Classical Mechanics,Richard Fitzpatrick.pdf

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<ul><li><p>Classical MechanicsAn introductory course</p><p>Richard Fitzpatrick</p><p>Associate Professor of Physics</p><p>The University of Texas at Austin</p></li><li><p>Contents</p><p>1 Introduction 7</p><p>1.1 Major sources: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7</p><p>1.2 What is classical mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7</p><p>1.3 mks units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9</p><p>1.4 Standard prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10</p><p>1.5 Other units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11</p><p>1.6 Precision and significant figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12</p><p>1.7 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12</p><p>2 Motion in 1 dimension 18</p><p>2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18</p><p>2.2 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18</p><p>2.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19</p><p>2.4 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21</p><p>2.5 Motion with constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23</p><p>2.6 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 24</p><p>2.7 Free-fall under gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26</p><p>3 Motion in 3 dimensions 32</p><p>3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32</p><p>3.2 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32</p><p>3.3 Vector displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33</p><p>3.4 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34</p><p>3.5 Vector magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35</p><p>2</p></li><li><p>3.6 Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35</p><p>3.7 Diagonals of a parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36</p><p>3.8 Vector velocity and vector acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 37</p><p>3.9 Motion with constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39</p><p>3.10 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 39</p><p>3.11 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41</p><p>3.12 Relative velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44</p><p>4 Newtons laws of motion 53</p><p>4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53</p><p>4.2 Newtons first law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53</p><p>4.3 Newtons second law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54</p><p>4.4 Hookes law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55</p><p>4.5 Newtons third law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56</p><p>4.6 Mass and weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57</p><p>4.7 Strings, pulleys, and inclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60</p><p>4.8 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67</p><p>4.9 Frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70</p><p>5 Conservation of energy 78</p><p>5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78</p><p>5.2 Energy conservation during free-fall . . . . . . . . . . . . . . . . . . . . . . . . . . 78</p><p>5.3 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81</p><p>5.4 Conservative and non-conservative force-fields . . . . . . . . . . . . . . . . . . . . . 88</p><p>5.5 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92</p><p>3</p></li><li><p>5.6 Hookes law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93</p><p>5.7 Motion in a general 1-dimensional potential . . . . . . . . . . . . . . . . . . . . . . 96</p><p>5.8 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99</p><p>6 Conservation of momentum 107</p><p>6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107</p><p>6.2 Two-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107</p><p>6.3 Multi-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112</p><p>6.4 Rocket science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115</p><p>6.5 Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118</p><p>6.6 Collisions in 1-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121</p><p>6.7 Collisions in 2-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127</p><p>7 Circular motion 136</p><p>7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136</p><p>7.2 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136</p><p>7.3 Centripetal acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138</p><p>7.4 The conical pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141</p><p>7.5 Non-uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142</p><p>7.6 The vertical pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148</p><p>7.7 Motion on curved surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150</p><p>8 Rotational motion 160</p><p>8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160</p><p>8.2 Rigid body rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160</p><p>8.3 Is rotation a vector? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162</p><p>4</p></li><li><p>8.4 The vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166</p><p>8.5 Centre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168</p><p>8.6 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172</p><p>8.7 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179</p><p>8.8 Power and work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184</p><p>8.9 Translational motion versus rotational motion . . . . . . . . . . . . . . . . . . . . . 186</p><p>8.10 The physics of baseball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186</p><p>8.11 Combined translational and rotational motion . . . . . . . . . . . . . . . . . . . . . 190</p><p>9 Angular momentum 204</p><p>9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204</p><p>9.2 Angular momentum of a point particle . . . . . . . . . . . . . . . . . . . . . . . . . 204</p><p>9.3 Angular momentum of an extended object . . . . . . . . . . . . . . . . . . . . . . . 206</p><p>9.4 Angular momentum of a multi-component system . . . . . . . . . . . . . . . . . . . 209</p><p>10 Statics 217</p><p>10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217</p><p>10.2 The principles of statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217</p><p>10.3 Equilibrium of a laminar object in a gravitational field . . . . . . . . . . . . . . . . 220</p><p>10.4 Rods and cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223</p><p>10.5 Ladders and walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226</p><p>10.6 Jointed rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228</p><p>11 Oscillatory motion 237</p><p>11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237</p><p>11.