Non-inertial frame of reference

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... 

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 first term can be interpreted as the vibrational kinetic enei y, and the second and third terms are the effective potential energy for the vibrational motion. The final term, which is the rotational kinetic energy, becomes an addition to the potential energy in the non-inertial frame of reference of the rotating molecule. 

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 ... 

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 ... 

In centrifugal microfluidic systems, the Navier-Stokes equation is most conveniently expressed within the reference frame where the substrate rotating at a frequency = 27ri> is at rest (Fig. 1). Due to the non-inertial nature of this frame of reference, the centrifugal force density the Euler force density and the Coriolis force density/c additionally appear in the hydrodynamic equation... 

LeRoy, J. P., and Wallace, R. (1987), Form of the Quantum Kinetic Energy Operator for Relative Motion of A Group of Particles in A General Non-Inertial Reference Frame, Chem. Phys. 118, 379. 

Intrinsic molecular volume, or the volume of the envelope of atomic spheres, can easily be calculated. Let N be the number of atoms in a molecule, with nuclear positions Xj reckoned in some reference frame, say the inertial reference frame. Let Ri be the atomic intermolecular non-bonding radius of atom i, briefly called henceforth the atomic radius. Let nj be the distance between the nuclei of two atoms joined by a chemical bond. Whenever ry is smaller than the sum of atomic radii, the sphere of atom i cuts into the sphere of atom j a spherical cap of height /ly. Molecular volume, Vm, can be calculated [8,10] by computing the total volume of the atomic spheres and subtracting the volumes of the intersecting caps ... 

The measurement of the Lense-Thirring effect is the first scientific goal of the HYPER project (Fig. 3) and will be more detailed in this section. The Lense-Thirring effect consists of a precession of a local reference frame (realized by inertial gyroscopes) and a non-local one realized by pointing the direction of fixed stars. This Lense-Thirring precession is given by ... 

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