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Classical Mechanics

Classical Mechanics An introductory course Richard Fitzpatrick Associate Professor of physics The University of Texas at Austin Contents 1 Introduction 7. Major sources: .. 7. What is Classical Mechanics ? .. 7. mks units .. 9. Standard prefixes .. 10. Other units .. 11. Precision and significant figures .. 12. Dimensional analysis .. 12. 2 Motion in 1 dimension 18. Introduction .. 18. Displacement .. 18. Velocity .. 19. Acceleration .. 21. Motion with constant velocity .. 23. Motion with constant acceleration .. 24. Free-fall under gravity .. 26. 3 Motion in 3 dimensions 32. Introduction .. 32. Cartesian coordinates .. 32. Vector displacement .. 33. Vector addition .. 34. Vector magnitude .. 35. 2. Scalar multiplication .. 35. Diagonals of a parallelogram.

Physics for scientists and engineers: R.A. Serway, and R.J. Beichner, Fifth edition, Vol. 1 (Saunders College Publishing, Orlando FL, 2000). 1.2 What is classical mechanics? Classical mechanics is the study of the motion of bodies (including the special

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Transcription of Classical Mechanics

1 Classical Mechanics An introductory course Richard Fitzpatrick Associate Professor of physics The University of Texas at Austin Contents 1 Introduction 7. Major sources: .. 7. What is Classical Mechanics ? .. 7. mks units .. 9. Standard prefixes .. 10. Other units .. 11. Precision and significant figures .. 12. Dimensional analysis .. 12. 2 Motion in 1 dimension 18. Introduction .. 18. Displacement .. 18. Velocity .. 19. Acceleration .. 21. Motion with constant velocity .. 23. Motion with constant acceleration .. 24. Free-fall under gravity .. 26. 3 Motion in 3 dimensions 32. Introduction .. 32. Cartesian coordinates .. 32. Vector displacement .. 33. Vector addition .. 34. Vector magnitude .. 35. 2. Scalar multiplication .. 35. Diagonals of a parallelogram.

2 36. Vector velocity and vector acceleration .. 37. Motion with constant velocity .. 39. Motion with constant acceleration .. 39. Projectile motion .. 41. Relative velocity .. 44. 4 Newton's laws of motion 53. Introduction .. 53. Newton's first law of motion .. 53. Newton's second law of motion .. 54. Hooke's law .. 55. Newton's third law of motion .. 56. Mass and weight .. 57. Strings, pulleys, and inclines .. 60. Friction .. 67. Frames of reference .. 70. 5 Conservation of energy 78. Introduction .. 78. Energy conservation during free-fall .. 78. Work .. 81. Conservative and non-conservative force-fields .. 88. Potential energy .. 92. 3. Hooke's law .. 93. Motion in a general 1-dimensional potential .. 96. Power .. 99. 6 Conservation of momentum 107.

3 Introduction .. 107. Two-component systems .. 107. Multi-component systems .. 112. Rocket science .. 115. Impulses .. 118. Collisions in 1-dimension .. 121. Collisions in 2-dimensions .. 127. 7 Circular motion 136. Introduction .. 136. Uniform circular motion .. 136. Centripetal acceleration .. 138. The conical pendulum .. 141. Non-uniform circular motion .. 142. The vertical pendulum .. 148. Motion on curved surfaces .. 150. 8 Rotational motion 160. Introduction .. 160. Rigid body rotation .. 160. Is rotation a vector? .. 162. 4. The vector product .. 166. Centre of mass .. 168. Moment of inertia .. 172. Torque .. 179. Power and work .. 184. Translational motion versus rotational motion .. 186. The physics of baseball .. 186. Combined translational and rotational motion.

4 190. 9 Angular momentum 204. Introduction .. 204. Angular momentum of a point particle .. 204. Angular momentum of an extended object .. 206. Angular momentum of a multi-component system .. 209. 10 Statics 217. Introduction .. 217. The principles of statics .. 217. Equilibrium of a laminar object in a gravitational field .. 220. Rods and cables .. 223. Ladders and walls .. 226. Jointed rods .. 228. 11 Oscillatory motion 237. Introduction .. 237. Simple harmonic motion .. 237. 5. The torsion pendulum .. 241. The simple pendulum .. 242. The compound pendulum .. 245. Uniform circular motion .. 246. 12 Orbital motion 253. Introduction .. 253. Historical background .. 253. Gravity .. 262. Gravitational potential energy .. 265. Satellite orbits .. 268.

