Physics
Inertial and Gravitational Mass
Inertial mass refers to an object's resistance to changes in its state of motion, while gravitational mass relates to the strength of the gravitational force experienced by an object. These two types of mass are found to be equivalent in the theory of general relativity, which forms the basis for our understanding of gravity.
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eBook - PDFModern General Relativity
Black Holes, Gravitational Waves, and Cosmology
- Mike Guidry(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
106 107 6.2 Inertial and Gravitational Mass 6.2 Inertial and Gravitational Mass The principle of equivalence originates in the observation that in Newtonian physics there are two ways in which the mass enters dynamical equations: 1. The inertial mass m i is defined through Newton’s second law of motion: m i = F /a, where F is the magnitude of the force and a is the magnitude of the acceleration. 2. The gravitational mass m g is defined through Newton’s law of gravitation: m g = r 2 F /GM, where r is the separation of the mass from a gravitating sphere like the Earth, F is the magnitude of the gravitational force, M is the mass of the gravitating sphere, and G is the gravitational constant. These definitions are highly asymmetric in scope. The inertial mass is defined in terms of a response to any force, while the gravitational mass is defined in terms of response only to a particular force, gravity. The quantitative relationship between the Inertial and Gravitational Masses for a given object was suggested by Galileo’s experiments with inclined planes showing that different objects fall at the same rate in a gravitational field, but was first established to high precision in the Eötvös experiment of 1893, which is illustrated in Fig. 6.1. Two equal weights composed of different material A and B are suspended from a sensitive torsion balance. Except at the Earth’s equator and the poles, if the Inertial and Gravitational Masses differ a couple will be produced by the action on the inertial mass of the centrifugal force associated with Earth’s rotation. The results of such tests of the equivalence principle are often reported in terms of the Eötvös parameter η, which is defined by the difference in acceleration a relative to the average gravitational acceleration a g for spheres made from two substances A and B, η(A, B) = a a g = ( m g /m i ) A − ( m g /m i ) B 1 2 ( m g /m i ) A + ( m g /m i ) B .
No longer available |Learn moreWeight and Mass
Basic Physical Quantities (Concepts and Applications)
- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
As of 2009, the Earth’s mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine digits. Inertial and Gravitational Mass Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any diffe-rence between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact. Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational mass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). However, in the resulting theory, gravitation is not a force and thus not subject to Newton's third law, so the equality of inertial and active gravitational mass [...] remains as puzzling as ever. Inertial mass Inertial mass is the mass of an object measured by its resistance to acceleration. To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of mass in any essential way. ________________________ WORLD TECHNOLOGIES ________________________ According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion where F is the force acting on the body and a is the acceleration of the body.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
• Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater inertia. • Active gravitational mass is a measure of the strength of an object’s gravita-tional flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small ‘test object’ to freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower than the same object would if it were in free-fall near the earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass. • Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object’s weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weig ht) than the object with a larger passive gravitational mass. • Energy also has mass according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair production and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
• Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater inertia. • Active gravitational mass is a measure of the strength of an object’s gravi-tational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small ‘test object’ to freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower than the same object would if it were in free-fall near the earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass. • Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object’s weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weig ht) than the object with a larger passive gravitational mass. • Energy also has mass according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair prod-uction and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.
