Inertial Frame of Reference

11 Key excerpts on "Inertial Frame of Reference"

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  A Modern Approach to Classical Mechanics

  • Harald Iro(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  4 . The coordinate system is often considered to be the reference frame, but strictly speaking one should distinguish between these two notions. Of course, a change of reference frame is accompanied by a coordinate transformation (but not, in general, vice versa). In a certain sense, one may say that the concept of a reference frame emphasizes the physical world view, while the concept of coordinate systems emphasizes the mathematical point of view.

  8.1  Inertial frames

  In view of the possible dependence of physical laws on the frame of reference, it is natural to ask: given a particular frame of reference, do there exist ‘equivalent’ reference frames, in which a preferred physical law is the same as in that frame?

  The fundamental law in nonrelativistic mechanics is Newton’s equation of motion for a particle influenced by a force: m = F. In Newton’s first law, free motion, i.e. F = 0, is selected as the basic case, relative to which – by the second law – the presence of forces can be assessed. Thus the distinguished role of uniform and rectilinear motion is established. In former times the view was advanced that every motion needs a cause. Also to maintain free motion of a body there has to be a permanent push. Later on, instead of this view, free motion was explained by the property of inertia5 of a body.

  Returning to the question of reference frames: Indeed, an important class of reference frames is selected by the requirement that in each of the frames, motion – in the absence of a force – obeys the equation

  In each frame free motion is uniform and rectilinear. Such frames are called inertial frames. In 1886, L. Lange proposed6

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  Modern Classical Mechanics

  • T. M. Helliwell, V. V. Sahakian(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  Picture a set of three orthogonal meter sticks defining a set of Cartesian coordinates drifting through space with no forces applied. An inertial observer drifts with the coordinate system and uses it to make measurements of physical phenomena. This inertial frame and inertial observer are not unique, however: having established one inertial frame, any other frame moving at constant velocity relative to it is also inertial, as illustrated in Figure 1.1. Two of these inertial observers, along with their personal coordinate systems, are depicted in Figure 1.2: observer O describes positions of objects through a Cartesian system labeled (x, y, z), while observer O uses a system labeled (x , y , z ). 3 4 1 Newtonian Particle Mechanics Fig. 1.1 Various inertial frames in space. If one of these frames is inertial, any other frame moving at constant velocity relative to it is also inertial. z y x y’ z’ x’ Fig. 1.2 Two inertial frames, Oand O , moving relative to one another along their mutual x or x axes. An event of interest to an observer is characterized by the position in space at which the measurement is made – but also by the instant in time at which the observation occurs, according to clocks at rest in the observer’s inertial frame. For example, an event could be a snapshot in time of the position of a particle along its trajectory. Hence, the event is assigned four numbers by observer O : x, y, z, and t for time, while observer O labels the same event x , y , z , and t . Without loss of generality, observer O can choose her x axis along the direction of motion of O , and then the x axis of O can be aligned with that axis as well, as shown in Figure 1.2. It seems intuitively obvious that the coordinates of the event are related by x = x + Vt , y = y , z = z , t = t , (1.1) 5 1.2 Newton’s Laws of Motion where we assume that the origins of the two frames coincide at time t = t = 0.

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  Physics for Scientists and Engineers with Modern Physics

  • Raymond Serway, John Jewett(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame. When you and the train accel- erate, however, you are observing the puck from a noninertial reference frame because you and the train are accelerating relative to the inertial reference frame of the Earth’s surface. While the puck appears to be accelerating according to your observations, a reference frame can be identified in which the puck has zero accel- eration. For example, an observer standing outside the train on the ground sees the puck sliding relative to the table but always moving with the same velocity with respect to the ground as the train had before it started to accelerate (because there is almost no friction to “tie” the puck and the train together). Therefore, Newton’s first law is still satisfied even though your observations as a rider on the train show an apparent acceleration relative to you. A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame, and for our purposes we can consider the Earth as being such a frame. The Earth is not really an inertial frame because of its orbital motion around the Sun and its rotational motion about its own axis, both of which involve centripetal accelerations. These accelerations are small com- pared with g, however, and can often be neglected. For this reason, we model the Earth as an inertial frame, along with any other frame attached to it. Let us assume we are observing an object from an inertial reference frame. (We will return to observations made in noninertial reference frames in Section 6.3.) Before about 1600, scientists believed that the natural state of matter was the state of rest. Observations showed that moving objects eventually stopped moving. Galileo was the first to take a different approach to motion and the natural state of matter.

