Inertial frame

Coriolis Acceleration The Coriohs acceleration arises in a rotating frame, which has no parallel in an inertial frame. When a body moves at a linear velocity u in a. rotating frame with angular speed H, it experiences a Coriolis acceleration with magnitude ... 

As a rule, geophysical literature describes the rotation of a particle on the earth surface with the help of the attraction force and the centrifugal force. It turns out that the latter appears because we use a system of coordinates that rotates together with Earth. As we know Newton s second law, wa = F, is valid only in an inertial frame of reference, that is, the product of mass and acceleration is equal to the real force acting on the particle. However, it is not true when we study a motion in a system of coordinates that has some acceleration with respect to the inertial frame. For instance, it may happen that there is a force but the particle does not move. On the contrary, there are cases when the resultant force is zero but a particle moves. Correspondingly, replacement of the acceleration in the inertial frame by that in a non-inertial one gives a new relation between the acceleration, mass, particle, and an applied force ... 

We see that the acceleration in the inertial frame P can be represented in terms of the acceleration, components of the velocity and coordinates of the point p in the rotating frame, as well as the angular velocity. This equation is one more example of transformation of the kinematical parameters of a motion, and this procedure does not have any relationship to Newton s laws. Let us rewrite Equation (2.37) in the form... 

Substitution of Equation (2.38) into the Newton s second law in the inertial frame of reference ma = F gives... 

Here v, and a, are the velocity and acceleration of the point p in the rotating frame of reference, respectively. Substitution of Equation (2.55) into Newton s second law gives an equation of motion in the non-inertial frame ... 

Here p is the pressure, ga the field of attraction, 5 the density of the fluid, and r the vector directed away from the axis of rotation and it is equal in magnitude to the distance between a particle and this axis. The first two terms of Equation (2.332) characterize the real forces acting on the particle, namely the surface and attraction ones. At the same time the last term is a centrifugal force, and it is introduced because we consider a non-inertial frame of reference. It is convenient to represent Equation (2.332) as... 

In order to outline the main features of measuring the gravitational field with the help of ballistic gravimeter imagine that a small body falls inside a vacuum cylinder under the action of the gravitational field only. In accordance with Newton s second law in the inertial frame we have... 

As was pointed out earlier, when we have considered the physical principles of the ballistic gravimeter and the pendulum an influence of the Coriolis force was ignored. Now we will try to take into account this factor and consider the motion of a particle near the earth s surface. With this purpose in mind let us choose a non-inertial frame of reference, shown in Fig. 3.5a its origin 0 is located near the earth s surface and it rotates together with the earth with angular velocity a>. The unit vectors i, j, and k of this system are fixed relative to the earth and directed as follows i is horizontal, that is, tangential to the earth s surface and points south, j is also horizontal and points east, k is vertical and points upward. As is shown in Fig. 3.5a SN is the earth s axis, drawn from south to north, I is the unit vector along OiO, and K is a unit vector parallel to SN. 

First, we derive again but in a slightly different way than in Chapter 2 the equation of a motion in a non-inertial frame of reference. As before, r is the position of the moving particle with respect to 0 and OiO = ro. The position of the particle with respect to the origin 0i of the inertial frame is... 

The problem relates directly to the constancy of c, which implies that the velocity of light is independent of both the motion of its source and the direction of propagation, a condition that cannot hold in more than one Newtonian inertial frame if the Galilean principle of relativity applies. Since there is no evidence that the laws of physics are not identical in all inertial frames of reference the only conclusion is that the prescription for Galilean transformations needs modification to be consistent, not only with simple mechanics, but also with electromagnetic effects. 

It is obvious that gab z) is independent both of the choice of inertial frame at z, with its corresponding natural coordinate system (v), and the choice of curve x(X). The elements of g are known as the components of the metric tensor in this coordinate system. Expression (39) is the required generalization that allows evaluation of 4> at all points in terms of gab %) and the curve x(A). 

We consider particles of spin 1/2, mass m, and electric charge e in a fixed inertial frame. They shall be exposed to external electromagnetic fields that are static in the given frame,... 

Thus, it is a matter of record that a satisfactory incorporation of Mach s principle within general relativity can be attained when the constraint of closure is imposed. However, there is still the point of view that, because general relativity allows solutions that give an internally consistent discussion of an empty inertial spacetime—whereas it is operationally impossible to define an inertial frame in the absence of matter—then the theory (general relativity) must have a non-fundamental basis at the classical level. 

Although the conventional approach outlined above contains all the essential components of Mach s principle, it does not focus on what, in our view, is the essential point about the principle that it is impossible to define inertial frames in the absence of material. This fact is brought out most clearly in the following alternative approach. 

Specifically, rather than define inertial frames with respect to the universal rest frame, we can define an inertial frame as any frame of reference within which the series of collision experiments discussed above yields the ratio AVa/AVb to be a constant independently of the experiment s initial conditions. If this constant ratio is then termed the relative inertial mass of the two balls, then the whole idea of the inertial frame and inertial mass is arrived at without any reference whatsoever to distant galaxies —and, in fact, is given a local context. 

We have argued that the fundamental significance of Mach s principle arises from its implication of the impossibility of defining inertial frames in the absence of material or, as a generalization, we can say that it is impossible to... 

At this stage, since no notion of inertial frame has been introduced, the idea of inertial mass cannot be defined. However, we have assumed the model universe to be composed of a countable infinity of labeled—but otherwise indistinguishable—material particles so that we can associate with each individual particle a property called mass that quantifies the amount of material in the particle, and is represented by a scale constant, say, mo, having units of mass. 

An absolute inertial frame of reference E must be reintroduced. In plain words, introduce a modem version of the ether. 

More formally, consider any frame S with an observer at the origin, and let the acceleration of the origin relative to E be aj = 0. Let 5s be the class of inertial frames equivalent to E ... 

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