Dynamics and Relativity

by McComb, W. D.

• ISBN13: 9780198501121
• ISBN10: 0198501129
• Format: Paperback
• Copyright: 2000-02-10
• Publisher: Oxford University Press

Summary

Special relativity suffers from the myth that it is difficult. In order to overcome this barrier Dynamics and relativity presents an integrated treatment of classical mechanics and special relativity, by treating classical mechanics as Galilean relativity. This gives students the freedom toformulate a particular problem in one frame of reference and solve it in another, where it takes a simpler form. This strategy, which is central to special relativity, is applied to problems in classical mechanics, thus preparing the tools and thought patterns for a treatment of specialrelativity.

Table of Contents

Galilean relativity	1	(30)
Brief reminder of Newton's laws	1	(3)
Newton's second law as a definition of the concepts of force and mass	2	(1)
Newton's second law as an equation of motion	3	(1)
The nature of force	4	(4)
Conservative forces	4	(1)
The fundamental forces	5	(2)
Forces encountered in applications	7	(1)
Force of gravity	8	(7)
Newtonian field equations for gravity	10	(2)
Self-gravitating fluids	12	(1)
Example: Radial variation of pressure and density in model stars	13	(2)
The nature of mass	15	(3)
Inertial mass	15	(1)
Gravitational mass	16	(1)
The principle of equivalence	17	(1)
The space-time continuum	18	(3)
Measurement of length and time	19	(1)
Ideal measuring rods and ideal clocks	20	(1)
Relativity of motion in Newtonian mechanics	21	(2)
Inertial frames of reference	21	(1)
Inertial frames in standard configuration	22	(1)
Galilean transformation	22	(1)
Galilean invariance of Newton's laws	23	(1)
The principle of Galilean relativity	24	(4)
Relativity of speed	25	(1)
Example: Motion of a projectile relative to S and S'	26	(1)
Example: Motion of a particle sliding down a slope in S' relative to S	27	(1)
The instantaneous rest frame of a moving body	28	(1)
Example: A particle undergoing constant linear acceleration	28	(1)
Exercises	29	(2)
Conservation laws	31	(18)
Conservation of linear momentum	32	(1)
Conservation of energy	32	(6)
Graphical treatment of potentials	34	(1)
Example:Find the potentials for specified force fields	35	(1)
Example: Time taken for a particle to move a given distance under the influence of a potential field	36	(1)
Example: Elastic collision of two particles	36	(1)
Example: Simple pendulum oscillating with finite amplitude	37	(1)
Angular momentum	38	(3)
Analogous form of N2 for angular motion	39	(1)
N1 for angular motion: conservation of angular momentum	39	(1)
Angular momentum of a particle relative to S and S'	40	(1)
Example: Angular momentum of a particle with impact parameter b incident from infinity on a centre of force in S and S'	40	(1)
Two-particle systems	41	(6)
Momentum	42	(1)
Centre of mass (CM) and relative coordinates	42	(1)
Kinetic and potential energy	43	(2)
The reduced mass and equivalent single-body problem	45	(1)
Example: Conservation of energy of two particles in terms of the reduced mass	46	(1)
Verification of the Galilean invariance of N3	47	(1)
Exercises	47	(2)
Central forces	49	(28)
Definition of a central force field	50	(1)
The two-dimensional problem	51	(3)
Constant angular momentum	51	(1)
Motion in a plane	52	(1)
Velocity of a particle in plane polar coordinates	52	(1)
Angular momentum in plane polar coordinates	53	(1)
Conservation of energy for motion under a central force	54	(1)
Conservation of angular momentum and the equivalent one-dimensional problem	55	(3)
The effective potential	55	(1)
Classification of orbits	56	(1)
Example: An attractive force vanishing at infinity	57	(1)
Equation for the orbit from conservation of energy	58	(6)