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  The Mechanical Universe

  Mechanics and Heat, Advanced Edition

  • Steven C. Frautschi, Richard P. Olenick, Tom M. Apostol, David L. Goodstein(Authors)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  CHAPTER FORCES IN ACCELERATING REFERENCE FRAMES From the beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer [moving relative to the earth], everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. Albert Einstein. Autobiographical Notes (1949) 9.1 INERTIAL AND NONINERTIAL REFERENCE FRAMES We have already introduced Galileo's ideas on relative motion in Chapter 4 . We defined inertial frames - frames in which the law of inertia holds - and remarked that an observer in any inertial frame deduces the same laws of motion, and has no way of determining whether he is at rest or moving in an absolute sense, Galileo was able to provide striking examples of these ideas, such as a stone dropped from the mast of a moving boat, and 203 204 FORCES IN ACCELERATING REFERENCE FRAMES to deduce a vitally important application - the earth need not be considered the stationary hub around which the heavens revolve. However, Galileo did not have a clear-cut dynamical framework within which to derive his ideas. And exactly how to treat motion in a rotating frame, or indeed in any noninertial frame - one that is accelerated relative to an inertial frame - remained obscure. It was only after Newton's second law was discovered that Galileo's ideas could be derived in a clear-cut way. Moreover, Newton's laws could be used in accelerated as well as inertial frames. This allowed Newton to supplement his description of circular motion as viewed from an inertial frame (where, as we have seen, some physical force must supply a centripetal acceleration) with a treatment of circular motion as felt by an observer riding along with the circling object.

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  Game Physics

  • David H. Eberly(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  The equations of motion F = ma will be used to establish the path of motion for an object by numerically solving the second-order differential equations for position. Each of the vector quantities of position, velocity, and acceleration is measured with respect to some coordinate system. This system is referred to as the inertial frame. If x = (x 1 , x 2 , x 3 ) is the representation of the position in the inertial frame, the components x 1 , x 2 , and x 3 are referred to as the inertial coordinates. Although in many cases the inertial frame is considered to be fixed (relative to the stars as it were), the frame can have a constant linear velocity and no rotation and still be inertial. Any other frame of reference is referred to as a noninertial frame. In many situations it is important to know whether the coordinate system you use is inertial or noninertial. In particular, we will see later that kinetic energy must be measured in an inertial system. 2.4 Forces A few general categories of forces are described here. We restrict our attention to those forces that are used in the examples that occur throughout this book. For example, we are not going to discuss forces associated with electromagnetic fields. 32 Chapter 2 Basic Concepts from Physics 2.4.1 Gravitational Forces Given two point masses m and M that have gravitational interaction, they attract each other with forces of equal magnitude but opposite direction, as indicated by Newton’s third law. The common magnitude of the forces is F gravity = GmM r 2 (2.46) where r is the distance between the points and G . = 6.67 × 10 −11 newton-meters squared per kilogram squared. The units of G are selected, of course, so that F gravity has units of newtons. The constant is empirically measured and is called the universal gravitational constant.

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  Foundations of Classical Mechanics

  • P. C. Deshmukh(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  They are introduced when motion is observed in a frame of reference that is either linearly accelerated with respect to an Inertial Frame of Reference (as in Eq. 3.6), or in a frame of reference that is rotating with respect to the inertial frame (as in Eq. 3.16b), or both (as in Eq. 3.20). • The principle of causality and determinism, which states that the rate of change of momentum of an object is equal to the force (stimulus) acting on it, refers to a physical fundamental force (gravity, electromagnetic-nuclear-weak, nuclear-strong). It is tacitly valid only in an Inertial Frame of Reference. In fact, an inertial frame is often defined as one as one in which Newton’s laws hold. The forces invoked in Newton’s fundamental laws are real forces resulting from physical interactions. The force acting on an object in the inertial frame is exactly equal to the rate of change of momentum of the object:    F dp dt ma = = . Pseudo-forces are invoked in a frame of reference that is accelerated Foundations of Classical Mechanics 90 with respect to an Inertial Frame of Reference, only in order to express its acceleration (a ′′ or a R ) as proportional to a force (F ′′ or F R ), which is, however, a combination of physical forces and pseudo-forces. Pseudo-forces thereby enable an interpretation of the perceived acceleration of an object as resulting from the application of a force, albeit an unreal one. They enable us write an equation of motion in an accelerated frame of reference that is isomorphous to Newton’s second law in an Inertial Frame of Reference. • The conservation of linear momentum expressed in Newton’s third law (Chapter 1) involves action–reaction forces which represent a pair of physical interactions in an Inertial Frame of Reference. Pseudo-forces are naturally irrelevant to the first and the third laws of Newton.

