DOCSLIB.ORG

Considerations on Gravitational Effects Stimulated by Gravitational Fields Via Classical Field Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Considerations on Gravitational Effects Stimulated by Gravitational Fields Via Classical Field Theory Considerations on Gravitational Effects Stimulated by Gravitational Fields via Classical Field Theory Helmut Kling * (Dated: October 7, 2013) Abstract A mass distribution is analyzed in terms of classical gravitational field theory. Newton’s law of gravitation is consistently applied on the assumption that the equivalence of energy and mass according to Einstein’s theory of relativity is valid for gravitational fields as well. Differ- ent from standard approaches the gravitational field, via its associated field energy, is han- dled as a source of gravitation by itself. Starting from these principles a gravitational self- shielding phenomenon is derived as a common characteristic of all mass/energy distribu- tions. Moreover, it is demonstrated that even in the absence of any stimulating mass the ex- istence of independent static gravitational fields is fully consistent with Newton’s law of grav- itation as long as the equivalence of energy and mass is respected. From a distance such gravitational fields appear as negative masses. Keywords: Gravitation, Gravitational Field, Gravitational Self-shielding 1. Introduction When analyzing the gravitation of a mass distribution by means of classical field theory New- ton’s law of gravitation is applied to the stimulating mass distribution. According to Einstein’s theory of special relativity energy and inertial mass are equivalent. Einstein’s theory of gen- eral relativity extended this principle to the gravitational mass. Consequently, the term stimu- lating mass has not only to include classical masses but any energy distribution. In classical field theory, however, the associated gravitational field is usually not seen as a source of gravitation although it represents an energy distribution. In this study a mass distribution is analyzed in terms of classical gravitational field theory in which the gravitational field energy is handled as a mass, which is acting as a source of gravitation by itself. In that sense equivalence of energy and mass is applied more conse- quently as is usually done. Based on these principles a field equation for static gravitational 1 fields is developed and applied to a spherical mass distribution and also to the special case where no stimulating mass is present at all, i.e. to vacuum. Beyond the described equivalence of energy and gravitational mass further aspects of Ein- stein’s theory of general relativity, its formalism including Einstein’s field equations are not in the scope of this work. 2. Gravitational Field Strength and Energy Density Analysis shall be started from Newton’s law of gravitation, which gives the following force to a particle of mass 푚 at the position 풙, which is exposed to the gravitation of a second mass 푚2 at the position 풙ퟐ 풙 − 풙2 푭품 = − 퐺 푚 푚2 3 , (1) |풙 − 풙2| where G is the gravitational constant. We get 풙 − 풙′ 푭 (풙) = − 퐺 푚 ∫ 휌(풙′) 푑3푥′ , (2) 품 |풙 − 풙′|3 if the origin of the gravitational force is not a single point mass but a mass distribution repre- sented by 휌(푥). This mass distribution is associated with a gravitational field described by its field strength 풙 − 풙′ 푬 (풙) = − 퐺 ∫ 휌(풙′) 푑3푥′ (3) 품 |풙 − 풙′|3 3 so that 푭푔 = 푚 푬푔. Using 훻(1/|풙 − 풙′ |) = −(풙 − 풙′)/|풙 − 풙′| leads to 휌(풙′) 푬 (풙) = 퐺 훻 ∫ 푑3푥′ . (4) 푔 |풙 − 풙′| The gravitational field carries energy at a density of 1 푢 = − 푬2 . (5) 푔 8휋퐺 푔 This expression can easily be obtained