scholarly journals Post-Newtonian treatise on the rotational motion of a finite body

1986 ◽  
Vol 114 ◽  
pp. 35-40 ◽  
Author(s):  
T. Fukushima

The definition of the angular momentum of a finite body is given in the post-Newtonian framework. The non-rotating and the rigidly rotating proper reference frame(PRF)s attached to the body are introduced as the basic coordinate systems. The rigid body in the post-Newtonian framework is defined as the body resting in a rigidly rotating PRF of the body. The feasibility of this rigidity is assured by assuming suitable functional forms of the density and the stress tensor of the body. The evaluation of the time variation of the angular momentum in the above two coordinate systems leads to the post-Newtonian Euler's equation of motion of a rigid body. The distinctive feature of this equation is that both the moment of inertia and the torque are functions of the angular velocity and the angular acceleration. The obtained equation is solved for a homogeneous spheroid suffering no torque. The post-Newtonian correction to the Newtonian free precession is a linear combination of the second, fourth and sixth harmonics of the precessional frequency. The relative magnitude of the correction is so small as of order of 10−23 in the case of the Earth.

1975 ◽  
Vol 42 (3) ◽  
pp. 552-556 ◽  
Author(s):  
A. J. Padgaonkar ◽  
K. W. Krieger ◽  
A. I. King

The computation of angular acceleration of a rigid body from measured linear accelerations is a simple procedure, based on well-known kinematic principles. It can be shown that, in theory, a minimum of six linear accelerometers are required for a complete definition of the kinematics of a rigid body. However, recent attempts in impact biomechanics to determine general three-dimensional motion of body segments were unsuccessful when only six accelerometers were used. This paper demonstrates the cause for this inconsistency between theory and practice and specifies the conditions under which the method fails. In addition, an alternate method based on a special nine-accelerometer configuration is proposed. The stability and superiority of this approach are shown by the use of hypothetical as well as experimental data.


1996 ◽  
Vol 172 ◽  
pp. 309-320
Author(s):  
S.A. Klioner

We consider rotational motion of an arbitrarily composed and shaped, deformable weakly self-gravitating body being a member of a system of N arbitrarily composed and shaped, deformable weakly self-gravitating bodies in the post-Newtonian approximation of general relativity. Considering importance of the notion of angular velocity of the body (Earth, pulsar) for adequate modelling of modern astronomical observations, we are aimed at introducing a post-Newtonian-accurate definition of angular velocity. Not attempting to introduce a relativistic notion of rigid body (which is well known to be ill-defined even at the first post-Newtonian approximation) we consider bodies to be deformable and introduce the post-Newtonian generalizations of the Tisserand axes and the principal axes of inertia.


1978 ◽  
Vol 33 (10) ◽  
pp. 1133-1150
Author(s):  
Eberhard Kern

The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result.The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory.The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.


2021 ◽  
Vol 18 (1) ◽  
pp. 136
Author(s):  
V. Tanriverdi

Euler derived equations for rigid body rotations in the body reference frame and in the stationary reference frame by considering an infinitesimal part of the rigid body.Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame.However, their equivalence is not shown explicitly.In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and in the stationary reference frame and torque-angular momentum relation.We also calculated equations of motion from Lagrangian.These calculations show that all four of them are equivalent.


1987 ◽  
Vol 109 (2) ◽  
pp. 124-127 ◽  
Author(s):  
Jorge Angeles

The computation of the angular acceleration of a rigid body from measurements of accelerations of three noncollinear points of the body is presented in this paper. This is based on algorithms presented previously for the computation of the orientation and the angular velocity of a rigid body from measurements of position and velocity of three noncollinear points of the body. Moreover, compatibility conditions that the said point measurements should verify are introduced. These are necessary to verify the rigidity assumption on the one hand; on the other hand, they are introduced as a means of filtering roundoff and/or measurement errors, which is particularly useful if redundant measurements are taken, i.e., on more than three points. The procedure is illustrated with a fully solved example.