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  A First Course in Rational Continuum Mechanics

  General Concepts

  • C. Truesdell, Samuel Eilenberg, Hyman Bass(Authors)
  • 2016(Publication Date)
  • Academic Press

    (Publisher)

  In a more general system of mechanics, we should have to lay down (8) 2 , or some other axiom, independently of (8)^ The forces and torques given by (8) are called inertial Providing the framing § be an inertial one, these forces and torques are those exerted upon the bodies of the great system Z by the bodies, whatever they may be, that are outside Z. When we choose instead to use a general framing j * , we think of the unknown motions of the exterior Z e as being subjected to the same change from the framing § to the framing §* as are the motions of Z. Therefore, the second axiom of inertia, while it refers to a particular class of framings, is itself a frame-indifferent statement. While we follow tradition in stating it as we have, in terms of an inertial frame, we need not do so. Axiom A2 of §1.12 asserts that all forces are frame-indifferent. Thus the quantity on the left-hand side of (S) 1 is frame-indifferent. Accordingly, a frame-indifferent statement that reduces to (S) 1 in an inertial frame is df(^,Z e )= -a * d M , (1.13-9) a$ being that frame-indifferent vector field over $ which in the inertial frame § reduces to x- We have already calculated a$ and recorded it in (1.11-3). In the remainder of this book we shall follow the tradition of mechanics in assuming tacitly that the frame used is an inertial one, so (8) holds. Our use of an inertial frame rests on more than respect for tradition. An essential feature of classical mechanics is the existence of special frames in which the relation between forces and the motions they produce is especially simple. Since we have these felicitous frames, it would be simply foolish not to use them. When for purposes of interpretation in a particular case we need to employ some frame that is not inertial, as for example in problems referred to a rotating earth, we formulate the laws of mechanics first in an inertial frame and then transform them to the other frame of interest.

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  Forces in Physics

  A Historical Perspective

  • Steven N. Shore(Author)
  • 2008(Publication Date)
  • Greenwood

    (Publisher)

  It follows that the notion of force as defined to this point in our discussions must be completely re-examined. Since force is defined through acceleration, which depends on both space and time, without an absolute frame the whole concept of force becomes murky. This 160 Forces in Physics is the problem with which Albert Einstein and Hermann Minkowski started and where we will as well. THE RESTRICTED, OR SPECIAL, THEORY Galileo, in the Dialogs, used a principle of relative motion to refute the arguments of the contra-Copernicans regarding the orbital motion of the Earth, a notion that is now referred to as Galilean invariance. If we’re in a moving system, and for the moment we’ll assume it is inertial in the sense that it moves without accelerating, any measurement we make of forces in that system will be unaffected by its motion. Otherwise stated, there is no way we can find the motion of the system by force measurements on bodies in the system. The example he used, dropping a body on the Earth, was based on simple low velocity experiences that are perfectly valid as a first approximation. Thus the location of a body moving in an “enclosure” is seen by a stationary observer as a sum of the two motions. Calling x the position in the stationary frame and x  that in the frame moving with a constant relative speed v, the relation between the two descriptions of the motion in a time interval t is x  = x − v t . The time, t , is the same for the two observers, a universal rate of ticking of a cosmic clock that they both agree to use. This could also be two identical pendula; whether on a moving Earth or an observer at rest with respect to the fixed stars, the pendulum will swing identically and therefore the only way to tell which frame the measurement was in is to ask some other question than what is the law of the pendulum.

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  Basic Concepts in Relativistic Astrophysics

  • L Z Fang, R Ruffini;;;(Authors)
  • 1983(Publication Date)
  • WSPC

    (Publisher)