in full analogy to the corresponding electrostatic case. Detailed evaluations of the electrical case are given in any textbook on electrody- namics such as [1] or [2]. However, even if well known [3] it has to be noted that different from electrostatics gravitational field energy is always negative, which is a consequence of 2 Newton’s law of gravitation when respecting the law of the conservation of energy due to the opposite sign compared to Coulomb’s law. 3. Evaluation of the Gravitational Field Strength Every mass/energy distribution is accompanied by a gravitational field carrying an energy density, which is, according to Einstein’s correlation E = mc2 (E energy, m mass, c vacuum speed of light), equivalent to a mass distribution. As Visser pointed out in his paper “A Clas- sical Model for the Electron” [4]1, it is logic and consistent to assume that the gravitational field acts again as a source of gravitation and the overall mass density to be taken into ac- 2 count is given by 휌푡표푡 = 휌 + 휌푓푖푒푙푑 = 휌 + 푢푔/푐 , where 휌 represents the density of the stimulating mass as introduced in section 1 and 휌푓푖푒푙푑 the equivalent mass density resulting from the gravitational field energy given by equation (5). It is obvious that this field-induced contribution, due to the property of the gravitational field energy to be negative, is equivalent to a negative mass distribution and acts opposite to the stimulating mass distribution: It weakens its gravitational effect. The gravitational field strength can therefore be expressed as 3 ′ 푢푔 푑 푥 푬 (풙) = 퐺 훻 ∫ [휌(풙′) + ] (6) 푔 푐2 |풙 − 풙′| resulting in the following field equation on 푬품 1 푑3푥′ 푬 (풙) = 퐺 훻 ∫ [휌(풙′) − 푬2(풙′)] . (7) 푔 8휋퐺푐2 푔 |풙 − 풙′| Let us now concentrate our analysis on a spherical mass distribution having a radius a and a mass M, whose center is located at the origin of our coordinate system. Spherical symmetry implies that 푬푔(풙) is not dependent on polar angles and always pointing into the direction of 풆푟 , which is the radial unit vector. When orientating our coordinate system in such a way that 풙 points along the z axis, i.e. 휗 =0, equation (7) can be rewritten as ∞ 2휋 휋 푀 1 2 2 푠푖푛 휗′ 푑 휗′ 퐸푔(푟) = − 퐺 − 훻푟 ∫ 퐸푔 (푟′)푟′ 푑푟′ ∫ 푑휑′ ∫ (8) 푟2 8휋푐2 √푟2 − 2 푟 푟′ 푐표푠 휗′ + 푟′2 푎 0 0 for r ≥ a, where r =|x|, 휑 and 휗 spherical coordinates and 푬푔(풙) = 퐸푔(푟) 풆푟. 1 In [4] M. Visser considers the gravitational field as well as the electrical field as a source of gravita- tion and applies this principle to his classical model for the electron, which assumes the electron to be a point particle. Other arrangements are not investigated in this paper. 3 Performing integration on 휑 and using 휋 ′ 2 ′ √ 2 ′ ′ ′ ′ ′ ′ (9) ∫0 (푠푖푛 휗 / 푟 − 2 푟 푟 푐표푠 휗 + 푟 ) 푑 휗 = (푟 + 푟 − |푟 − 푟 |)/푟 푟 we end up at 푟 ∞ 푀 1 1 퐸 (푟) = − 퐺 − 훻 [ ∫ 퐸2(푟′)푟′ 2푑푟′ + ∫ 퐸2(푟′)푟′푑푟′] 품 푟2 2푐2 푟 푟 푔 푔 푎 푟 푟 푀 1 1 = − 퐺 + ∫ 퐸2(푟′)푟′ 2푑푟′ . (10) 푟2 2푐2 푟2 푔 푎 Equation (10) reflects the well known fact that the gravitational field at a distance r from the center of a spherical mass distribution is equal to the field of a point mass located at the cen- ter whose magnitude is identical to the total mass enclosed by the sphere of radius r. In or- 2 der to simplify the evaluation of this integral equation we substitute 휅(푟) = − 푟 퐸푔(푟)/퐺푀 assuming M>0, which leads to 푟 2 ′ 2 휅(푟) = 1 − 푅푔 ∫ 휅 (푟 )/푟′ 푑푟′ , (11) 푎 2 where 푅푔 = 퐺푀/2푐 is one fourth of the Schwarzschild radius [5]. Differentiation yields 푑휅(푟) 휅2(푟) = −푅 . (12) 푑푟 푔 푟 2 Separation of the variables 휅 and r results in 푑휅(푟) 푑푟 = −푅 . (13) 휅2 푔 푟 2 Integration then leads to 1 휅(푟) = , (14) 푑 − 푅푔/r where d is an integration constant. It can easily be calculated by turning back to equation (11). Finally we get 1 휅(푟) = (15) 푅푔 푅푔 1 + 푎 − 푟 4 and 1 푀 퐸푔(푟) = − 퐺 2 . (16) 푅푔 푅푔 푟 1 + 푎 − 푟 Evidently 휅(푟) describes the influence of the gravitational field itself. Figure 1 shows 휿 ver- sus r/a for different Rg/a. Obviously at the surface of the sphere no self-contribution of the 1 R /a 0,9 g 0,8 0.01 0,7 0.1 0,6 0.2 0,5 0.5 0,4 1 0,3 2 5 0,2 10 0,1 0 1 10 100 Figure 1: 휿 versus r/a for different Rg/a gravitational field is at work, which is confirmed by equation (15), i.e. 