2374: 2366: 2361: 2358: 2355: 2351: 2347: 2341: 2338: 2330: 2324: 2319: 2313: 2307: 2304: 2298: 2293: 2285: 2281: 2277: 2269: 2263: 2256: 2252: 2230: 2223: 2217: 2216: 2205: 2199: 2196: 2189: 2184: 2179: 2171: 2167: 2161: 2156: 2153: 2150: 2146: 2142: 2136: 2130: 2125: 2119: 2116: 2110: 2105: 2099: 2096: 2088: 2082: 2077: 2059: 2053: 2052: 2041: 2034: 2031: 2024: 2019: 2014: 2006: 2002: 1996: 1991: 1988: 1985: 1981: 1977: 1972: 1967: 1959: 1956: 1949: 1945: 1941: 1933: 1928: 1925: 1922: 1918: 1914: 1908: 1905: 1897: 1894: 1888: 1883: 1877: 1871: 1868: 1860: 1854: 1849: 1828: 1819: 1813: 1812: 1801: 1795: 1790: 1783: 1779: 1773: 1768: 1765: 1762: 1758: 1754: 1748: 1745: 1739: 1734: 1728: 1722: 1701: 1695: 1694: 1683: 1677: 1672: 1665: 1661: 1655: 1650: 1647: 1644: 1640: 1636: 1630: 1624: 1601: 1587: 1576: 1570:inertial frame 1561: 1541: 1538: 1532: 1523: 1514: 1505: 1490: 1475: 1471:is located at 1460: 1449: 1438: 1435: 1434: 1433: 1430: 1429: 1413: 1412: 1354:Main article: 1351: 1348: 1323: 1320: 1312:Coriolis force 1308: 1307: 1304: 1303: 1277: 1276: 1249: 1248: 1244: 1201:contact forces 1177:Coriolis force 1168: 1165: 1161:Coriolis force 1091: 1088: 1057:Coriolis force 1044: 1041: 1002: 971: 968: 967: 966: 959: 949: 918:inertial force 877:Coriolis force 865:Harald Iro in 862: 812: 809: 779: 778: 776: 775: 768: 761: 753: 750: 749: 748: 747: 734: 717: 716: 712: 711: 706: 701: 696: 691: 686: 681: 676: 671: 666: 661: 656: 651: 646: 641: 636: 631: 626: 621: 616: 611: 606: 601: 596: 590: 589: 586: 585: 582: 581: 578: 577: 558: 557: 552: 547: 542: 540:Coriolis force 537: 536: 535: 525: 520: 515: 509: 508: 503: 502: 499: 498: 494: 493: 488: 483: 482: 481: 476: 466: 461: 456: 449: 438: 437: 436: 431: 418: 417: 412: 407: 402: 395: 390: 385: 379: 378: 375: 374: 371: 370: 367: 366: 363: 362: 357: 352: 347: 342: 337: 331: 325: 323: 316: 313: 310: 309: 306: 305: 301: 300: 295: 290: 285: 280: 275: 270: 265: 259: 254: 248: 243: 232: 227: 222: 217: 212: 211: 210: 205: 195: 190: 185: 180: 174: 173: 170: 169: 166: 165: 161: 160: 155: 150: 145: 140: 135: 130: 125: 119: 118: 115: 114: 111: 110: 108: 107: 102: 97: 91: 88: 87: 82: 68: 65: 59: 55: 49: 31: 30: 24: 23: 15: 9: 6: 4: 3: 2: 11888: 11877: 11874: 11872: 11869: 11867: 11864: 11863: 11861: 11851: 11848: 11845: 11842: 11840: 11837: 11835: 11834: 11829: 11827: 11824: 11823: 11813: 11811:0-521-78780-7 11807: 11803: 11802: 11796: 11792: 11790:0-12-472252-0 11786: 11782: 11781: 11775: 11771: 11769:0-201-07392-7 11765: 11761: 11756: 11752: 11750:0-7506-2896-0 11746: 11742: 11738: 11737: 11732: 11731:Lev D. Landau 11728: 11727: 11715: 11711: 11707: 11703: 11699: 11695: 11694:0-07-033484-6 11691: 11687: 11683: 11665: 11662: 11653: 11626: 11622: 11599: 11595: 11591: 11586: 11579: 11576: 11567: 11560: 11557: 11548: 11545: 11536: 11528: 11523: 11516: 11513: 11498: 11491: 11487: 11481: 11473: 11471:981-256-182-X 11467: 11463: 11462: 11453: 11451:0-7506-6169-0 11447: 11443: 11442: 11432: 11428: 11425: 11421: 11417: 11413: 11408: 11400: 11394: 11390: 11389: 11380: 11378:0-7506-7403-2 11374: 11370: 11369: 11360: 11352: 11351: 11343: 11335: 11333:0-8031-0446-4 11329: 11325: 11324: 11316: 11308: 11306:90-5809-361-1 11302: 11298: 11297: 11289: 11281: 11275: 11271: 11270: 11263: 11257: 11249: 11247:0-387-98643-X 11243: 11239: 11238: 11230: 11222: 11220:0-486-65067-7 11216: 11212: 11211: 11203: 11196: 11192: 11188: 11187: 11182: 11176: 11168: 11166:0-486-65227-0 11162: 11158: 11157: 11149: 11141: 11139:0-521-82678-0 11135: 11131: 11130: 11122: 11114: 11110: 11109: 11100: 11091: 11083: 11081:1-891389-22-X 11077: 11073: 11072: 11064: 11062: 11053: 11051:981-02-1298-4 11047: 11043: 11042: 11034: 11026: 11024:0-486-43261-0 11020: 11016: 11015: 11007: 10999: 10997:0-521-35578-8 10993: 10989: 10988: 10980: 10973: 10969: 10967:1-56347-224-4 10963: 10959: 10954: 10953: 10944: 10936: 10934:1-85233-695-1 10930: 10926: 10925: 10917: 10911: 10910:0-201-38423-X 10907: 10903: 10897: 10889: 10887:9781489963338 10883: 10879: 10878: 10869: 10860: 10852: 