eBook - PDFThe Curious History of Relativity
How Einstein's Theory of Gravity Was Lost and Found Again
- Jean Eisenstaedt(Author)
- 2018(Publication Date)
- Princeton University Press(Publisher)
Simply put, the gravitational mass is the gravitational charge. The inertial mass plays a role in the fundamental equation of dynamics ( f = m i a, where a is the acceleration), while the gravi- tational mass is involved in the definition of the gravitational force exerted by a body of (gravitational) mass M g upon a body C H A P T E R 4 78 of (gravitational) mass m g ( f = −Gm g M g /r 2 ). And it is because the gravitational mass of a body is—exactly (but why?)—the same as its inertial mass (m i = m g ) that the trajectory of a particle in a gravitational field is independent of its own mass; the accelera- tion g depends only on the (gravitational) mass M g that creates the gravitational field and not on the mass of the particle that is subject to it: g = −GM g /r 2 . This remark seems almost innocuous. For Newton, it was no doubt a fundamental observation but ab- solutely not a principle, and we could quite easily assume, with- out changing anything in the structure of his theory, that the gravitational and the inertial masses are not the same. If such were the case then, in vacuum, a grain of lead would not react in the same way as a grain of wheat (or a small piece of cotton) to a gravitational field. Their orbits would be different for the same initial conditions, as if the grains would be subject to slightly dif- ferent gravitational fields. That is precisely what happens in electromagnetism, where the ratio e/m is everywhere present; but such is not the case for gravitation. We must turn to experience if we wish to be convinced. Do two different substances, whose respective physicochemical composi- tions are as different as possible, react in the same way to a given gravitational field? Well before Galileo, the answer was certainly yes. But Newton wanted to check this by himself. Using wood and gold, he showed that the period of a pendulum was inde- pendent of its composition.
eBook - PDFHypersymmetry
Physics of the Isotopic Field-Charge Spin Conservation
- György Darvas(Author)
- 2020(Publication Date)
- De Gruyter(Publisher)
2 MASS Mass is a physical quantity familiar to the reader. It plays an odd role in physics. Therefore, we start the discussion with mass. The difference between gravitational and inertial masses was known for over three hundred years. Why did not one distinguish them in the equations of physics? There were different reasons for it. First , according to a simplified formulation of the second law of Newton, the mass can be defined as the ratio between a force that affects it and its acceleration caused by that force. This definition does not take into consideration whether the force in question originates from the gravity of another object or it is an inertial force. The so defined mass could be measured by the same unit in both cases. Second , the proportion between the measures of the two kinds of masses was fixed to “ 1 ” . This made the appearance like the two masses were identical, and at the available test procedures, the results seemed satisfactory for centuries. Third , the so-called equivalence principle was formulated. The quantitative equivalence between the gravitational and inertial masses of the same body (at least at rest) was measured accurately first by R. Eötvös (and his colleagues, 1910, 1922) in 1906 – 1909. The experimental proof of their equal value made available for A. Einstein to formulate the equivalence principle in the early 1910s. When Einstein (1915, 1916) formulated the general theory of gravity, known as the general theory of relativity (GTR), he really considered that the equivalence principle meant iden-tity between the equivalent value masses. Although he changed his mind in a few years and formulated more precisely, the so-called “ weak formulation of the equiv-alence principle ” inspired many textbook writers to identify them, even up to now. This was pragmatically correct, but theoretically not justified.
eBook - PDFOn General Relativity
An Analysis of the Fundamentals of the Theory of General Relativity and Gravitation
- A. Mercier, H.-J. Treder, W. Yourgrau(Authors)
- 1979(Publication Date)
- De Gruyter(Publisher)
102 5. Einstein's Principles of Equivalence Thus, two senses of mass refer to the behaviour of a body B in a given gravitational field of components jF„= —dv
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The result is a subtle shift in the interpreta- tion of m in the equation. The mass m in Equation 5.6 determines the strength of the gravitational attraction between the object and the Earth. This role is com- pletely different from that previously described for mass, that of measuring the resistance to changes in motion in response to an external force. In that role, mass is also called inertial mass. We call m in Equation 5.6 the gravitational mass. Even though this quantity is different in behavior from inertial mass, it is one of the PITFALL PREVENTION 5.4 “Weight of an Object” We are familiar with the everyday phrase, the “weight of an object.” Weight, however, is not an inherent prop- erty of an object; rather, it is a measure of the gravitational force between the object and the Earth (or other planet). Therefore, weight is a property of a system of items: the object and the Earth. PITFALL PREVENTION 5.5 Kilogram Is Not a Unit of Weight You may have seen the “con- version” 1 kg 5 2.2 lb. Despite popular statements of weights expressed in kilograms, the kilo- gram is not a unit of weight, it is a unit of mass. The conversion statement is not an equality; it is an equivalence that is valid only on the Earth’s surface. We first raised this issue in the Chapter 1 storyline. 3 This statement ignores that the mass distribution of the Earth is not perfectly spherical. Figure 5.5 Astronaut Harrison Schmitt carries a backpack on the Moon. NASA/Eugene Cernan Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDF- 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.
