Reference system Noninertial

Noninertial reference systems. An inertia force D Alembert principle... 

So far we have considered the motion of a body in inertial reference systems. However, there exist many problems where it is necessary to use noninertial reference systems such as, for example, the motion of molecules in a centrifuge along a circular path or accelerating motion in the rocket. In noninertial reference systems expressions (1.3.14) are not fulfilled. Reference systems in which the motion of a free body is not rectilinear and uniform are referred to as noninertial systems. Consequently, any reference system moving with acceleration relative to any inertial reference system is a noninertial one. The acceleration can be both translational (a 0) or rotational (a 0). In the general case, the acceleration of different points of a moving body can be different. This means that the space connected with the noninertial reference systems is neither uniform, nor isotropic. 

The equation of a MP motion regarding the noninertial system looks different from an inertial one. Consider the specific example of a noninertial reference system K (x, y, z ) moving with an acceleration comparative to a certain inertial system K x, y, z). Suppose then that a MP in this inertial system moves with an acceleration a, and this acceleration is caused by the action of forces F, F, F. According to eq. (1.3.13) we can write... 

In the system K the same MP will have an acceleration a (relative acceleration), which is the sum of a and the acceleration of the noninertial reference system Uq, i.e., a = a - a. We can multiply the right- and left-hand sides of this expression by the mass of an MP (m) ... 

The expression obtained differs from the equation of motion in the inertial reference system (1.3.7) by the term -truiQ. The noncompliance with the second Newton law is caused by the appearance of that additional term. Moreover, if the geometric snm of acting forces is equal to zero, then a = Uq, whereas according to the second Newton law it also has to be zero. The product of the body s mass and the acceleration of the noninertial reference system taken with the opposite sign is called the force of inertia. 

Inertia forces are the uncommon forces that disobey the laws of classical Newton mechanics. Indeed, in a noninertia reference system we are unable to indicate a body whose action can explain the appearance of inertia forces. This signifies that Newtonian laws are not executed in noninertial reference systems. Figuratively speaking, there exists a force of actions (the force of inertia), but no force of counteraction. In noninertial reference systems, these particularities of inertia forces do not allow the selection of a closed system of bodies (refer to 1.3.7), since for any body in a noninertial system the inertia forces are the internal ones. Thus, in the noninertial reference system the conservation laws of energy and momentum, which will be considered below (see Section 1.5), are not valid. 

This task can be solved in a noninertial system. In this case a reference systan can be connected to the elevator. It means that to all forces the D Alambert force (F, = ma should be added (a, is the acceleration of the elevator movement relative to the earth). Three forces are acting on a body in this case gravitational force mg, elasticity force N and the inertia force F,. In the reference system connected with the elevator the body is at rest. Therefore, the sum of the forces is zero mg + N + F, = 0. After projection on the z-axis the equation is transformed toN g ma = 0, whereas the support reaction is N = mg + ma = m(g + a) we arrive at the same equation as in the Part A of this example. 

The conservation of momentum or Newton s second law applies to a particle or fixed set of particles, namely a system. The velocity used must always be defined relative to a fixed or inertial reference plane. The Earth is a sufficient inertial reference. Therefore, any control volume associated with accelerating aircraft or rockets must account for any differences associated with how the velocities are measured or described. We will not dwell on these differences, since we will not consider such noninertial applications. 

Of course, the Sagnac-type experiments were not made in a perfect inertial systems. The earth s orbital motion around the sun is also a noninertial system. But the circumference velocities in both cases are extremely low, v/c[Pg.398]

In the subsequent treatment the electron coordinate will be measured from the accelerated target nucleus and is the only dynamical variable. Thus the target system is the frame of reference [31,32], In such a noninertial system non-Newtonian forces arise. The corresponding Hamiltonian is... 

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