10850:0-521-01069-1 10846: 10842: 10841: 10833: 10825: 10819: 10815: 10814: 10806: 10798: 10796:0-07-035048-5 10792: 10788: 10784: 10783: 10775: 10768: 10766: 10759: 10751: 10744: 10737: 10733: 10729: 10722: 10714: 10712:0-387-96890-3 10708: 10704: 10703: 10695: 10693: 10685: 10681: 10676: 10671: 10667: 10660: 10652: 10650:90-277-0107-5 10646: 10642: 10641: 10633: 10626: 10620: 10612: 10605: 10597: 10590: 10584: 10579: 10565: 10558: 10550: 10548:0-486-65067-7 10544: 10540: 10539: 10531: 10525: 10520: 10513: 10509: 10507:0-486-60769-0 10503: 10499: 10494: 10493: 10484: 10476: 10470: 10462: 10455: 10447: 10440: 10432: 10425: 10417: 10415:981-238-213-5 10411: 10407: 10406: 10398: 10390: 10384: 10370: 10366: 10360: 10356: 10345: 10339: 10334: 10329: 10326: 10324: 10321: 10319: 10316: 10314: 10311: 10309: 10306: 10304: 10301: 10299: 10296: 10294: 10291: 10289: 10286: 10284: 10281: 10279: 10276: 10274: 10271: 10269: 10266: 10264: 10261: 10259: 10256: 10254: 10251: 10248: 10245: 10243: 10240: 10238: 10235: 10233: 10230: 10228: 10225: 10223: 10220: 10218: 10215: 10213: 10210: 10208: 10205: 10204: 10197: 10195: 10191: 10181: 10177: 10175: 10171: 10166: 10147: 10142: 10129: 10123: 10118: 10114: 10110: 10107: 10102: 10092: 10089: 10086: 10083: 10080: 10077: 10047: 10046: 10045: 10028: 10022: 10012: 10009: 10006: 10001: 9991: 9979: 9978: 9977: 9958: 9948: 9945: 9940: 9930: 9918: 9917: 9916: 9914: 9895: 9890: 9877: 9871: 9868: 9847: 9846: 9845: 9840: 9821: 9816: 9806: 9803: 9782: 9781: 9780: 9763: 9746: 9740: 9737: 9727: 9721: 9718: 9698: 9687: 9679: 9676: 9659: 9651: 9648: 9645: 9642: 9612: 9611: 9610: 9607: 9604: 9593: 9589: 9587: 9583: 9579: 9575: 9552: 9546: 9530: 9524: 9519: 9515: 9511: 9506: 9493: 9487: 9484: 9481: 9479: 9469: 9460: 9456: 9452: 9449: 9446: 9443: 9437: 9434: 9424: 9420: 9416: 9413: 9410: 9407: 9401: 9398: 9392: 9384: 9378: 9373: 9369: 9365: 9360: 9351: 9347: 9343: 9340: 9337: 9334: 9328: 9325: 9315: 9311: 9307: 9304: 9301: 9298: 9292: 9289: 9286: 9280: 9275: 9272: 9269: 9266: 9264: 9254: 9245: 9241: 9237: 9234: 9231: 9228: 9222: 9219: 9209: 9205: 9201: 9198: 9195: 9192: 9186: 9183: 9177: 9169: 9163: 9158: 9154: 9150: 9145: 9136: 9132: 9128: 9125: 9122: 9119: 9113: 9110: 9100: 9096: 9092: 9089: 9086: 9083: 9077: 9074: 9071: 9065: 9060: 9054: 9051: 9046: 9043: 9037: 9034: 9029: 9020: 9016: 9012: 9009: 9006: 9003: 8997: 8994: 8984: 8980: 8976: 8973: 8970: 8967: 8961: 8958: 8952: 8942: 8938: 8934: 8929: 8924: 8920: 8913: 8911: 8903: 8884: 8883: 8882: 8880: 8876: 8869: 8862: 8839: 8834: 8821: 8815: 8812: 8809: 8804: 8791: 8788: 8783: 8780: 8774: 8772: 8762: 8753: 8749: 8745: 8742: 8739: 8736: 8730: 8727: 8717: 8713: 8709: 8706: 8703: 8700: 8694: 8691: 8688: 8682: 8674: 8668: 8665: 8662: 8657: 8648: 8644: 8640: 8637: 8634: 8631: 8625: 8622: 8612: 8608: 8604: 8601: 8598: 8595: 8589: 8586: 8580: 8572: 8569: 8564: 8561: 8555: 8553: 8545: 8526: 8525: 8524: 8522: 8503: 8500: 8497: 8494: 8488: 8482: 8475: 8474: 8473: 8471: 8467: 8463: 8458: 8454: 8435: 8430: 8421: 8417: 8413: 8410: 8407: 8404: 8398: 8395: 8389: 8383: 8373: 8369: 8365: 8362: 8359: 8356: 8350: 8347: 8341: 8335: 8329: 8324: 8319: 8310: 8304: 8294: 8288: 8282: 8277: 8272: 8259: 8253: 8250: 8244: 8229: 8228: 8227: 8225: 8221: 8211: 8209: 8205: 8201: 8197: 8187: 8182: 8172: 8170: 8164: 8162: 8161:Lamm equation 8158: 8157:sedimentation 8151: 8144: 8137: 8130: 8125: 8122: 8118: 8111: 8104: 8100: 8093: 8076: 8071: 8065: 8055: 8046: 8042: 8017: 8012: 8003: 7993: 7988: 7985: 7971: 7963: 7958: 7924: 7915: 7905: 7894: 7891: 7886: 7876: 7872: 7863: 7853: 7848: 7845: 7831: 7823: 7818: 7808: 7806: 7797: 7791: 7781: 7772: 7768: 7757: 7746: 7736: 7725: 7722: 7717: 7707: 7703: 7698: 7695: 7687: 7679: 7674: 7664: 7662: 7653: 7647: 7637: 7628: 7624: 7611: 7607: 7601: 7596: 7593: 7590: 7586: 7579: 7562: 7550: 7546: 7540: 7535: 7532: 7529: 7525: 7518: 7515: 7510: 7500: 7496: 7491: 7488: 7480: 7472: 7467: 7457: 7455: 7442: 7438: 7434: 7427: 7415: 7411: 7399: 7395: 7389: 7384: 7381: 7378: 7374: 7367: 7361: 7358: 7351: 7341: 