The inverse-square law: orbits as conic sections	60	(2)
Example: α-particle scattering	62	(2)
The inverse problem: obtain the force given the orbit	64	(1)
Examples: Obtain the forces given the orbits	65	(1)
Equation of the orbit using N2	65	(6)
Acceleration in polar coordinates	66	(1)
The equations of radial and circumferential motion	66	(1)
Example: Verify that an elliptic orbit corresponds to an inverse-square law of force	67	(1)
The orbit equation	68	(1)
Example: Given the orbit, obtain the force	69	(1)
Example: Given the law of force, find the orbit	69	(2)
Kepler's laws of planetary motion	71	(4)
Kepler's third law	72	(1)
The two-body problem and the Galilean-relativistic corrections to Kepler's third law	73	(2)
Exercises	75	(2)
Mechanical vibrations and waves	77	(26)
Stable and unstable equilibrium	77	(2)
Small oscillations	78	(1)
Systems with one degree of freedom	79	(7)
The simple harmonic oscillator	79	(1)
The simple pendulum	80	(1)
The spring--mass system	81	(1)
A mass on a stretched string	82	(1)
The effect of damping	82	(3)
Forced vibrations and resonance	85	(1)
Systems with several degrees of freedom	86	(5)
Two masses on a stretched string	87	(1)
Normal modes of vibration	88	(1)
Example: The double pendulum	89	(2)
Waves	91	(10)
SHM in a moving reference frame	91	(1)
The mathematical representation of a travelling wave	92	(2)
Wave speeds in elastic media as obtained by Galilean transformation	94	(2)
The general form of the wave equation and its solutions	96	(2)
Travelling waves and standing waves	98	(1)
Acoustics in moving reference frames: the classical Doppler effect	99	(2)
Exercises	101	(2)
Systems of N particles	103	(14)
The centre of mass of a system	103	(2)
Example: Centre of mass of three given masses at specified positions	104	(1)
Linear momentum of a system	105	(2)
Example: Velocity of the centre of mass of three particles	106	(1)
Energy of a system	107	(4)
Kinetic energy	108	(1)
Potential energy	108	(1)
Example: Energy of N particles in CM and relative coordinates	108	(2)
Example: Kinetic energy of two particles in terms of the reduced mass	110	(1)
The angular momentum of a system	111	(2)
Torque	113	(2)
Form of N2 for angular motion	113	(2)
Continuous distributions of mass	115	(1)
Example: Centre of mass of a solid hemisphere	115	(1)
Exercises	116	(1)
Solid-body motion	117	(18)
Motion about a fixed axis	117	(1)
Moments of inertia	118	(10)
The moment of inertia / of a body	119	(1)
The radius of gyration of a body	120	(1)
Example: Moment of inertia of a slender rod about an axis perpendicular to its length	121	(1)
Example: Moment of inertia of a solid cylinder about its axis of symmetry	122	(1)
Parallel axes theorem	122	(1)
Perpendicular axes theorem	123	(2)
Example: Radius of gyration of a cylinder about its axis and about one of its generators	125	(1)
The compound pendulum	125	(2)
Example: Circular cylinder oscillating about one of its generators	127	(1)
Motion about a fixed point	128	(2)
Inertia tensor	130	(3)
Example: Inertia tensor for four point masses in a plane	130	(1)
Example: Verification of the perpendicular axes theorem for a lamina using the inertia tensor	131	(1)
Example: Inertia tensor for a thin disc	132	(1)
Exercises	133	(2)
Non-inertial frames of reference	135	(12)
Coordinate system moving with constant linear acceleration	135	(2)
Example: A bucket of water undergoing constant acceleration	136	(1)
Coordinate system rotating with constant angular velocity	137	(8)
Rate of change of a vector relative to a rotating coordinate system	137	(1)
Interpretation of acceleration in a rotating frame	138	(1)
Newton's second law in a rotating frame: the centrifugal and Coriolis forces	139	(1)
Example: A bucket of water in a rotating frame	140	(1)
Example: A particle falling freely to earth	141	(2)
