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Mechanics 84 6.1 2D Particle in Cartesian Coordinates A Hostetler Handbook v 0.82 Contents Preface v 1 Newton's Laws1 1.1 Many Particles ................................. 2 1.2 Cartesian Coordinates............................. 3 1.3 Polar Coordinates ............................... 4 1.4 Cylindrical Coordinates ............................ 7 1.5 Air Resistance, Friction, and Buoyancy.................... 8 1.6 Charged Particle in a Magnetic Field..................... 17 1.7 Summary: Newton's Laws........................... 20 2 Momentum and Center of Mass 23 2.1 Rocket with no External Force ........................ 24 2.2 Multistage Rockets............................... 25 2.3 Rocket with External Force.......................... 25 2.4 Center of Mass................................. 26 2.5 Angular Momentum of a Single Particle................... 28 2.6 Angular Momentum of a System of Particles ................ 29 2.7 Angular Momentum of a Continuous Mass Distribution .......... 30 2.8 Summary: Momentum and Center of Mass ................. 34 3 Energy 36 3.1 General One-dimensional Systems ...................... 40 3.2 Atwood Machine................................ 43 3.3 Spherically Symmetric Central Forces .................... 44 3.4 The Energy of a System of Particles ..................... 45 3.5 Summary: Energy ............................... 47 4 Oscillations 49 4.1 Simple Harmonic Oscillators.......................... 49 4.2 Damped Harmonic Oscillators......................... 52 4.3 Driven Oscillators ............................... 57 4.4 Summary: Oscillations............................. 69 5 Calculus of Variations 72 5.1 The Euler-Lagrange Equation......................... 72 5.2 The Brachistochrone Problem......................... 77 5.3 General Parametrization............................ 80 5.4 Summary: Calculus of Variations....................... 83 6 Lagrangian Mechanics 84 6.1 2D Particle in Cartesian Coordinates..................... 84 6.2 2D Particle in Polar Coordinates....................... 86 iv Contents 6.3 Unconstrained Particles in 3D......................... 87 6.4 Constrained Particles in 3D.......................... 87 6.5 Noether's Theorem............................... 95 6.6 The Hamiltonian................................ 96 6.7 Lagrange Multipliers.............................. 98 6.8 Summary: Lagrangian Mechanics....................... 100 7 Orbits 102 7.1 The Kepler Problem.............................. 105 7.2 Kepler's Laws.................................. 109 7.3 Transfer Orbits................................. 109 7.4 Summary: Orbits................................ 113 8 Noninertial Frames 115 8.1 Frame with Linear Acceleration........................ 115 8.2 Rotating Frames ................................ 116 8.3 Centrifugal Force................................ 118 8.4 Coriolis Force.................................. 119 8.5 Summary: Noninertial Frames......................... 123 9 Rigid Rotations 125 9.1 Rotation About a Fixed Axis......................... 126 9.2 The Inertia Tensor............................... 128 9.3 Principal Axes ................................. 133 9.4 Torque-free Motion............................... 136 9.5 Summary: Rigid Rotations .......................... 139 10 Coupled Oscillators 141 10.1 Two Masses and Three Springs........................ 141 10.2 The General Problem ............................. 145 10.3 Double Pendulum................................ 147 10.4 Triatomic Molecule............................... 149 10.5 Parallel and Series Springs........................... 151 10.6 Summary: Coupled Oscillators ........................ 152 11 Collision and Scattering 154 11.1 Hard Sphere Scattering ............................ 155 11.2 The General Case................................ 156 11.3 Rutherford Scattering ............................. 158 12 Math Reference 160 12.1 Complex Numbers ............................... 160 12.2 Vectors ..................................... 160 12.3 Curvilinear Coordinates............................ 160 12.4 Differential Equations ............................. 163 12.5 Taylor Series .................................. 163 12.6 Approximations................................. 164 Index 165 Preface About These Notes Cover image: NASA (The International Space Station) These are my class notes from two courses on classical mechanics (PHY 3221 and PHY 4222) that I took at Florida State University. Our textbook was Classical Mechanics by John R. Taylor. My class notes can be found at www.leonhostetler.com/classnotes Please bear in mind that these notes will contain errors. Any errors are certainly my own. If you find one, please email me at [email protected] with the name of the class notes, the page on which the error is found, and the nature of the error. This work is currently licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License. That means you are free to copy and distribute this document in whole for noncommercial use, but you are not allowed to distribute derivatives of this document or to copy and distribute it for commercial