5 Planetary orbits .. 269. 13 Wave motion 279. Introduction .. 279. Waves on a stretched string .. 279. General waves .. 284. Wave-pulses .. 285. Standing waves .. 289. The Doppler effect .. 291. 6. 1 INTRODUCTION. 1 Introduction Major sources: The sources which I consulted most frequently whilst developing this course are: Analytical Mechanics : Fowles, Third edition (Holt, Rinehart, & Winston, New York NY, 1977). physics : R. Resnick, D. Halliday, and Krane, Fourth edition, Vol. 1 (John Wiley & Sons, New York NY, 1992). Encyclop dia Brittanica: Fifteenth edition (Encyclop dia Brittanica, Chicago IL, 1994). physics for scientists and engineers: Serway, and Beichner, Fifth edition, Vol. 1 (Saunders College Publishing, Orlando FL, 2000). What is Classical Mechanics ?

6 Classical Mechanics is the study of the motion of bodies (including the special case in which bodies remain at rest) in accordance with the general principles first enunciated by Sir Isaac Newton in his Philosophiae Naturalis Principia Math- ematica (1687), commonly known as the Principia. Classical Mechanics was the first branch of physics to be discovered, and is the foundation upon which all other branches of physics are built. Moreover, Classical Mechanics has many im- portant applications in other areas of science, such as Astronomy ( , celestial Mechanics ), Chemistry ( , the dynamics of molecular collisions), Geology ( , the propagation of seismic waves, generated by earthquakes, through the Earth's crust), and Engineering ( , the equilibrium and stability of structures).

7 Classi- cal Mechanics is also of great significance outside the realm of science. After all, the sequence of events leading to the discovery of Classical Mechanics starting with the ground-breaking work of Copernicus, continuing with the researches of Galileo, Kepler, and Descartes, and culminating in the monumental achievements 7. 1 INTRODUCTION What is Classical Mechanics ? of Newton involved the complete overthrow of the Aristotelian picture of the Universe, which had previously prevailed for more than a millennium, and its replacement by a recognizably modern picture in which humankind no longer played a privileged role. In our investigation of Classical Mechanics we shall study many different types of motion, including: Translational motion motion by which a body shifts from one point in space to another ( , the motion of a bullet fired from a gun).

8 Rotational motion motion by which an extended body changes orientation, with respect to other bodies in space, without changing position ( , the motion of a spinning top). Oscillatory motion motion which continually repeats in time with a fixed period ( , the motion of a pendulum in a grandfather clock). Circular motion motion by which a body executes a circular orbit about another fixed body [ , the (approximate) motion of the Earth about the Sun]. Of course, these different types of motion can be combined: for instance, the motion of a properly bowled bowling ball consists of a combination of trans- lational and rotational motion, whereas wave propagation is a combination of translational and oscillatory motion. Furthermore, the above mentioned types of motion are not entirely distinct: , circular motion contains elements of both rotational and oscillatory motion.

9 We shall also study statics: , the subdivision of Mechanics which is concerned with the forces that act on bodies at rest and in equilibrium. Statics is obviously of great importance in civil engineering: for instance, the principles of statics were used to design the building in which this lecture is taking place, so as to ensure that it does not collapse. 8. 1 INTRODUCTION mks units mks units The first principle of any exact science is measurement. In Mechanics there are three fundamental quantities which are subject to measurement: 1. Intervals in space: , lengths. 2. Quantities of inertia, or mass, possessed by various bodies. 3. Intervals in time. Any other type of measurement in Mechanics can be reduced to some combina- tion of measurements of these three quantities.

10 Each of the three fundamental quantities length, mass, and time is mea- sured with respect to some convenient standard. The system of units currently used by all scientists, and most engineers, is called the mks system after the first initials of the names of the units of length, mass, and time, respectively, in this system: , the meter, the kilogram, and the second. The mks unit of length is the meter (symbol m), which was formerly the dis- tance between two scratches on a platinum-iridium alloy bar kept at the Inter- national Bureau of Metric Standard in S`evres, France, but is now defined as the distance occupied by 1, 650, wavelengths of light of the orange-red spectral line of the isotope Krypton 86 in vacuum. The mks unit of mass is the kilogram (symbol kg), which is defined as the mass of a platinum-iridium alloy cylinder kept at the International Bureau of Metric Standard in S`evres, France.


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