7330: 7326: 7320: 7315: 7312: 7309: 7305: 7298: 7295: 7290: 7280: 7275: 7272: 7262: 7260: 7250: 7246: 7242: 7235: 7223: 7219: 7204: 7203: 7202: 7200: 7181: 7170: 7160: 7157: 7154: 7140: 7133: 7128: 7115: 7112: 7108: 7083: 7075: 7065: 7062: 7048: 7041: 7036: 7023: 7020: 7016: 7007: 6993: 6986: 6983: 6980: 6977: 6974: 6968: 6965: 6962: 6959: 6956: 6950: 6945: 6915: 6906: 6903: 6900: 6897: 6894: 6888: 6885: 6882: 6879: 6876: 6873: 6867: 6862: 6848: 6847: 6846: 6841: 6834: 6827: 6820: 6813: 6809: 6805: 6801: 6793: 6789: 6782: 6778: 6773: 6768: 6758: 6756: 6752: 6748: 6744: 6740: 6736: 6732: 6713: 6709: 6704: 6700: 6696: 6693: 6653: 6652: 6651: 6634: 6628: 6625: 6613: 6609: 6605: 6602: 6572: 6571: 6570: 6567: 6550: 6545: 6539: 6529: 6520: 6516: 6491: 6486: 6481: 6478: 6470: 6462: 6457: 6427: 6421: 6418: 6406: 6402: 6398: 6395: 6391: 6386: 6383: 6375: 6367: 6362: 6352: 6347: 6344: 6329: 6325: 6321: 6315: 6311: 6305: 6300: 6297: 6283: 6282: 6281: 6279: 6260: 6252: 6249: 6241: 6233: 6228: 6225: 6212: 6209: 6205: 6200: 6195: 6192: 6178: 6177: 6176: 6174: 6155: 6148: 6141: 6138: 6132: 6129: 6123: 6117: 6114: 6108: 6105: 6101: 6097: 6094: 6089: 6086: 6072: 6071: 6070: 6068: 6064: 6041: 6033: 6023: 6015: 6012: 6002: 6000: 5987: 5983: 5979: 5972: 5960: 5956: 5944: 5940: 5934: 5929: 5926: 5923: 5919: 5915: 5909: 5906: 5899: 5889: 5878: 5874: 5868: 5863: 5860: 5857: 5853: 5846: 5843: 5838: 5828: 5823: 5820: 5810: 5808: 5798: 5794: 5790: 5783: 5771: 5767: 5752: 5751: 5750: 5748: 5744: 5740: 5736: 5728: 5724: 5720: 5716: 5712: 5708: 5704: 5700: 5693: 5689: 5684: 5674: 5671: 5663: 5653: 5649: 5645: 5639: 5638: 5634: 5629:This section 5627: 5623: 5618: 5617: 5609: 5607: 5606: 5601: 5600: 5595: 5594: 5574: 5557: 5551: 5548: 5538: 5532: 5529: 5509: 5498: 5490: 5487: 5470: 5462: 5459: 5456: 5453: 5423: 5422: 5421: 5416: 5409: 5402: 5395: 5388: 5385: 5378: 5359: 5342: 5336: 5333: 5323: 5317: 5297: 5286: 5278: 5261: 5253: 5250: 5233: 5212: 5211: 5210: 5205: 5198: 5194: 5187: 5184: 5165: 5160: 5142: 5133: 5129: 5121: 5104: 5098: 5095: 5085: 5079: 5062: 5054: 5051: 5034: 5013: 5012: 5011: 5009: 4986: 4982: 4975: 4967: 4955: 4951: 4945: 4940: 4937: 4934: 4930: 4926: 4917: 4913: 4905: 4900: 4888: 4884: 4878: 4873: 4870: 4867: 4863: 4859: 4853: 4850: 4840: 4834: 4828: 4820: 4808: 4804: 4798: 4793: 4790: 4787: 4783: 4779: 4771: 4768: 4751: 4749: 4740: 4733: 4725: 4715: 4706: 4702: 4692: 4688: 4682: 4677: 4674: 4671: 4667: 4663: 4655: 4645: 4639: 4636: 4626: 4618: 4614: 4608: 4603: 4600: 4597: 4593: 4589: 4583: 4575: 4565: 4555: 4551: 4545: 4540: 4537: 4534: 4530: 4526: 4520: 4503: 4501: 4474: 4460: 4452: 4448: 4444: 4437: 4425: 4421: 4412: 4408: 4402: 4397: 4394: 4391: 4387: 4383: 4377: 4374: 4367: 4357: 4346: 4342: 4336: 4331: 4328: 4325: 4321: 4317: 4314: 4297: 4289: 4285: 4281: 4260: 4256: 4245: 4244: 4243: 4241: 4234: 4230: 4226: 4207: 4203: 4196: 4188: 4178: 4169: 4165: 4157: 4152: 4142: 4136: 4133: 4123: 4117: 4111: 4108: 4100: 4092: 4082: 4076: 4068: 4063: 4053: 4047: 4044: 4034: 4028: 4020: 4016: 4012: 4004: 3996: 3984: 3980: 3969: 3968: 3967: 3950: 3944: 3936: 3926: 3918: 3912: 3909: 3901: 3893: 3883: 3873: 3872: 3871: 3854: 3848: 3842: 3839: 3833: 3830: 3825: 3822: 3816: 3794: 3793: 3792: 3790: 3786: 3780: 3771: 3769: 3764: 3759: 3740: 3731: 3727: 3723: 3716: 3704: 3700: 3691: 3687: 3681: 3676: 3673: 3670: 3666: 3662: 3659: 3653: 3650: 3643: 3633: 3625: 3621: 3615: 3610: 3607: 3604: 3600: 3596: 3593: 3590: 3570: 3567: 3564: 3516: 3515: 3514: 3497: 3453: 3436: 3415: 3414: 3413: 3408: 3401: 3398: 3391: 3372: 3361: 3357: 3353: 3346: 3334: 3330: 3321: 3317: 3311: 3306: 3303: 3300: 3296: 3292: 3289: 3283: 3280: 3273: 3263: 3255: 3251: 3245: 3240: 3237: 3234: 3230: 3226: 3223: 3220: 3200: 3197: 3180: 3159: 3158: 3157: 3154: 3152: 3133: 3130: 3126: 3120: 3110: 3104: 3100: 3096: 3088: 3081: 3078: 3072: 3067: 3060: 3051: 3044: 3042: 3028: 3020: 3016: 3012: 3005: 2993: 2989: 2980: 2976: 2970: 2965: 2962: 2959: 2955: 2951: 2945: 2942: 2935: 2925: 2917: 2913: 2907: 2902: 2899: 2896: 2892: 