reasons. Updates Last Updated: July 10, 2019 Version 0.82: (Jul. 10, 2019) Updated layout, spell-checked Version 0.81: (Dec. 21, 2017) First upload. Conventions I will represent a vector by a bold letter topped with an arrow. For example: F~. I will represent a unit vector (i.e. a direction vector with magnitude 1) by a bold leter topped with a hat. For example: x^. In physics, a time derivative is often represented with a dot. For example, instead dx d2x of writing dt or dt2 , one might writex _ orx ¨. This convention will generally be used in these notes. Chapter 1 Newton's Laws Mechanics is the study of the motion of material bodies in space as a function of time. We'll be studying classical mechanics which assumes v << c (i.e. non-relativistic) and Planck's constant is treated as h = 0 (i.e. non-quantum). First Law: In the absence of forces, a body moves with uniform velocity. It's the ability to visualize the ideal case that enabled Newton to achieve this break- through since in the real world, moving bodies tend to slow down. Remarkably, a state- ment about an ideal situation can allow us to make arbitrarily precise predictions. Second Law: F~ = m~a This law does not hold relativistically. Here, m is the inertial mass, which is an attribute of the body. Inertial mass quantifies a body's resistance to force. F~ is not an attribute of the body, but rather it quantifies the interaction of the body with its environment. Newton also introduced a \quantity of motion" that is known today as momentum, ~p = m~v. Using this, we can write Newton's law in a more general form that also holds relativistically: d~p F~ = = ~p_: dt Third law: For every action, there's an equal but opposite reaction. The notation F~ ij denotes the force exerted on particle i by particle j. Read it as \force on i by j". According to Newton's third law, for two particles i and j, F~ ij = −F~ ji: ext If a system of two particles is isolated with no outside forces, that is, F~ = 0, then the only force on any of the two particles is the force exerted by the other particle. Since ~ ~ ~ _ ~ _ F ij = −F ji and F 12 = ~p1 and F 21 = −~p2, we have that d (~p + ~p ) = 0; dt 1 2 or conservation of momentum. This is what the third law is really telling us. The weight force of a body is ~w = m~g: The mass m in this case is gravitational mass. Although gravitational mass and inertial mass are defined differently, for any object ever measured, the two always had the same value. In fact, Einstein's principle of equivalence states that gravitational mass = inertial mass: Some important concepts implied by Newton's laws are 2 Newton's Laws Inertial frames: An inertial frame is any frame in which Newton's first law holds. Fic- titious forces arise when not in an inertial frame. Galilean invariance: Any frame in uniform motion with respect to an inertial frame is also an inertial frame. Equation of motion: The equation of motion for bodies is Newton's second law. The second law can be written as a second order differential equation with respect to position F~ = m~r¨. To get the position equation for a body, we integrate this twice _ with respect to time|getting two constants of integration ~r0 and ~r0 = ~v0. These constants are often fixed by initial conditions. 1.1 Many Particles Suppose you have many particles spread throughout some space. Each particle, in this case, is experiencing an external force from outside the system as well as a force from every other particle. If you define some origin O, then the position of the ith particle can be represented by ~ri, and its momentum by ~pi = mi~ri. Newton's second law implies d X ext ~p = F~ + F~ : dt i ij i j6=i What this states is that the time derivative of the momentum of the ith particle is equal to the sum of all the forces due to the other particles plus the external force on the particle. The index j 6= i simply means to sum over all j not equal to i. That is, sum over all other particles|ignoring the case where F~ ii since a particle doesn't exert a force on itself. The total momentum of the system, which we denote with a capital P~ is the sum of the momenta of all the particles ~ X P = ~pi: i Since we can differentiate sums term by term, we know that ~_ X _ P = ~pi: i _ Replacing ~pi with our result from above gives us 0 1 ~_ X X ~ ~ ext X X ~ X ~ ext P = @ F ij + F i A = F ij + F i : i j6=i i j6=i i Looking at the first term on the right, we can expand this as X X X X X X F~ ij = F~ ij + F~ ij: i j6=i i j<i i j>i We can write the right side more compactly by combining the two terms and switching the subscripts on one of the force vectors X X X X F~ ij = F~ ij + F~ ji : i j6=i i j<i Now we're still getting every term with no double counting. By Newton's third law, every term F~ ij + F~ ji = 0, so this entire sum is zero. Therefore, returning to our total momentum derivative, we have that ~_ X X ~ X ~ ext X ~ ext P = F ij + F i = F i : i j6=i i i 1.2. Cartesian Coordinates 3 This simplifies to _ ext P~ = F~ : In other words, the rate of change of the total momentum of the system depends only on the external forces on the system|not on the internal forces.
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