eBook - PDF- Joaquim A. Batlle, Ana Barjau Condomines(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
(1.11) are the gravitational masses of the particles. They are conceptually different from the inertial masses, but there is no empirical evidence so far that they should differ, so they will be treated as one same thing. The constant parameter G 0 in Eq. (1.11) is the universal constant of gravitation P Q U m P m Q gravitational attraction = G 0 m P m Q U 2 Fig. 1.11 P Q Intermediate Element (IE) F PoQ F QoP negligible mass through the IE through the IE F QoP = F PoQ Fig 1.10 17 1.9 Formulation of Interaction Forces G 0 ¼ 6:67 10 11 m 3 = kg s 2 ð Þ in the International System of Units, SI), and is inde- pendent from the particles (that is why it is a universal constant). 9 The vector F grav P!Q =m Q is called the gravitational field g P!Q created by P at Q location. Appendix 1.A presents the formulation of the Earth’s gravitational field. The universal law of gravitation has been widely discussed because of some very particular features. On the one hand, and as a result of the identification of inertial mass and gravitational mass, it predicts a same acceleration (relative to a Galilean reference frame) for any particle under the same gravitational field. On the other hand, Eq. (1.11) describes an instantaneous interaction. The direction and intensity of the gravitational attraction is adjusted instantaneously whatever the distance between the interacting particles might be: the gravitational field propagates with an infinite speed! 10 Interaction Force through Springs The term spring is used to designate any element with negligible mass responsible for an interaction force depending exclusively on the distance between particles (called “spring length” from now on): F spring P$Q ¼ f ρ ðÞ. Springs may introduce either an attrac- tion or a repulsion between their endpoints.
eBook - PDF- L Z Fang, R Ruffini;;;(Authors)
- 1983(Publication Date)
- WSPC(Publisher)
Although the electrostatic force is long-range, it can be screened. Electric charges can be positive or negative, but a celes-tial body as a whole is neutral, so that all long-range electrostatic forces are ineffective. Gravity is very different from the electrosta-tic force in that it cannot be screened, since all material bodies attract each other. It is inevitable that problems on a large scale in spacetime should encounter gravity and a correct theory of gravitation is nec-essary for relating the events on the astronomical scale. Newton's theory of gravitation is quite good in the case of weak gravitational fields and at low speeds, but the situations discussed in relativistic astrophysics often involve strong gravitational fields, or high speeds, or both at the same time. These problems are peculiar to and charac-teristic of astrophysics. 1 -2 Newtonian Mechanics and Absolute Space Newton's second law, which is the nucleus of the whole of Newtonian mechanics, is represented most generally as F = m a , (1.1) where m is the mass of the body, a is its acceleration relative to a certain frame of reference, and F is the resultant force acting on the body due to all other bodies. The types of frames in which Newtonian mechanics can be employed are called inertial frames, moving relative to each other with uniform velocities. It follows that the inertial frames occupy a special posi-tion in Newtonian mechanics. We wish to ask more fundamental questions like why Nature selects such frames preferentially and how does one 3 determine whether a given frame is inertial. It is generally agreed that inertial frames are unaccelerated and non-rotating. If they are not, we are then led to ask what the acceleration and rotation are relative to. Newton himself had given his answers to these questions.
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