휅(푟 = 푎) = 1. For Rg > a another interesting reference point is r = Rg. In order to achieve a given gravitational field strength at this point according to equation (16) a mass is needed which is a factor Rg/a big- ger than it would be expected when ignoring the contribution of the gravitational field itself. Let us now investigate the resulting behavior at long distances from the sphere, in particular for r ≫ Rg. Expansion of 퐸푔(푟) in powers of 푅푔⁄푟 yields 2 1 1 푅푔 1 푅푔 푀 퐸푔(푟) = − [1 + + ( ) + . ] 퐺 2 . (15) 푅푔 푅푔 푟 푅푔 푟 푟 1 + 푎 1 + 푎 1 + 푎 Restriction to the first term leads to 1 푀 퐸푔(푟) = − 퐺 2 . (16) 푅푔 푟 1 + 푎 5 When observing the gravitational interaction at a long distance the sphere will appear as an object having a mass Mobs according to classical theory of gravitation, i.e. when ignoring the contribution of the gravitational field itself, which is 푀 푀표푏푠 = = (1 − 휎) 푀 , (17) 푅푔 1 + 푎 2 where 푅푔 = 퐺푀/2푐 and 휎 = 1 – 1/(1 + 푅푔⁄푎). 휎 is a self-shielding coefficient measur- ing, which amount of the gravitational field strength is shielded by the gravitational field itself.
Load more

Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • Rotational Motion (The Dynamics of a Rigid Body)
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat­ ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de­ scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates.
    [Show full text]
  • Basic Galactic Dynamics
    5 Basic galactic dynamics 5.1 Basic laws govern galaxy structure The basic force on galaxy scale is gravitational • Why? The other long-range force is electro-magnetic force due to electric • charges: positive and negative charges are difficult to separate on galaxy scales. 5.1.1 Gravitational forces Gravitational force on a point mass M1 located at x1 from a point mass M2 located at x2 is GM M (x x ) F = 1 2 1 − 2 . 1 − x x 3 | 1 − 2| Note that force F1 and positions, x1 and x2 are vectors. Note also F2 = F1 consistent with Newton’s Third Law. Consider a stellar− system of N stars. The gravitational force on ith star is the sum of the force due to all other stars N GM M (x x ) F i j i j i = −3 . − xi xj Xj=6 i | − | The equation of motion of the ith star under mutual gravitational force is given by Newton’s second law, dv F = m a = m i i i i i dt and so N dvi GMj(xi xj) = − 3 . dt xi xj Xj=6 i | − | 5.1.2 Energy conservation Assuming M2 is fixed at the origin, and M1 on the x-axis at a distance x1 from the origin, we have m1dv1 GM1M2 = 2 dt − x1 1 2 Using 2 dv1 dx1 dv1 dv1 1 dv1 = = v1 = dt dt dx1 dx1 2 dx1 we have 2 1 dv1 GM1M2 m1 = 2 , 2 dx1 − x1 which can be integrated to give 1 2 GM1M2 m1v1 = E , 2 − x1 The first term on the lhs is the kinetic energy • The second term on the lhs is the potential energy • E is the total energy and is conserved • 5.1.3 Gravitational potential The potential energy of M1 in the potential of M2 is M Φ , − 1 × 2 where GM2 Φ2 = − x1 is the potential of M2 at a distance x1 from it.
    [Show full text]
  • THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
    GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P.