  As u is proportional to d, the smaller the dimen-sions of a laboratory the more difficult it would be to carry out these observations. Strictly speaking, to state that all phenomena of gravity can be eliminated would be correct only for a point-like system in free-fall. This is the reason why we must emphasize the word local. 1 -7 Local Inertial Frames For any local region in free fall, the principle of equivalence ensures that there exists a frame in which all effects of gravity are eliminated, and all particles without external forces (i.e. forces other than gravitational forces) acting on them are in rectilinear motion. They are called local inertial frames and conform to the definition of an inertial frame. In general, in Newtonian mechanics, one can assess whether a frame is inertial by using the law of inertia: a body must be in inertial mo-tion if no external force acts upon it. However, there can never ac-tually be a laboratory without any external forces, and inertial frames based on these abstractions have no practical value, because the 17 ubiquitous gravity cannot be screened. Only for local inertial frames within which the law of inertia does indeed hold, can we really find the environment without any external forces. Therefore, the local inertia! frames are closer to the original meaning of an inertial frame. Within local limits in spacetime, there is an infinite number of local inertial frames, each moving with constant velocity relative to another. Special relativity applies rigorously within these frames. Local inertial frames are more restricted and more general than Newton's inertial frames. First of all, the inhomogeneity of real gravitational fields makes them only applicable locally instead of in-finite in extent. Secondly, it is now obvious to us that the reason why inertial frames are in an advantageous position is because of the elimination of the effects of gravity.

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  Principles of Dynamics

  • Rodney Hill(Author)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R ΙΓ F O U N D A T I O N S O F M E C H A N I C S : G E N E R A L P R I N C I P L E S 2.1 Frames of Reference and Rates of Change In the preceding account of celestial mechanics the frame of reference was generally sidereal, and in particular Newtonian, since the basic equations for an arbitrary system are thereby reduced to their simplest form. Exceptionally, in the restricted problem of three bodies, a frame revolving with the primary masses was shown to have certain advantages. In terrestrial mechanics, on the other hand, it is clearly convenient to adopt a 'natural' frame of reference, rigidly attached to the earth or at least moving relatively to it in an assigned way. Yet again, in space travel the natural frame is pro-vided by the vehicle, while this itself is navigated by reference to the stars or the nearest planets. We need, therefore, to be able to relate the apparent motions of a body or a system seen from different frames. This analysis is moreover indispensable subsequently in clarifying the concept of force. Suppose that the path of a moving point is observed from some particular frame, denoted by α ('acting' or 'adopted'). From the apparent velocity and acceleration in α it is required to determine their apparent values in some other frame, perhaps more funda-mental, denoted by β ('background' or 'basic'). For example, α might be terrestrial and β sidereal. The relative motion of the two frames themselves will usually have both translational and rota-tional components. For the present it is enough to suppose that the translation is arbitrary but that the axis of rotation has a constant direction with respect to either frame.f Select orthogonal triads of right-handed axes (I, η , ζ) and (|', η , ζ') fixed in α and β respec-tively, and for convenience let ζ and ζ' be parallel to the axis of rotation. The spin of α with respect to β is then 'ψ where ψ is the angle between ξ and ξ or η' and η (Fig. 9).

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  Diffusion and Mass Transfer

  • James S. Vrentas, Christine M. Vrentas(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  567 17 Transport with Moving Reference Frames The analyses of a majority of transport problems are carried out using a laboratory frame attached to the earth. As noted in Sections 2.6 and 4.1, such a laboratory frame can gener-ally be regarded as an Inertial Frame of Reference since it is approximately stationary with respect to the fixed stars. Although this type of inertial frame does provide a satisfactory reference frame for most transport problems, there are problems which are analyzed more efficiently using reference frames which are translating and/or rotating with respect to the fixed Inertial Frame of Reference. In this chapter, various aspects of moving reference frames are discussed, and the utilization of a rotating frame is illustrated in the analysis of an ultracentrifuge. 17.1 Relationships between Fixed and Moving Reference Frames For a fixed frame of reference, it follows from the discussion in Section A.5 that the velocity vector v can be calculated using the equations v p p i i i = ∂ ∂       = = = = t D Dt Dp Dt Dx Dt v Xj i i i i i i (17.1) v Dx Dt i i = (17.2) where p is the position vector of a particle. Note that, in this chapter, all of the material time derivatives are strictly valid for one-component systems or for binary systems which are primarily solvent since the solute mass fraction is very low. For a moving frame of reference, a velocity v *(Rel) can be defined as the velocity which an observer attached to the moving frame of reference measures relative to a set of base vectors which are moving with respect to the fixed frame but which are considered to be stationary in the moving frame. The velocity v *(Rel) can be calculated using the equations v i i i * * * * * * * * * Re l R el ( ) = = = ( ) Dp Dt Dx Dt v α α α α α α (17.3) v Dx Dt α α * * * Rel ( ) = (17.4) 568 Diffusion and Mass Transfer The type of derivative used in Equation 17.3 is what Johns (2005, p. 194) calls a body deriva-tive.

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