2885: 2882: 2865: 2845: 2837: 2833: 2829: 2808: 2804: 2793: 2792: 2789: 2772: 2766: 2763: 2756: 2746: 2738: 2734: 2728: 2723: 2720: 2717: 2713: 2709: 2692: 2686: 2683: 2676: 2666: 2658: 2654: 2648: 2643: 2640: 2637: 2633: 2629: 2624: 2611: 2608: 2601: 2597: 2593: 2585: 2580: 2577: 2574: 2570: 2566: 2560: 2557: 2538: 2528: 2527: 2526: 2521: 2502: 2494: 2490: 2486: 2479: 2467: 2463: 2454: 2450: 2444: 2439: 2436: 2433: 2429: 2425: 2419: 2416: 2409: 2399: 2390: 2387: 2380: 2376: 2372: 2364: 2359: 2356: 2353: 2349: 2345: 2339: 2336: 2317: 2311: 2291: 2283: 2279: 2275: 2254: 2250: 2239: 2238: 2237: 2234: 2229: 2222: 2203: 2197: 2194: 2187: 2177: 2169: 2165: 2159: 2154: 2151: 2148: 2144: 2140: 2123: 2103: 2097: 2094: 2075: 2065: 2064: 2063: 2058: 2039: 2032: 2029: 2022: 2012: 2004: 2000: 1994: 1989: 1986: 1983: 1979: 1975: 1970: 1957: 1954: 1947: 1943: 1939: 1931: 1926: 1923: 1920: 1916: 1912: 1906: 1903: 1881: 1875: 1869: 1866: 1847: 1837: 1836: 1835: 1832: 1827: 1822: 1818: 1799: 1793: 1781: 1777: 1771: 1766: 1763: 1760: 1756: 1752: 1732: 1711: 1710: 1709: 1707: 1700: 1681: 1675: 1663: 1659: 1653: 1648: 1645: 1642: 1638: 1634: 1613: 1612: 1611: 1609: 1604: 1600: 1595: 1593: 1586: 1582: 1575: 1571: 1567: 1560: 1557: 1554: 1550: 1547: 1531: 1530: 1522: 1521: 1513: 1512: 1504: 1500: 1496: 1489: 1485: 1481: 1474: 1470: 1466: 1459: 1455: 1448: 1443: 1426: 1419: 1415: 1414: 1410: 1407: 1406: 1403: 1402: 1401: 1399: 1395: 1391: 1387: 1383: 1379: 1375: 1371: 1367: 1363: 1357: 1347: 1345: 1341: 1337: 1333: 1329: 1319: 1317: 1316:Rossby number 1313: 1302: 1300: 1296: 1292: 1283: 1279: 1278: 1274: 1271: 1270: 1267: 1266: 1265: 1263: 1257: 1254: 1245: 1242: 1238: 1234: 1230: 1226: 1222: 1221: 1220: 1217: 1213: 1211: 1207: 1202: 1197: 1193: 1183: 1178: 1174: 1164: 1162: 1158: 1153: 1151: 1145: 1143: 1139: 1135: 1132: 1128: 1124: 1120: 1116: 1110: 1108: 1103: 1097: 1087: 1085: 1084:Eötvös effect 1080: 1077: 1073: 1072:Léon Foucault 1070:For example, 1068: 1066: 1062: 1058: 1054: 1050: 1040: 1038: 1034: 1030: 1025: 1021: 1017: 1008: 1001: 999: 995: 989: 987: 981: 977: 964: 960: 958: 954: 950: 947: 943: 942: 941: 938: 936: 932: 931:contact force 927: 925: 924: 919: 914: 912: 908: 905: 902: = 901: 897: 892: 890: 886: 882: 878: 868: 861: 859: 853: 850: 847: 843: 839: 835: 830: 826: 822: 817: 808: 806: 802: 798: 794: 790: 786: 774: 769: 767: 762: 760: 755: 754: 752: 751: 745: 735: 732: 727: 721: 720: 719: 718: 710: 707: 705: 702: 700: 697: 695: 692: 690: 687: 685: 682: 680: 677: 675: 672: 670: 667: 665: 662: 660: 657: 655: 652: 650: 647: 645: 642: 640: 637: 635: 632: 630: 627: 625: 622: 620: 617: 615: 612: 610: 607: 605: 602: 600: 597: 595: 592: 591: 584: 583: 576: 572: 568: 564: 561: 560: 556: 553: 551: 548: 546: 543: 541: 538: 534: 531: 530: 529: 526: 524: 521: 519: 516: 514: 511: 510: 506: 501: 500: 492: 489: 487: 484: 480: 477: 475: 472: 471: 470: 467: 465: 462: 460: 457: 455: 450: 447: 443: 440: 439: 435: 432: 429: 425: 421: 420: 416: 413: 411: 408: 406: 403: 401: 396: 394: 391: 389: 386: 384: 381: 380: 373: 372: 361: 358: 356: 353: 351: 348: 346: 343: 341: 338: 336: 333: 332: 330: 329: 324: 321: 320: 315: 314: 308: 307: 299: 296: 294: 291: 289: 286: 284: 281: 279: 276: 274: 271: 269: 266: 264: 260: 258: 255: 253: 249: 247: 244: 241: 237: 233: 231: 228: 226: 223: 221: 218: 216: 213: 209: 206: 204: 201: 200: 199: 196: 194: 191: 189: 186: 184: 181: 179: 176: 175: 168: 167: 159: 156: 154: 151: 149: 146: 144: 141: 139: 136: 134: 131: 129: 126: 124: 121: 120: 113: 112: 106: 103: 101: 98: 96: 93: 92: 90: 89: 85: 66: 63: 53: 47: 33: 32: 29: 26: 25: 21: 20: 11832: 11800: 11779: 11759: 11735: 11697: 11685: 11497: 11480: 11460: 11439: 11430: 11426: 11419: 11415: 11411: 11407: 11387: 11367: 11359: 11349: 11342: 11322: 11315: 11295: 11288: 11268: 11256: 11236: 11229: 11209: 11202: 11194: 11185: 11175: 11155: 11148: 11128: 11121: 11107: 11099: 11090: 11070: 11040: 11033: 11013: 11006: 10986: 10979: 10971: 10951: 10943: 10923: 10916: 10901: 10896: 10880:. 