Surprising facts about gravitational forces

Mallmann, A. James; Hock, J.L.; Ogden, Kent M.

doi:10.1119/1.2344089

Cited by 4 publications

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“…where the total mass m T = m 1 + m 2 . This equation is derived in [6], starting from Newton's second law. Thus, the time taken for the two masses to collide is given by…”

Section: Case 2b: Eom For Two Unequal Masses (M 1 = M 2 )mentioning

confidence: 99%

“…The calculation needed to give a good approximation of the time required is a direct application of the inverse square law to the two-body problem. A literature search using keywords such as 'inverse square' and 'fall' leads to a number of references motivated by different questions, such as the time a body takes to fall from a large elevation h to the Earth [1], time taken for the Earth to fall into the Sun [2][3][4][5], time required for two initially separated 1.0 g spheres to come into contact under free fall [6] and probably the most intriguing, the distance from heaven to hell [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

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From Moon-fall to motions under inverse square laws

Foong¹

2008

Eur. J. Phys.

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The motion of two bodies, along a straight line, under the inverse square law of gravity is considered in detail, progressing from simpler cases to more complex ones: (a) one body fixed and one free, (b) both bodies free and identical mass, (c) both bodies free and different masses and (d) the inclusion of electrostatic forces for both bodies free and different masses. The equations of motion (EOM) are derived starting from Newton's second law or from conservation of energy. They are then reduced to dimensionless EOM using appropriate scales for time and distance. Solutions of the dimensionless EOM as well as the original EOM are given. The time interval for the bodies to fall is expressed as a function of the distance fallen. Formulae for the inverse were obtained. The coalescence times for the different cases are (a) where L is the initial separation of the two bodies and m1 is the mass of the fixed body, (b) and (c) where mT is the total mass of the two bodies and (d) where and is a measure of the ratio of the electrostatic force to gravity. The last formula may also be used when with the interpretation that there is no collision if t is infinity or imaginary. We also discuss this motion along the straight line as a special case of the general elliptic motion of two bodies. I believe that this paper will be useful to university tutors as well as undergraduate and even graduate students who prefer to consider the special case before the general case, and their relationship.

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“…where the total mass m T = m 1 + m 2 . This equation is derived in [6], starting from Newton's second law. Thus, the time taken for the two masses to collide is given by…”

Section: Case 2b: Eom For Two Unequal Masses (M 1 = M 2 )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

From Moon-fall to motions under inverse square laws

Foong¹

2008

Eur. J. Phys.

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“…(2). Thereby, the issues of coordinate freedom and absence of gauge invariance are ignored [104,105,119].…”

Section: Analysis Of the Journals In Physics Educationmentioning

confidence: 99%

Gauge dependence and self-force from Galilean to Einsteinian free fall, compact stars falling into black holes, Hawking radiation and the Pisa tower at the general relativity centennial

Spallicci

Putten

2016

Int. J. Geom. Methods Mod. Phys.

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Obviously, in Galilean physics, the universality of free fall implies an inertial frame, which in turns implies that the mass m of the falling body is omitted (because it is a test mass; put otherwise, the centre of mass of the system coincides with the centre of the main, and fixed, mass M ; or else, we consider only an homogeneous gravitational field). Otherwise, an additional (in the same or opposite direction) acceleration proportional to m/M would rise either for an observer at the centre of mass of the system, or for an observer at a fixed distance from the centre of mass of M . These elementary, but overlooked, considerations fully respect the equivalence principle and the (local) identity of an inertial or a gravitational pull for an observer in the Einstein cabin. They value as fore-runners of the self-force and gauge dependency in general relativity. Because of its importance in teaching and in the history of physics, coupled to the introductory role to Einstein's equivalence principle, the approximate nature of Galilei's law of free fall is explored herein. When stepping into general relativity, we report how the geodesic free fall into a black hole was the subject of an intense debate again centred on coordinate choice. Later, we describe how the infalling mass and the emitted gravitational radiation affect the free fall motion of a body. The general relativistic self-force might be dealt with to perfectly fit into a geodesic conception of motion. Then, embracing quantum mechanics, real black holes are not classical static objects any longer. Free fall has to handle the Hawking radiation, and leads us to new perspectives on the varying mass of the evaporating black hole and on the varying energy of the falling mass. Along the paper, we also estimate our findings for ordinary masses being dropped from a Galilean or Einsteinian Pisa-like tower with respect to the current state of the art drawn from precise measurements in ground and space laboratories, and to the constraints posed by quantum measurements. The appendix describes how education physics and high impact factor journals discuss the free fall. Finally, case studies conducted on undergraduate students and teachers are reviewed.

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“…We can now succinctly state the problem: Find the distance r between the particles as a function of time t. An exact solution for the inverse relationship t(r) has been published in several places [2][3][4][5][6] and is reviewed in the appendix. The purpose of the present paper is to develop an approximate formula using ideas that would be more appropriate in an introductory course.…”

Section: Problem Statementmentioning

confidence: 99%

Radial Motion of Two Mutually Attracting Particles

Mungan

2009

The Physics Teacher

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A pair of masses or opposite-sign charges released from rest will move directly toward each other under the action of the inverse-distance-squared force of attraction between them. An exact expression for the separation distance as a function of time can only be found by numerically inverting the solution of a differential equation. A simpler, approximate formula can be obtained by combining dimensional analysis, Kepler's third law, and the familiar quadratic dependence of distance on time for a mass falling near Earth's surface. These exact and approximate results are applied to several interesting examples: the flight time and maximum altitude attained by an object fired straight upward from Earth's surface; the time required for an asteroid of known starting position and speed to cross Earth's orbit if it is bearing toward the Sun; and the collision time of two oppositely charged particles starting from rest.

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Surprising facts about gravitational forces

Cited by 4 publications

References 0 publications

From Moon-fall to motions under inverse square laws

From Moon-fall to motions under inverse square laws

Gauge dependence and self-force from Galilean to Einsteinian free fall, compact stars falling into black holes, Hawking radiation and the Pisa tower at the general relativity centennial

Radial Motion of Two Mutually Attracting Particles

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