Example: Foucault's pendulum	143	(2)
Inertial and fictitious forces	145	(1)
Exercises	145	(2)
Variable-mass problems	147	(12)
Derivation of the general equation of motion from dp/dt = F	147	(1)
Accretion-of-mass problems	148	(4)
A raindrop falling through a cloud	149	(1)
Example: Falling raindrop with given rate of mass increase	150	(1)
Example: An open-topped railway wagon in a shower of rain	151	(1)
The rocket-propulsion equation	152	(6)
Example: A rocket with no external force acting	154	(1)
Example: Rocket taking off in a gravitational field	154	(1)
Multi-stage rockets	155	(2)
Example: Final speed of a two-stage rocket	157	(1)
Exercises	158	(1)
Collisions	159	(12)
Elastic collisions	159	(3)
Example: Elastic collision of identical particles: particles move off at right angles to each other	159	(2)
Example: Relationship between initial and final speeds of an elastically scattered particle	161	(1)
Conservation laws for a general two-body collision	162	(1)
LAB and centre-of-mass (CM) frames	163	(3)
Conservation of energy and momentum in the LAB frame	164	(1)
Conservation of energy and momentum in the CM frame	164	(2)
Galilean transformation between the Lab and CM frames	166	(4)
Relationship between &thetas;2 and &thetas;'	167	(1)
Relationship between &thetas;1 and &thetas;'	168	(1)
Relation between scattering angles in elastic collision of equal-mass particles as a special case	169	(1)
Exercises	170	(1)
Scattering of particles	171	(15)
Scattering from a fixed centre of force	172	(1)
Impact parameter and scattering angle	172	(2)
Differential and total cross-sections for scattering	174	(3)
Calculation of the differential cross-section in the CM frame	174	(2)
Transformation to the LAB frame	176	(1)
Hard-sphere scattering as a special case	177	(5)
Example: Scattering of a small sphere by a large, fixed sphere	177	(2)
Example: Scattering by identical spherical particles which are fixed in space	179	(2)
Example: Scattering by identical spherical particles which are free to recoil	181	(1)
Scattering in a central-force field	182	(1)
Repulsive inverse-square law of force	182	(1)
Rutherford's model of the atom	183	(1)
Attractive scattering forces and capture processes	183	(2)
Example: A planet passing through a cloud of meteors	184	(1)
Exercises	185	(1)
Special relativity	186	(18)
The need for a new principle of relativity	186	(1)
Einstein's axioms	187	(1)
Derivation of the Lorentz transformations	188	(4)
Implications of replacing Galilean with Lorentz invariance	192	(4)
Relativity of simultaneity	192	(1)
Lorentz--Fitzgerald contraction	193	(1)
Example: Rod at an angle to the x-axis moving in the x-direction	194	(1)
Example: Moving particle track at an angle to the x-axis in S relative to S'	195	(1)
Time dilation	196	(2)
Experimental evidence	197	(1)
Example: Decay of high-energy pions	197	(1)
The twins paradox	198	(2)
The experiment	199	(1)
The paradox or apparent contradiction	199	(1)
The resolution of the paradox	200	(1)
Example: Analogy of Lorentz transformation with rotations	200	(1)
General Lorentz transformation	201	(1)
Exercises	202	(2)
Relativistic kinematics	204	(18)
Lorentz transformation of intervals between events	204	(1)
Velocity transformations	205	(9)
Composition of velocities and verification of the Lorentz invariance of the speed of light	207	(1)
Example: A problem involving two spaceships and the Earth	207	(1)
Example: Another problem involving two spaceships and the Earth	208	(1)
Fresnel drag coefficient	209	(1)
Stellar aberration	210	(2)
Doppler effect	212	(1)
Thermal Doppler effect	213	(1)
Rapidity	214	(4)
Example: Lorentz transformations as hyperbolic functions	214	(1)
Example: Transitive property of the Lorentz transformations	215	(1)