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This led 1241:banked turn 1065:Euler force 1051:. To solve 963:Euler force 709:von Neumann 376:Core topics 11860:Categories 11714:four-force 11262:barycenter 10569:2007-09-03 10374:2021-12-14 10351:References 10293:Kinematics 8179:See also: 6765:See also: 6280:given by: 3761:fictitious 1409:Animation: 1396:. See the 1332:trajectory 1273:Animation: 1171:See also: 1094:See also: 1024:privileged 998:fictitious 974:See also: 970:Background 825:body force 807:inertial. 644:d'Alembert 624:Maupertuis 587:Scientists 469:Rigid body 143:Kinematics 11760:Mechanics 11736:Mechanics 11650:Γ 11580:˙ 11561:˙ 11533:Γ 11517:¨ 10623:The term 10115:ω 10103:θ 10090:ω 10081:− 10044:we find: 10013:ω 10010:− 10002:θ 9992:× 9988:Ω 9959:θ 9949:ω 9931:× 9927:Ω 9747:× 9732:Ω 9719:− 9699:× 9695:Ω 9688:× 9684:Ω 9677:− 9660:× 9656:Ω 9646:− 9516:ω 9512:− 9507:θ 9494:ω 9453:π 9444:ω 9438:⁡ 9417:π 9408:ω 9402:⁡ 9370:ω 9366:− 9344:π 9335:ω 9329:⁡ 9308:π 9299:ω 9293:⁡ 9287:− 9276:ω 9238:π 9229:ω 9223:⁡ 9202:π 9193:ω 9187:⁡ 9155:ω 9151:− 9129:π 9120:ω 9114:⁡ 9093:π 9084:ω 9078:⁡ 9072:− 9061:ω 9013:π 9004:ω 8998:⁡ 8977:π 8968:ω 8962:⁡ 8835:θ 8813:ω 8746:π 8737:ω 8731:⁡ 8710:π 8701:ω 8695:⁡ 8689:− 8666:ω 8641:π 8632:ω 8626:⁡ 8605:π 8596:ω 8590:⁡ 8414:π 8405:ω 8399:⁡ 8366:π 8357:ω 8351:⁡ 8190:carousel. 8056:× 8052:Ω 8043:× 8039:Ω 7972:× 7969:Ω 7959:× 7953:Ω 7906:× 7902:Ω 7832:× 7829:Ω 7819:× 7813:Ω 7782:× 7778:Ω 7769:× 7765:Ω 7737:× 7733:Ω 7688:× 7685:Ω 7675:× 7669:Ω 7638:× 7634:Ω 7625:× 7621:Ω 7587:∑ 7563:× 7560:Ω 7526:∑ 7481:× 7478:Ω 7468:× 7462:Ω 7375:∑ 7306:∑ 7161:ω 7158:− 7141:× 7138:Ω 7066:ω 7049:× 7046:Ω 6984:ω 6981:⁡ 6975:− 6963:ω 6960:⁡ 6904:ω 6901:⁡ 6895:− 6883:ω 6880:⁡ 6874:− 6701:ω 6610:ω 6530:× 6526:Ω 6517:× 6513:Ω 6471:× 6468:Ω 6458:× 6452:Ω 6403:ω 6399:− 6376:× 6373:Ω 6363:× 6357:Ω 6242:× 6239:Ω 6139:ω 6133:⁡ 6115:ω 6109:⁡ 5920:∑ 5854:∑ 5660:July 2008 5631:does not 5558:× 5543:Ω 5530:− 5510:× 5506:Ω 5499:× 5495:Ω 5488:− 5471:× 5467:Ω 5457:− 5343:× 5328:Ω 5318:− 5298:× 5294:Ω 5287:× 5283:Ω 5279:− 5262:× 5258:Ω 5251:− 5143:× 5139:Ω 5130:× 5126:Ω 5105:× 5090:Ω 5063:× 5059:Ω 4931:∑ 4927:× 4923:Ω 4914:× 4910:Ω 4864:∑ 4860:× 4845:Ω 4784:∑ 4780:× 4776:Ω 4716:× 4712:Ω 4703:× 4699:Ω 4668:∑ 4646:× 4631:Ω 4594:∑ 4566:× 4562:Ω 4531:∑ 4388:∑ 4322:∑ 4179:× 4175:Ω 4166:× 4162:Ω 4143:× 4128:Ω 4077:× 4073:Ω 4054:× 4039:Ω 3927:× 3923:Ω 3843:ω 3826:θ 3808:Ω 3667:∑ 3660:− 3601:∑ 3591:− 3568:− 3297:∑ 3231:∑ 3097:∑ 2956:∑ 2893:∑ 2714:∑ 2634:∑ 2571:∑ 2430:∑ 2350:∑ 2145:∑ 1980:∑ 1917:∑ 1757:∑ 1639:∑ 1478:in frame 1390:spacetime 1386:curvature 1340:potential 1225:direction 1129: = − 1074:used his 1037:spacetime 1033:geodesics 894:Assuming 885:spacetime 689:Liouville 571:frequency 491:Vibration 208:potential 133:Continuum 128:Celestial 105:Textbooks 11680:are the 11614:, where 11490:See note 10200:See also 10194:kinetics 10168:are the 9592:walker. 8873:at time 8196:carousel 8159:and the 8119:, where 1553:position 1237:friction 1192:the bend 1136:, where 1102:velocity 1043:On Earth 1003:— 986:Tonnelat 935:reaction 863:— 744:Category 669:Hamilton 654:Lagrange 649:Clairaut 614:Horrocks 575:velocity 545:Pendulum 533:reactive 505:Rotation 474:dynamics 424:Inertial 410:Friction 293:Velocity 268:Momentum 148:Kinetics 138:Dynamics 116:Branches 100:Timeline 10680:Bibcode 10323:Statics 10283:Gravity 7096: 6931: 5709:. Axis 5652:removed 5637:sources 3513:where: 1366:gravity 1344:kinetic 1262:equator 1210:inertia 1009:, p.113 829:inertia 704:Koopman 664:Poisson 659:Laplace 604:Huygens 599:Galileo 444: ( 383:Damping 236:Inertia 230:Impulse 203:kinetic 153:Statics 123:Applied 95:History 11808: 11787: 11766: 11747: 11692: 11468: 11448: 11395: 11383:, and 11375: 11330: 11303: 11276: 11244: 11217: 11163: 11136: 11078: 11048: 11021: 10994: 10964: 10931: 10908: 10884: 10847: 10820: 10793: 10709: 10647: 10545: 10504: 10412: 9537: 9497: 9491: 8115:| = ω 7956: 7921: 7898: 7816: 7761: 7755: 7752: 7729: 7672: 7617: 7583: 7577: 7556: 7522: 7465: 7405: 7371: 7336: 7302: 7179: 7176: 7104: 7081: 6972: 6919: 6913: 6892: 6455: 6360: 6258: 6153: 6127: 6039: 6021: 5950: 5884: 5850: 4661: 4524: 4352: 3370: 2889: 1556:vector 1336:energy 1059:, the 869:p. 180 742: 694:Appell 679:Cauchy 674:Jacobi 619:Halley 609:Newton 594:Kepler 446:linear 442:Motion 288:Torque 263:Moment 198:Energy 188:Couple 11876:Force 11115:–349. 