Earth-Fixed Reference Frame

The first question that we may ask is the form of the relationship between D/Dt and the ordinary partial time derivative 3/31. The so-called sky-diver problem illustrated in Fig. 2-4 provides a simple physical example that may serve to clarify the nature of this relationship without the need for notational complexity. A sky diver leaps from an airplane at high altitude and begins to record the temperature T of the atmosphere at regular intervals of time as he falls toward the Earth. We denote his velocity as —Uc/iveri-, where izis a unit vector in the vertical direction, and the time derivative of the temperature he records as D T/Dt. Here, D /Dt represents the time rate of change (of 7) measured in a reference frame that moves with the velocity of the diver. Evidently there is a close relationship between this derivative and the convected derivative that was introduced in the preceding paragraph. Let us now suppose, for simplicity, that the temperature of the atmosphere varies with the distance above the Earth s surface but is independent of time at any fixed point, say, z = z. In this case, the partial time derivative 3 T/dt is identically equal to zero. Nevertheless, in the frame of reference of the sky diver, D T/Dt is not zero. Instead,... 

Using these relations it is mandatory that, B,a,A refer to an identical (conventional) earth fixed reference frame, eg to the CIO/BIH geocentric tripod. Both astronomical and GPS measurements must therefore be reduced to this frame. For this purpose the GPS measurement has to be linked to at least one fiducial point of the CIGNET (= Cooperative International GPS Network) which has a well known SLR (= Satellite Laser Ranging) derived absolute position. 

A reference frame whose axes are fixed relative to the earth s surface is what this book will call a lab frame. A lab frame for all practical purposes is inertial (Sec. G.IO on page 503). It is in this kind of stationary frame that the laws of thermodynamics have been found by experiment to be valid. 

The internal energy, U, is the energy of the system measured in a reference frame that allows U to be a state function—that is, at each instant the value of U depends only on the state of the system. This book will call a reference frame with this property a local frame. A local frame may also be, but is not necessarily, an earth-fixed lab frame. 

Newton s laws of motion are obeyed only in an inertial reference frame. A reference frame that is fixed or moving at a constant velocity relative to local stars is practically an inertial reference frame. To a good approximation, a reference frame fixed relative to the earth s surface is also an inertial system (the necessary corrections are discussed in Sec. G.IO). This reference frame will be called simply the lab frame, and treated as an inertial frame in order that we may apply Newton s laws. 

APPENDIX G FORCES, ENERGY, AND WORK G. 10 Earth-Fixed Reference frame... 

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. 

In the eyes of a distant observer using a fixed coordinate system, a meteorite falling in the gravitational field of the earth describes a parabolic path. An observer standing on earth uses the rotating frame of reference of the earth. For him, the complicated path of the falling meteorite simplifies to a straight vertical line. 

Example 1 Most discussions involving the Bloch model introduce the concept of the rotating frame. The concept of a rotating coordinate system is a familiar one because in real life positions and motion are referred to the earth, a coordinate system that is rotating. Similarly rather than refer the motion of the magnetization vectors to the fixed laboratory coordinate system, it is simpler to refer their motion to a rotating frame of reference which rotates at the NMR transmitter frequency of the nucleus under study. 

It is understood that the distances and velocities associated with these two laws are determined relative to an inertial frame and that the torque and angular momentum are measured relative to the same fixed point. It is important to note that an inertial frame is a frame in which these laws hold, thus, it must be found by experiment. In his study of the motion of Mars about the sun, Newton found that the stars provided a satisfactory inertial frame. For many engineering problems, a frame fixed relative to the earth can be used as an inertial frame however, this is not the case for large scale meteorological phenomena for which the rotation of the earth produces an acceleration referred to as the Coriolis force (Dutton, 1976). 

---