Velocity composition in terms of rapidities	216	(1)
Example: Speed of multi-stage rocket relative to Earth	216	(1)
Example: Relationship between gamma factors for a rocket moving relative to S', where S' is moving relative to S	217	(1)
Acceleration transformation and proper acceleration	218	(1)
Apparent rotations and changes in shape	219	(1)
Exercises	220	(2)
Space--time geometry	222	(23)
Vector spaces: background and motivation	222	(3)
Four-space: the old-fashioned version	223	(1)
Four-space: the modern version	224	(1)
Lorentz-invariant interval between events in 4-space	225	(1)
Causality	225	(2)
The light cone	226	(1)
Four-vectors: spacelike, timelike and null	227	(2)
Scalar product of 4-vectors	227	(1)
Classification of 4-vectors	228	(1)
Proper time element dτ	229	(1)
Four-velocity of a body	230	(1)
Invariant scalar product	230	(1)
Four-acceleration of a body	231	(2)
Orthogonality of 4-velocity and 4-acceleration	231	(1)
Example: Express 4-acceleration in terms of 3-acceleration and 3-velocity	232	(1)
Transformation rules for 4-vectors	233	(4)
Example: Obtain 3-velocity transformation laws from the rules for 4-vectors	234	(1)
Example: Obtain linear acceleration in the comoving frame	235	(1)
Example: Central acceleration in the comoving frame	236	(1)
Usefulness of 4-vectors	237	(1)
Reconsideration of special relativity	237	(6)
The metric tensor	237	(1)
Example: Metric tensors for Euclidean 3-space and Minkowski 4-space	238	(2)
Geodesics (timelike and null)	240	(1)
Restatement of special relativity	241	(2)
Exercises	243	(2)
Relativistic dynamics	245	(25)
Particle collisions	245	(1)
Derivation of Lorentz-invariant 4-momentum conservation law	246	(1)
Relativistic momentum	246	(1)
Kinetic energy and the mass-energy relation	247	(1)
Momentum--energy relationship	248	(2)
Derivation from definitions of energy and momentum	248	(1)
Verification of Lorentz invariance	249	(1)
Example: Coalescence of two moving particles	249	(1)
Lorentz invariants	250	(1)
Massless particles	251	(4)
Example: Compton scattering using conservation of 4-momentum	252	(2)
Compton scattering: alternative calculation using Lorentz invariants	254	(1)
Examples of high-energy collisions	255	(5)
Example: Photon absorbed by stationary proton	256	(1)
Example: Two electrons scattering at right angles in the CM frame	257	(2)
Example: Proton makes head-on collision with photon	259	(1)
Example: Hard-sphere collisions revisited	260	(2)
Rockets revisited	262	(4)
The photon rocket	262	(1)
Example: Conventional rockets	263	(2)
Example: Comparison of conventional and photon rockets	265	(1)
Relativistic treatment of forces	266	(3)
The 4-vector form of N2	268	(1)
Exercises	269	(1)
Towards general relativity	270	(25)
Principle of equivalence	270	(4)
Einstein's lift experiments	271	(1)
Gravitation as an inertial force	272	(1)
The inertial frame revisited	273	(1)
Curved space	274	(4)
The necessity for a curved metric	274	(2)
Equivalence of a non-inertial frame and curved space	276	(1)
Signature of a vector space	277	(1)
Curved vector spaces	278	(4)
Geometry of space curves in 3D	278	(2)
Curvature in (1 + 3)D	280	(2)
The Einstein tensor	282	(1)
The field equations of general relativity	282	(4)
Einstein field equations for gravity	283	(2)
The full field equations of general relativity	285	(1)
Solving the field equations	286	(2)
The Schwarzschild metric	286	(1)
The matter tensor	287	(1)
The predictions of general relativity	288	(5)
Precession of the planetary orbits	288	(1)
The gravitational redshift	289	(1)
Black holes	290	(3)
Exercises	293	(2)
Appendices	295	(74)
A Integration using a dummy variable	297	(2)
B Solid angles	299	(2)
C Relativistic inertial mass	301	(1)
D The Penrose--Terrell rotation	302	(2)
E Solutions to exercises	304	(65)
Index	369