10670:arXiv 10500:–78. 9837:with 8859:with 8519:with 8169:tides 7199:Eq. 1 6741:from 5747:Eq. 1 4240:Eq. 1 3412:, by 1606:with 1493:with 1425:Note: 1338:from 1299:Note: 1150:below 789:force 787:is a 699:Gibbs 684:Routh 639:Euler 278:Speed 273:Space 215:Force 11806:ISBN 11785:ISBN 11764:ISBN 11745:ISBN 11690:ISBN 11466:ISBN 11456:and 11446:ISBN 11412:Note 11393:ISBN 11373:ISBN 11328:ISBN 11301:ISBN 11274:ISBN 11242:ISBN 11215:ISBN 11161:ISBN 11134:ISBN 11076:ISBN 11046:ISBN 11019:ISBN 10992:ISBN 10962:ISBN 10929:ISBN 10906:ISBN 10882:ISBN 10845:ISBN 10818:ISBN 10791:ISBN 10707:ISBN 10645:ISBN 10543:ISBN 10502:ISBN 10410:ISBN 9976:and 9603:zero 8113:fict 8097:| = 6747:same 6175:as: 6069:at: 5635:any 5633:cite 5418:fict 5397:fict 5181:The 3966:and 1551:and 1546:mass 1508:= ( 1501:are 1328:work 1175:and 1018:and 994:real 978:and 955:and 283:Time 246:Mass 11708:by 11191:118 11113:348 10958:172 10787:363 9580:of 9435:sin 9399:cos 9326:cos 9290:sin 9220:sin 9184:cos 9111:cos 9075:sin 8995:sin 8959:cos 8728:cos 8692:sin 8623:sin 8587:cos 8396:sin 8348:cos 8226:): 8101:≫ | 7201:): 6978:cos 6957:sin 6898:sin 6877:cos 6751:all 6743:its 6130:sin 6106:cos 5749:): 5699:not 5646:by 4242:): 3153:). 1388:of 1342:to 1035:of 805:not 11862:: 11739:. 11193:. 11060:^ 10970:. 10960:. 10789:. 10691:^ 10678:, 10668:, 10510:. 10498:76 10367:. 9586:sω 8171:. 8163:. 8095:AB 6838:, 6757:. 5608:. 5413:+ 5406:= 5382:= 5010:: 4236:AB 3770:. 3395:= 3148:x 3085:/d 3069:AB 2231:AB 1829:AB 1577:AB 1535:). 1526:, 1517:, 1476:AB 1400:. 1264:. 1243:.) 1086:. 988:: 948:); 913:. 891:. 783:A 11814:. 11793:. 11772:. 11753:. 11666:k 11663:l 11654:j 11627:j 11623:F 11600:j 11596:F 11592:= 11587:k 11577:x 11568:l 11558:x 11549:k 11546:l 11537:j 11529:+ 11524:j 11514:x 11492:. 11474:. 11454:. 11434:R 11431:u 11416:R 11401:. 11381:. 11336:. 11309:. 11282:. 11250:. 11223:. 11169:. 11142:. 11084:. 11054:. 11027:. 11000:. 10937:. 10890:. 10853:. 10826:. 10799:. 10767:, 10752:. 10738:. 10715:. 10682:: 10672:: 10653:. 10613:. 10572:. 10551:. 10477:. 10463:. 10448:. 10433:. 10418:. 10391:. 10377:. 10148:. 10143:R 10138:u 10133:) 10130:t 10127:( 10124:R 10119:2 10111:m 10108:+ 10098:u 10093:s 10087:m 10084:2 10078:= 10072:t 10069:c 10066:i 10063:f 10057:F 10029:, 10023:R 10018:u 10007:= 9997:u 9954:u 9946:= 9941:R 9936:u 9913:Ω 9896:, 9891:R 9886:u 9881:) 9878:t 9875:( 9872:R 9869:= 9863:B 9857:x 9842:R 9839:u 9822:, 9817:R 9812:u 9807:s 9804:= 9798:B 9792:v 9764:. 9758:B 9752:x 9741:t 9738:d 9728:d 9722:m 9716:) 9710:B 9704:x 9691:( 9680:m 9671:B 9665:v 9652:m 9649:2 9643:= 9637:t 9634:c 9631:i 9628:f 9622:F 9574:R 9553:. 9547:R 9542:u 9534:) 9531:t 9528:( 9525:R 9520:2 9502:u 9488:s 9485:2 9482:= 9470:] 9464:) 9461:4 9457:/ 9450:+ 9447:t 9441:( 9428:) 9425:4 9421:/ 9414:+ 9411:t 9405:( 9393:[ 9388:) 9385:t 9382:( 9379:R 9374:2 9361:] 9355:) 9352:4 9348:/ 9341:+ 9338:t 9332:( 9319:) 9316:4 9312:/ 9305:+ 9302:t 9296:( 9281:[ 9273:s 9270:2 9267:= 9255:] 9249:) 9246:4 9242:/ 9235:+ 9232:t 9226:( 9213:) 9210:4 9206:/ 9199:+ 9196:t 9190:( 9178:[ 9173:) 9170:t 9167:( 9164:R 9159:2 9146:] 9140:) 9137:4 9133:/ 9126:+ 9123:t 9117:( 9104:) 9101:4 9097:/ 9090:+ 9087:t 9081:( 9066:[ 9055:t 9052:d 9047:R 9044:d 9038:2 9035:+ 9030:] 9024:) 9021:4 9017:/ 9010:+ 9007:t 9001:( 8988:) 8985:4 8981:/ 8974:+ 8971:t 8965:( 8953:[ 8943:2 8939:t 8935:d 8930:R 8925:2 8921:d 8914:= 8907:) 8904:t 8901:( 8897:a 8875:t 8871:R 8868:u 8864:θ 8861:u 8840:, 8830:u 8825:) 8822:t 8819:( 8816:R 8810:+ 8805:R 8800:u 8792:t 8789:d 8784:R 8781:d 8775:= 8763:] 8757:) 8754:4 8750:/ 8743:+ 8740:t 8734:( 8721:) 8718:4 8714:/ 8707:+ 8704:t 8698:( 8683:[ 8678:) 8675:t 8672:( 8669:R 8663:+ 8658:] 8652:) 8649:4 8645:/ 8638:+ 8635:t 8629:( 8616:) 8613:4 8609:/ 