    [Show full text]
  • Chapter 7: Gravitational Field
    H2 Physics (9749) A Level Updated: 03/07/17 Chapter 7: Gravitational Field I Gravitational Force Learning Objectives 퐺푚 푚 recall and use Newton’s law of gravitation in the form 퐹 = 1 2 푟2 1. Newton’s Law of Gravitation Newton’s law of gravitation states that two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The magnitude of the gravitational force 퐹 between two particles of masses 푀 and 푚 which are separated by a distance 푟 is given by: 푭 = 퐠퐫퐚퐯퐢퐭퐚퐭퐢퐨퐧퐚퐥 퐟퐨퐫퐜퐞 (퐍) 푮푴풎 푀, 푚 = mass (kg) 푭 = ퟐ 풓 퐺 = gravitational constant = 6.67 × 10−11 N m2 kg−2 (Given) 푀 퐹 −퐹 푚 푟 Vector quantity. Its direction towards each other. SI unit: newton (N) Note: ➢ Action and reaction pair of forces: The gravitational forces between two point masses are equal and opposite → they are attractive in nature and always act along the line joining the two point masses. ➢ Attractive force: Gravitational force is attractive in nature and sometimes indicate with a negative sign. ➢ Point masses: Gravitational law is applicable only between point masses (i.e. the dimensions of the objects are very small compared with other distances involved). However, the law could also be applied for the attraction exerted on an external object by a spherical object with radial symmetry: • spherical object with constant density; • spherical shell of uniform density; • sphere composed of uniform concentric shells. The object will behave as if its whole mass was concentrated at its centre, and 풓 would represent the distance between the centres of mass of the two bodies.
    [Show full text]
  • Einstein's Gravitational Field
    Einstein’s gravitational field Abstract: There exists some confusion, as evidenced in the literature, regarding the nature the gravitational field in Einstein’s General Theory of Relativity. It is argued here that this confusion is a result of a change in interpretation of the gravitational field. Einstein identified the existence of gravity with the inertial motion of accelerating bodies (i.e. bodies in free-fall) whereas contemporary physicists identify the existence of gravity with space-time curvature (i.e. tidal forces). The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with. 1 Author: Peter M. Brown e-mail: [email protected] 2 INTRODUCTION Einstein’s General Theory of Relativity (EGR) has been credited as the greatest intellectual achievement of the 20th Century. This accomplishment is reflected in Time Magazine’s December 31, 1999 issue 1, which declares Einstein the Person of the Century. Indeed, Einstein is often taken as the model of genius for his work in relativity. It is widely assumed that, according to Einstein’s general theory of relativity, gravitation is a curvature in space-time. There is a well- accepted definition of space-time curvature. As stated by Thorne 2 space-time curvature and tidal gravity are the same thing expressed in different languages, the former in the language of relativity, the later in the language of Newtonian gravity. However one of the main tenants of general relativity is the Principle of Equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept.
    [Show full text]
  • Center of Mass, Torque and Rotational Inertia
    Center of Mass, Torque and Rotational Inertia Objectives 1. Define center of mass, center of gravity, torque, rotational inertia 2. Give examples of center of gravity 3. Explain how center of mass affects rotation and toppling 4. Write the equations for torque and rotational inertia 5. Explain how to change torque and rotational inertia Center of Mass/Gravity • COM – point which is at the center of the objects mass – Middle of meter stick – Center of a ball – Middle of a donut • Donuts are yummy • COG – center of object’s weight distribution – Usually the same point as center of mass Rotation and Center of Mass • Objects rotate around their center of mass – Wobbling baseball bat tossed through the air • They also balance there – Balancing a broom on your finger • Things topple over when their center of gravity is not above the base – Leaning Tower of Pisa Torque • Force that causes rotation • Equation τ = Fr τ = torque (in Nm) F = force (in N) r = arm radius (in m) • How do you increase torque – More force – Longer radius Rotational Inertia • Resistance of an object to a change in its rotational motion • Controlled by mass distribution and location of axis • Equation I = mr2 I = rotational inertia (in kgm2) m = mass (in kg) r = radius (in m) • How do you increase rotational inertia? – Even mass distribution Angular Momentum • Measure how difficult it is to start or stop a rotating object • Equations L = Iω = mvtr • It is easier to balance on a moving bicycle • Precession – Torque applied to a spinning wheel changes the direction of its angular momentum – The wheel rotates about the axis instead of toppling over Satellite Motion John Glenn 6 min • Satellites “fall around” the object they orbit • The tangential speed must be exact to match the curvature of the surface of the planet – On earth things fall 4.9 m in one second – Earth “falls” 4.9 m every 8000 m horizontally – Therefore a satellite must have a tangential speed of 8000 m/s to keep from hitting the ground – Rockets are launched vertically and rolled to horizontal during their flights Sputnik-30 sec .