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237</p><p>5</p></li><li><p>11.3 The torsion pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241</p><p>11.4 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242</p><p>11.5 The compound pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245</p><p>11.6 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246</p><p>12 Orbital motion 253</p><p>12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253</p><p>12.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253</p><p>12.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262</p><p>12.4 Gravitational potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265</p><p>12.5 Satellite orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268</p><p>12.6 Planetary orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269</p><p>13 Wave motion 279</p><p>13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279</p><p>13.2 Waves on a stretched string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279</p><p>13.3 General waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284</p><p>13.4 Wave-pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285</p><p>13.5 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289</p><p>13.6 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291</p><p>6</p></li><li><p>1 INTRODUCTION</p><p>1 Introduction</p><p>1.1 Major sources:</p><p>The sources which I consulted most frequently whilst developing this course are:</p><p>Analytical Mechanics: G.R. Fowles, Third edition (Holt, Rinehart, &amp; Winston, NewYork NY, 1977).</p><p>Physics: R. Resnick, D. Halliday, and K.S. Krane, Fourth edition, Vol. 1 (John Wiley&amp; Sons, New York NY, 1992).</p><p>Encyclopdia Brittanica: Fifteenth edition (Encyclopdia Brittanica, Chicago IL,1994).</p><p>Physics for scientists and engineers: R.A. Serway, and R.J. Beichner, Fifth edition,Vol. 1 (Saunders College Publishing, Orlando FL, 2000).</p><p>1.2 What is classical mechanics?</p><p>Classical mechanics is the study of the motion of bodies (including the special</p><p>case in which bodies remain at rest) in accordance with the general principles</p><p>first enunciated by Sir Isaac Newton in his Philosophiae Naturalis Principia Math-</p><p>ematica (1687), commonly known as the Principia. Classical mechanics was the</p><p>first branch of Physics to be discovered, and is the foundation upon which all</p><p>other branches of Physics are built. Moreover, classical mechanics has many im-</p><p>portant applications in other areas of science, such as Astronomy (e.g., celestial</p><p>mechanics), Chemistry (e.g., the dynamics of molecular collisions), Geology (e.g.,</p><p>the propagation of seismic waves, generated by earthquakes, through the Earths</p><p>crust), and Engineering (e.g., the equilibrium and stability of structures). Classi-</p><p>cal mechanics is also of great significance outside the realm of science. After all,</p><p>the sequence of events leading to the discovery of classical mechanicsstarting</p><p>with the ground-breaking work of Copernicus, continuing with the researches of</p><p>Galileo, Kepler, and Descartes, and culminating in the monumental achievements</p><p>7</p></li><li><p>1 INTRODUCTION 1.2 What is classical mechanics?</p><p>of Newtoninvolved the complete overthrow of the Aristotelian picture of the</p><p>Universe, which had previously prevailed for more than a millennium, and its</p><p>replacement by a recognizably modern picture in which humankind no longer</p><p>played a privileged role.</p><p>In our investigation of classical mechanics we shall study many different types</p><p>of motion, including:</p><p>Translational motionmotion by which a body shifts from one point in space toanother (e.g., the motion of a bullet fired from a gun).</p><p>Rotational motionmotion by which an extended body changes orientation, withrespect to other bodies in space, without changing position (e.g., the motion</p><p>of a spinning top).</p><p>Oscillatory motionmotion which continually repeats in time with a fixed period(e.g., the motion of a pendulum in a grandfather clock).</p><p>Circular motionmotion by which a body executes a circular orbit about anotherfixed body [e.g., the (approximate) motion of the Earth about the Sun].</p><p>Of course, these different types of motion can be combined: for instance, the</p><p>motion of a properly bowled bowling ball consists of a combination of trans-</p><p>lational and rotational motion, whereas wave propagation is a combination of</p><p>translational and oscillatory motion. Furthermore, the above mentioned types of</p><p>motion are not entirely distinct: e.g., circular motion contains elements of both</p><p>rotational and oscillatory motion. We shall also study statics: i.e., the subdivision</p><p>of mechanics which is concerned with the forces that act on bodies at rest and</p><p>in equilibrium. Statics is obviously of great importance in civil engineering: for</p><p>instance, the principles of statics were used to design the building in which this</p><p>lecture is taking place, so as to ensure that it does not collapse.</p><p>8</p></li><li><p>1 INTRODUCTION 1.3 mks units</p><p>1.3 mks units</p><p>The first principle of any exact science is measurement. In mechanics there are</p><p>three fundamental quantities which are subject to measurement:</p><p>1. Intervals in space: i.e., lengths.</p><p>2. Quantities of inertia, or mass, possessed by various bodies.</p><p>3. Intervals in time.</p><p>Any other type of measurement in mechanics can be reduced to some combina-</p><p>tion of measurements of these three quantities.</p><p>Each of the three fundamental quantitieslength, mass, and timeis mea-</p><p>sured with respect to some convenient standard. The system of units currently</p><p>used by all scientists, and most engineers, is called the mks systemafter the first</p><p>initials of the names of the units of length, mass, and time, respectively, in this</p><p>system: i.e., the meter, the kilogram, and the second.</p><p>The mks unit of length is the meter (symbol m), which was formerly the dis-</p><p>tance between two scratches on a platinum-iridium alloy bar kept at the Inter-</p><p>national Bureau of Metric Standard in Se`vres, France, but is now defined as the</p><p>distance occupied by 1, 650, 763.73 wavelengths of light of the orange-red spectral</p><p>line of the isotope Krypton 86 in vacuum.</p><p>The mks unit of mass is the kilogram (symbol kg), which is defined as the mass</p><p>of a platinum-iridium alloy cylinder kept at the International Bureau of Metric</p><p>Standard in Se`vres, France.</p><p>The mks unit of time is the second (symbol s), which was formerly defined in</p><p>terms of the Earths rotation, but is now defined as the time for 9, 192, 631, 770</p><p>oscillations associated with the transition betwe...</p></li></ul>