2012 ◽  
Vol 2 (1) ◽  
pp. 28
Author(s):  
Luigi T. Bercades ◽  
Willy Pieter

<p class="MsoNormal" style="text-align: justify; margin: 0cm 0cm 6pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Calibri;">This study is a theoretical analysis of the kinematic and kinetic aspects of the modified taekwondo axe kick. The traditional or classical axe kick has the whole kicking leg (the thigh and the shank) considered as a rigid body on both the upswing and downswing phases of the kick, which is speculated to have sufficient angular momentum to increase the risk of some forms of injuries in competition. The present study seeks to present an alternate version that will decrease the moment of inertia on the downswing, reduce the subsequent angular momentum<em style="mso-bidi-font-style: normal;">, </em>and finally decrease the resultant impulse to the target. Theoretically, this will reduce the chances of certain types of injury caused by the kick.</span></span></span></p>


10.14311/664 ◽  
2005 ◽  
Vol 45 (1) ◽  
Author(s):  
O. Brůha ◽  
T. Brůha

This paper deals with an effect which appears when heating or cooling a rotating body. No external forces acting on the body are supposed. Due to thermal expansion, the moment of inertia of the body varies together with the temperature changes. In agreement with the principle of conservation of angular momentum [1], the angular momentum is constant. This results in angular velocity changes and subsequently in kinetic energy changes. Also the stress energy varies together with the changes in thermal dimension. To satisfy the principle of energy conservation we have to suppose that the changes in kinetic and stress energy are compensated by the changes in internal energy, which is correlated with temperature changes of the body. This means that the rules for the heating or cooling process of a rotating body are not the same as those for a body at rest. This idea, applied to a cylinder rotating around its geometric axis under specific parameters, has been mathematically treated. As a result, the difference between the final temperature of the rotating cylinder and the temperature of the cylinder at rest has been found. 


2000 ◽  
Vol 15 (17) ◽  
pp. 1107-1115
Author(s):  
F. I. COOPERSTOCK

Recently we demonstrated that Eddington's spinning rod lost "kinetic" energy but preserved its total energy while emitting gravitational waves. Additional information is now determined by analyzing the angular momentum of the rod. We develop an expression for the angular momentum loss of the material system alone with an integration over the source region. As expected, it is found that the moment of inertia of the rod increases as the rotation rate diminishes in analogy with typical classical systems. Its rate of change is determined in relation to the angular deceleration of the body.


Author(s):  
Fred Feldman

Reflection on death gives rise to a variety of philosophical questions. One of the deepest of these is a question about the nature of death. Typically, philosophers interpret this question as a call for an analysis or definition of the concept of death. Plato, for example, proposed to define death as the separation of soul from body. However, this definition is not acceptable to those who think that there are no souls. It is also unacceptable to anyone who thinks that plants and lower animals have no souls, but can nonetheless die. Others have defined death simply as the cessation of life. This too is problematic, since an organism that goes into suspended animation ceases to live, but may not actually die. Death is described as ‘mysterious’, but neither is it clear what this means. Suppose we cannot formulate a satisfactory analysis of the concept of death: in this respect death would be mysterious, but no more so than any other concept that defies analysis. Some have said that what makes death especially mysterious and frightening is the fact that we cannot know what it will be like. Death is typically regarded as a great evil, especially if it strikes someone too soon. However, Epicurus and others argued that death cannot harm those who die, since people go out of existence when they die, and people cannot be harmed at times when they do not exist. Others have countered that the evil of death may lie in the fact that death deprives us of the goods we would have enjoyed if we had lived. On this view, death may be a great evil for a person, even if they cease to exist at the moment of death. Philosophers have also been concerned with the question of whether people can survive death. This is open to several interpretations, depending on what we understand to be people and what we mean by ‘survive’. Traditional materialists take each person to be a purely physical object – a human body. Since human bodies generally continue to exist after death, such materialists presumably must say that we generally survive death. However, such survival would be of little value to the deceased, since the surviving entity is just a lifeless corpse. Dualists take each person to have both a body and a soul. A dualist may maintain that at death the soul separates from the body, thereby continuing to enjoy (or suffer) various experiences after the body has died. Some who believe in survival think that the eternal life of the soul after bodily death can be a good beyond comparison. But Bernard Williams has argued that eternal life would be profoundly unattractive. If we imagine ourselves perpetually stuck at a given age, we may reasonably fear that eternal life will eventually become rather boring. On the other hand, if we imagine ourselves experiencing an endless sequence of varied ‘lives’, each disconnected from the others, then it is questionable whether it will in fact be ‘one person’ who lives eternally. Finally, there are questions about death and the meaning of life. Suppose death marks the end of all conscious experience – would our lives be then rendered meaningless? Or would the fact of impending death help us to recognize the value of our lives, and thereby give deeper meaning to life?


1973 ◽  
Vol 74 (2) ◽  
pp. 365-367 ◽  
Author(s):  
P. C. Chatwin

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equationwhere u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.


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