    [Show full text]
  • Lecture 18, Structure of Spiral Galaxies
    GALAXIES 626 Upcoming Schedule: Today: Structure of Disk Galaxies Thursday:Structure of Ellipticals Tuesday 10th: dark matter (Emily©s paper due) Thursday 12th :stability of disks/spiral arms Tuesday 17th: Emily©s presentation (intergalactic metals) GALAXIES 626 Lecture 17: The structure of spiral galaxies NGC 2997 - a typical spiral galaxy NGC 4622 yet another spiral note how different the spiral structure can be from galaxy to galaxy Elementary properties of spiral galaxies Milky Way is a ªtypicalº spiral radius of disk = 15 Kpc thickness of disk = 300 pc Regions of a Spiral Galaxy · Disk · younger generation of stars · contains gas and dust · location of the open clusters · Where spiral arms are located · Bulge · mixture of both young and old stars · Halo · older generation of stars · contains little gas and dust · location of the globular clusters Spiral Galaxies The disk is the defining stellar component of spiral galaxies. It is the end product of the dissipation of most of the baryons, and contains almost all of the baryonic angular momentum Understanding its formation is one of the most important goals of galaxy formation theory. Out of the galaxy formation process come galactic disks with a high level of regularity in their structure and scaling laws Galaxy formation models need to understand the reasons for this regularity Spiral galaxy components 1. Stars · 200 billion stars · Age: from >10 billion years to just formed · Many stars are located in star clusters 2. Interstellar Medium · Gas between stars · Nebulae, molecular clouds, and diffuse hot and cool gas in between 3. Galactic Center ± supermassive Black Hole 4.
    [Show full text]
  • Gravitation (Symon Chapter Six)
    Gravitation (Symon Chapter Six) Physics A301∗ Summer 2006 Contents 0 Preliminaries 2 0.1 Course Outline . 2 0.2 Composite Properties in Curvilinear Co¨ordinates . 2 0.2.1 The Volume Element in Spherical Co¨ordinates. 3 0.3 Calculating the Center of Mass in Spherical Co¨ordinates . 5 1 The Gravitational Force 7 1.1 Force Between Two Point Masses . 7 1.2 Force Due to a Distribution of Masses . 9 2 Gravitational Field and Potential 10 2.1 Gravitational Field . 10 2.2 Gravitational Potential . 10 2.2.1 Warnings about Symon . 11 2.3 Example: Gravitational Field of a Spherical Shell . 12 3 Gauss's Law 15 A Appendix: Correspondence to Class Lectures 17 ∗Copyright 2006, John T. Whelan, and all that 1 Tuesday, May 9, 2006 0 Preliminaries 0.1 Course Outline 1. Gravity and Moving Co¨ordinateSystems Ch. 6 Gravitation Ch. 7 Moving Co¨ordinateSystems 2. Lagrangian and Hamlionian Mechanics (Ch. 9) 3. Tensor Analysis and Rigid Body Motion Ch. 10 Tensor Algebra Ch. 11 Rotation of a Rigid Body Subject to modification, as Carl Brans takes over after chapter 7. 0.2 Composite Properties in Curvilinear Co¨ordinates As a precursor to our analysis of the gravitational influence of extended bodies, let's return to the calculation of quantities such as total mass and center-of-mass position vector, for a composite or extended body. Recall from last semester the definitions of total mass M and the center-of-mass position vector R~ for either a collection of point masses or an extended solid body: Point Masses Mass Distribution N X 3 M = mk M= y ρ(~r) d V k=1 N 1 X
    [Show full text]
  • Chapter 8: Rotational Motion
    TODAY: Start Chapter 8 on Rotation Chapter 8: Rotational Motion Linear speed: distance traveled per unit of time. In rotational motion we have linear speed: depends where we (or an object) is located in the circle. If you ride near the outside of a merry-go-round, do you go faster or slower than if you ride near the middle? It depends on whether “faster” means -a faster linear speed (= speed), ie more distance covered per second, Or - a faster rotational speed (=angular speed, ω), i.e. more rotations or revolutions per second. • Linear speed of a rotating object is greater on the outside, further from the axis (center) Perimeter of a circle=2r •Rotational speed is the same for any point on the object – all parts make the same # of rotations in the same time interval. More on