8602:+ 8599:t 8593:( 8581:[ 8573:t 8570:d 8565:R 8562:d 8556:= 8549:) 8546:t 8543:( 8539:v 8521:s 8504:, 8501:t 8498:s 8495:= 8492:) 8489:t 8486:( 8483:R 8470:t 8468:( 8466:R 8462:t 8457:R 8453:u 8436:, 8431:] 8425:) 8422:4 8418:/ 8411:+ 8408:t 8402:( 8393:) 8390:t 8387:( 8384:R 8377:) 8374:4 8370:/ 8363:+ 8360:t 8354:( 8345:) 8342:t 8339:( 8336:R 8330:[ 8325:= 8320:] 8314:) 8311:t 8308:( 8305:y 8298:) 8295:t 8292:( 8289:x 8283:[ 8278:= 8273:R 8268:u 8263:) 8260:t 8257:( 8254:R 8251:= 8248:) 8245:t 8242:( 8238:r 8224:t 8222:( 8220:r 8204:ω 8200:Ω 8153:B 8150:v 8146:2 8143:u 8139:1 8136:u 8132:B 8129:v 8121:R 8117:R 8110:F 8106:B 8103:x 8099:R 8092:X 8090:| 8077:, 8072:) 8066:B 8061:x 8047:( 8018:, 8013:) 8009:) 8004:B 7999:x 7994:+ 7989:B 7986:A 7981:X 7976:( 7964:( 7925:, 7916:B 7911:v 7895:2 7892:+ 7887:B 7882:a 7877:+ 7873:) 7869:) 7864:B 7859:x 7854:+ 7849:B 7846:A 7841:X 7836:( 7824:( 7809:= 7798:) 7792:B 7787:x 7773:( 7758:+ 7747:B 7742:v 7726:2 7723:+ 7718:B 7713:a 7708:+ 7704:) 7699:B 7696:A 7691:X 7680:( 7665:= 7654:) 7648:j 7643:u 7629:( 7612:j 7608:x 7602:3 7597:1 7594:= 7591:j 7580:+ 7571:j 7567:u 7551:j 7547:v 7541:3 7536:1 7533:= 7530:j 7519:2 7516:+ 7511:B 7506:a 7501:+ 7497:) 7492:B 7489:A 7484:X 7473:( 7458:= 7443:2 7439:t 7435:d 7428:j 7423:u 7416:2 7412:d 7400:j 7396:x 7390:3 7385:1 7382:= 7379:j 7368:+ 7362:t 7359:d 7352:j 7347:u 7342:d 7331:j 7327:v 7321:3 7316:1 7313:= 7310:j 7299:2 7296:+ 7291:B 7286:a 7281:+ 7276:B 7273:A 7268:a 7263:= 7251:2 7247:t 7243:d 7236:A 7231:x 7224:2 7220:d 7182:. 7171:1 7166:u 7155:= 7149:2 7145:u 7134:= 7129:2 7124:u 7116:t 7113:d 7109:d 7084:; 7076:2 7071:u 7063:= 7057:1 7053:u 7042:= 7037:1 7032:u 7024:t 7021:d 7017:d 6994:. 6990:) 6987:t 6969:, 6966:t 6954:( 6951:= 6946:2 6941:u 6916:; 6910:) 6907:t 6889:, 6886:t 6871:( 6868:= 6863:1 6858:u 6843:2 6840:u 6836:1 6833:u 6829:3 6826:u 6822:1 6819:u 6815:1 6812:u 6808:B 6804:A 6800:B 6794:. 6792:Ω 6788:B 6784:j 6781:u 6777:B 6755:B 6739:B 6735:B 6731:B 6714:. 6710:R 6705:2 6697:m 6694:= 6690:| 6683:t 6680:c 6677:i 6674:f 6668:F 6662:| 6635:, 6629:B 6626:A 6621:X 6614:2 6606:m 6603:= 6597:t 6594:c 6591:i 6588:f 6582:F 6566:B 6551:, 6546:) 6540:B 6535:x 6521:( 6492:, 6487:) 6482:B 6479:A 6474:X 6463:( 6428:. 6422:B 6419:A 6414:X 6407:2 6396:= 6392:) 6387:B 6384:A 6379:X 6368:( 6353:= 6348:B 6345:A 6340:X 6330:2 6326:t 6322:d 6316:2 6312:d 6306:= 6301:B 6298:A 6293:a 6278:B 6261:, 6253:B 6250:A 6245:X 6234:= 6229:B 6226:A 6221:X 6213:t 6210:d 6206:d 6201:= 6196:B 6193:A 6188:v 6173:B 6156:, 6149:) 6145:) 6142:t 6136:( 6124:, 6121:) 6118:t 6112:( 6102:( 6098:R 6095:= 6090:B 6087:A 6082:X 6067:A 6063:B 6042:, 6034:B 6029:a 6024:+ 6016:B 6013:A 6008:a 6003:= 5988:2 5984:t 5980:d 5973:j 5968:u 5961:2 5957:d 5945:j 5941:x 5935:3 5930:1 5927:= 5924:j 5916:+ 5910:t 5907:d 5900:j 5895:u 5890:d 5879:j 5875:v 5869:3 5864:1 5861:= 5858:j 5847:2 5844:+ 5839:B 5834:a 5829:+ 5824:B 5821:A 5816:a 5811:= 5799:2 5795:t 5791:d 5784:A 5779:x 5772:2 5768:d 5743:A 5739:R 5735:B 5729:. 5727:A 5723:R 5719:B 5715:A 5711:Ω 5707:Ω 5703:B 5695:j 5692:u 5688:B 5673:) 5667:( 5662:) 5658:( 5654:. 5640:. 5575:. 5569:B 5563:x 5552:t 5549:d 5539:d 5533:m 5527:) 5521:B 5515:x 5502:( 5491:m 5482:B 5476:v 5463:m 5460:2 5454:= 5448:t 5445:c 5442:i 5439:f 5433:F 5415:F 5411:A 5408:F 5404:B 5401:F 5394:F 5390:B 5387:a 5384:m 5380:B 5377:F 5360:. 5354:B 5348:x 5337:t 5334:d 5324:d 5315:) 5309:B 5303:x 5290:( 5273:B 5267:v 5254:2 5245:A 5239:a 5234:= 5228:B 5222:a 5207:B 5204:a 5200:B 5197:a 5189:A 5186:a 5166:. 5161:) 5154:B 5148:x 5134:( 5122:+ 5116:B 5110:x 5099:t 5096:d 5086:d 5080:+ 5074:B 5068:v 5055:2 5052:+ 5046:B 5040:a 5035:= 5029:A 5023:a 4987:. 4983:] 4979:) 4976:t 4973:( 4968:j 4963:u 4956:j 4952:x 4946:3 4941:1 4938:= 4935:j 4918:[ 4906:+ 4901:j 4896:u 4889:j 4885:x 4879:3 4874:1 4871:= 4868:j 4854:t 4851:d 4841:d 4835:+ 4832:) 4829:t 4826:( 4821:j 4816:u 4809:j 4805:v 4799:3 4794:1 4791:= 4788:j 4772:2 4769:+ 4763:B 4757:a 4752:= 4741:] 4737:) 4734:t 4731:( 4726:j 4721:u 4707:[ 4693:j 4689:x 4683:3 4678:1 4675:= 4672:j 4664:+ 4656:j 4651:u 4640:t 4637:d 4627:d 4619:j 4615:x 4609:3 4604:1 4601:= 4598:j 4590:+ 4587:) 4584:t 4581:( 4576:j 4571:u 4556:j 4552:v 4546:3 4541:1 4538:= 4535:j 4527:2 4521:+ 4515:B 4509:a 4504:= 4494:A 4488:a 4461:, 4453:2 4449:t 4445:d 4438:j 4433:u 4426:2 4422:d 4413:j 4409:x 4403:3 4398:1 4395:= 4392:j 4384:+ 4378:t 4375:d 4368:j 4363:u 4358:d 4347:j 4343:v 4337:3 4332:1 4329:= 4326:j 4318:2 4315:+ 4309:B 4303:a 4298:= 4290:2 4286:t 4282:d 4274:A 4268:x 4261:2 4257:d 4233:a 4229:t 4208:, 4204:] 4200:) 4197:t 4194:( 4189:j 4184:u 4170:[ 4158:+ 4153:j 4148:u 4137:t 4134:d 4124:d 4118:= 4112:t 4109:d 4104:) 4101:t 4098:( 4093:j 4088:u 4083:d 4069:+ 4064:j 4059:u 4048:t 4045:d 4035:d 4029:= 4021:2 4017:t 4013:d 4008:) 4005:t 4002:( 3997:j 3992:u 3985:2 3981:d 3951:, 3948:) 3945:t 3942:( 3937:j 3932:u 3919:= 3913:t 3910:d 3905:) 3902:t 3899:( 3894:j 3889:u 3884:d 3855:, 3852:) 3849:t 3846:( 3840:= 3834:t 3831:d 3823:d 3817:= 3813:| 3803:| 3785:Ω 3758:F 3741:. 3732:2 3728:t 3724:d 3717:j 3712:u 3705:2 3701:d 3692:j 3688:x 3682:3 3677:1 3674:= 3671:j 3663:m 3654:t 3651:d 3644:j 3639:u 3634:d 3626:j 3622:v 3616:3 3611:1 3608:= 3605:j 3597:m 3594:2 3585:B 3582:A 3576:a 3571:m 3565:= 3559:s 3556:u 3553:o 3550:i 3547:t 3544:i 3541:t 3538:c 3535:i 3532:f 3526:F 3498:, 3492:s 3489:u 3486:o 3483:i 3480:t 3477:i 3474:t 3471:c 3468:i 3465:f 3459:F 3454:+ 3448:A 3442:F 3437:= 3431:B 3425:F 3410:A 3407:F 3403:B 3400:a 3397:m 3393:B 3390:F 3373:. 3362:2 3358:t 3354:d 3347:j 3342:u 3335:2 3331:d 3322:j 3318:x 3312:3 3307:1 3304:= 3301:j 3293:m 3290:+ 3284:t 3281:d 3274:j 3269:u 3264:d 3256:j 3252:v 3246:3 3241:1 3238:= 3235:j 3227:m 3224:2 3221:+ 3215:B 3212:A 3206:a 3201:m 3198:+ 3192:B 3186:F 3181:= 3175:A 3169:F 3150:j 3134:t 3131:d 3127:/ 3121:j 3116:u 3111:d 3105:j 3101:x 3087:t 3083:B 3080:v 3077:d 3066:a 3062:B 3059:a 3050:) 3048:1 3046:( 3029:. 3021:2 3017:t 3013:d 3006:j 3001:u 2994:2 2990:d 2981:j 2977:x 2971:3 2966:1 2963:= 2960:j 2952:+ 2946:t 2943:d 2936:j 2931:u 2926:d 2918:j 2914:v 2908:3 2903:1 2900:= 2897:j 2886:2 2883:+ 2877:B 2871:a 2866:+ 2860:B 2857:A 2851:a 2846:= 2838:2 2834:t 2830:d 2822:A 2816:x 2809:2 2805:d 2773:. 2767:t 2764:d 2757:j 2752:u 2747:d 2739:j 2735:v 2729:3 2724:1 2721:= 2718:j 2710:+ 2704:B 2698:a 2693:= 2687:t 2684:d 2677:j 2672:u 2667:d 2659:j 2655:v 2649:3 2644:1 2641:= 2638:j 2630:+ 2625:j 2620:u 2612:t 2609:d 2602:j 2598:v 2594:d 2586:3 2581:1 2578:= 2575:j 2567:= 2561:t 2558:d 2550:B 2544:v 2539:d 2523:B 2520:x 2503:. 2495:2 2491:t 2487:d 2480:j 2475:u 2468:2 2464:d 2455:j 2451:x 2445:3 2440:1 2437:= 2434:j 2426:+ 2420:t 2417:d 2410:j 2405:u 2400:d 2391:t 2388:d 2381:j 2377:x 2373:d 2365:3 2360:1 2357:= 2354:j 2346:+ 2340:t 2337:d 2329:B 2323:v 2318:d 2312:+ 2306:B 2303:A 2297:a 2292:= 2284:2 2280:t 2276:d 2268:A 2262:x 2255:2 2251:d 2228:v 2224:B 2221:v 2204:. 2198:t 2195:d 2188:j 2183:u 2178:d 2170:j 2166:x 2160:3 2155:1 2152:= 2149:j 2141:+ 2135:B 2129:v 2124:+ 2118:B 2115:A 2109:v 2104:= 2098:t 2095:d 2087:A 2081:x 2076:d 2060:B 2057:v 2040:. 2033:t 2030:d 2023:j 2018:u 2013:d 2005:j 2001:x 1995:3 1990:1 1987:= 1984:j 1976:+ 1971:j 1966:u 1958:t 1955:d 1948:j 1944:x 1940:d 1932:3 1927:1 1924:= 1921:j 1913:+ 1907:t 1904:d 1896:B 1893:A 1887:X 1882:d 1876:= 1870:t 1867:d 1859:A 1853:x 1848:d 1826:X 1821:j 1817:u 1800:. 1794:j 1789:u 1782:j 1778:x 1772:3 1767:1 1764:= 1761:j 1753:+ 1747:B 1744:A 1738:X 1733:= 1727:A 1721:x 1706:t 1702:B 1699:x 1682:. 1676:j 1671:u 1664:j 1660:x 1654:3 1649:1 1646:= 1643:j 1635:= 1629:B 1623:x 1608:j 1603:j 1599:u 1592:t 1590:( 1588:B 1585:x 1581:t 1579:( 1574:X 1566:t 1564:( 1562:A 1559:x 1549:m 1533:3 1529:x 1524:2 1520:x 1515:1 1511:x 1506:B 1503:x 1499:B 1495:j 1491:j 1488:u 1484:B 1480:A 1473:X 1469:B 1465:B 1461:B 1458:x 1454:A 1450:A 1447:x 1144:. 1142:K 1138:a 1134:a 1131:m 1127:F 1123:m 1119:k 1115:K 911:m 907:a 904:m 900:F 860:. 842:a 772:e 765:t 758:v 448:) 67:t 64:d 58:p 54:d 48:= 43:F

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Index