rotational vs tangential speed For motion in a circle, linear speed is often called tangential speed – The faster the ω, the faster the v in the same way v ~ ω. directly proportional to − ω doesn’t depend on where you are on the circle, but v does: v ~ r He’s got twice the linear speed than this guy. Same RPM (ω) for all these people, but different tangential speeds. Clicker Question A carnival has a Ferris wheel where the seats are located halfway between the center and outside rim. Compared with a Ferris wheel with seats on the outside rim, your angular speed while riding on this Ferris wheel would be A) more and your tangential speed less. B) the same and your tangential speed less.
    [Show full text]
  • Galactic Mass Distribution Without Dark Matter Or Modified Newtonian
    1 astro-ph/0309762 v2, revised 1 Mar 2007 Galactic mass distribution without dark matter or modified Newtonian mechanics Kenneth F Nicholson, retired engineer key words: galaxies, rotation, mass distribution, dark matter Abstract Given the dimensions (including thickness) of a galaxy and its rotation profile, a method is shown that finds the mass and density distribution in the defined envelope that will cause that rotation profile with near-exact speed matches. Newton's law is unchanged. Surface-light intensity and dark matter are not needed. Results are presented in dimensionless plots allowing easy comparisons of galaxies. As compared with the previous version of this paper the methods are the same, but some data are presented in better dimensionless parameters. Also the part on thickness representation is simplified and extended, the contents are rearranged to have a logical buildup in the problem development, and more examples are added. 1. Introduction Methods used for finding mass distribution of a galaxy from rotation speeds have been mostly those using dark-matter spherical shells to make up the mass needed beyond the assumed loading used (as done by van Albada et al, 1985), or those that modify Newton's law in a way to approximately match measured data (for example MOND, Milgrom, 1983). However the use of the dark-matter spheres is not a correct application of Newton's law, and (excluding relativistic effects) modifications to Newton's law have not been justified. In the applications of dark-matter spherical shells it is assumed that all galaxies are in two parts, a thin disk with an exponential SMD loading (a correct solution for rotation speeds by Freeman, 1970) and a series of spherical shells centered on the galaxy center.
    [Show full text]
  • Multiple Integrals
    Chapter 11 Multiple Integrals 11.1 Double Riemann Sums and Double Integrals over Rectangles Motivating Questions In this section, we strive to understand the ideas generated by the following important questions: What is a double Riemann sum? • How is the double integral of a continuous function f = f(x; y) defined? • What are two things the double integral of a function can tell us? • Introduction In single-variable calculus, recall that we approximated the area under the graph of a positive function f on an interval [a; b] by adding areas of rectangles whose heights are determined by the curve. The general process involved subdividing the interval [a; b] into smaller subintervals, constructing rectangles on each of these smaller intervals to approximate the region under the curve on that subinterval, then summing the areas of these rectangles to approximate the area under the curve. We will extend this process in this section to its three-dimensional analogs, double Riemann sums and double integrals over rectangles. Preview Activity 11.1. In this activity we introduce the concept of a double Riemann sum. (a) Review the concept of the Riemann sum from single-variable calculus. Then, explain how R b we define the definite integral a f(x) dx of a continuous function of a single variable x on an interval [a; b]. Include a sketch of a continuous function on an interval [a; b] with appropriate labeling in order to illustrate your definition. (b) In our upcoming study of integral calculus for multivariable functions, we will first extend 181 182 11.1.
    [Show full text]