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Classical Mechanics

Classical Mechanics

Prof. Deepto Chakrabarty , Dr. Peter Dourmashkin , Dr. Michelle Tomasik , Prof. Anna Frebel and Prof. Vladan Vuletic via MIT OpenCourseWare Direct link

22.1 Path Independence - Gravity

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22.1 Path Independence - Gravity

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Classroom Contents

Classical Mechanics

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  1. 1 8.01SC Classical Mechanics Introduction
  2. 2 0.1 Vectors vs. Scalars
  3. 3 0.2 Vector Operators
  4. 4 0.3 Coordinate Systems and Unit Vectors
  5. 5 0.4 Vectors - Magnitude and Direction
  6. 6 0.5 Vector Decomposition into components
  7. 7 0.6 Going Between Representations
  8. 8 1.0 Week 1 Introduction
  9. 9 1.1 Coordinate Systems and Unit Vectors in 1D
  10. 10 1.2 Position Vector in 1D
  11. 11 1.3 Displacement Vector in 1D
  12. 12 1.4 Average Velocity in 1D
  13. 13 1.5 Instantaneous Velocity in 1D
  14. 14 1.7 Worked Example: Derivatives in Kinematics
  15. 15 2.1 Introduction to Acceleration
  16. 16 2.2 Acceleration in 1D
  17. 17 2.3 Worked Example: Acceleration from Position
  18. 18 2.4 Integration
  19. 19 3.1 Coordinate System and Position Vector in 2D
  20. 20 3.2 Instantaneous Velocity in 2D
  21. 21 3.3 Instantaneous Acceleration in 2D
  22. 22 3.4 Projectile Motion
  23. 23 3.5 Demo: Shooting an Apple
  24. 24 3.5 Demo: Relative Motion Gun
  25. 25 PS.1.1 Three Questions Before Starting
  26. 26 PS.1.2 Shooting the apple solution
  27. 27 P.1.3 Worked Example: Braking Car
  28. 28 P.1.4 Sketch the Motion
  29. 29 P.1.5 Worked Example: Pedestrian and Bike at Intersection
  30. 30 4.0 Week 2 Introduction
  31. 31 4.1 Newton's First and Second Laws
  32. 32 4.2 Newton's Third Law
  33. 33 4.3 Reference Frames
  34. 34 4.4 Non-inertial Reference Frames
  35. 35 5.1 Universal Law of Gravitation
  36. 36 5.2 Worked Example: Gravity - Superposition
  37. 37 5.3 Gravity at the surface of the Earth: The value of g.
  38. 38 6.1 Contact Forces
  39. 39 6.2 Static Friction
  40. 40 7.1 Pushing Pulling and Tension
  41. 41 7.2 Ideal Rope
  42. 42 7.3 Solving Pulley Systems
  43. 43 7.4 Hooke's Law
  44. 44 DD.1.1 Friction at the Nanoscale
  45. 45 PS.2.1 Worked Example - Sliding Block
  46. 46 PS.2.2 Worked Example - Stacked Blocks - Free Body Diagrams and Applying Newtons 2nd Law
  47. 47 PS.2.2 Worked Example - Stacked Blocks - Solve for the Maximum Force
  48. 48 PS.2.2 Worked Example - Stacked Blocks - Choosing the System of 2 Blocks Together
  49. 49 PS.2.3 Window Washer Free Body Diagrams
  50. 50 PS.2.3 Window Washer Solution
  51. 51 Newton's 3rd Law Pairs
  52. 52 Internal and External Forces
  53. 53 Applying Newton's 2nd Law
  54. 54 8.0 Week 3 Introduction
  55. 55 8.1 Polar Coordinates
  56. 56 8.2 Circular Motion: Position and Velocity Vectors
  57. 57 8.3 Angular Velocity
  58. 58 9.1 Uniform Circular Motion
  59. 59 9.2 Uniform Circular Motion: Direction of the Acceleration
  60. 60 10.1 Circular Motion - Acceleration
  61. 61 10.2 Angular Acceleration
  62. 62 10.3 Worked Example - Angular position from angular acceleration.
  63. 63 11.1 Newton's 2nd Law and Circular Motion
  64. 64 11.2 Worked Example - Car on a Banked Turn
  65. 65 11.3 Demo: Rotating Bucket
  66. 66 PS.3.1 Worked Example - Orbital Circular Motion - Radius
  67. 67 PS.3.1 Worked Example - Orbital Circular Motion - Velocity
  68. 68 PS.3.1 Worked Example - Orbital Circular Motion - Period
  69. 69 12.0 Week 4 Introduction
  70. 70 12.1 Pulley Problems
  71. 71 12.2 Constraint Condition
  72. 72 12.3 Virtual Displacement
  73. 73 12.4 Solve the System of Equations
  74. 74 12.5 Worked Example: 2 Blocks and 2 Pulleys
  75. 75 13.1 Rope Hanging Between Trees
  76. 76 13.2 Differential Analysis of a Massive Rope
  77. 77 13.3 Differential Elements
  78. 78 13.4 Density
  79. 79 13.5 Demo: Wrapping Friction
  80. 80 13.6 Summary for Differential Analysis
  81. 81 14.1 Intro to resistive forces
  82. 82 14.2 Resistive forces - low speed case
  83. 83 14.3 Resistive forces - high speed case
  84. 84 15.0 Week 5 Introduction
  85. 85 15.1 Momentum and Impulse
  86. 86 15.2 Impulse is a Vector
  87. 87 15.3 Worked Example - Bouncing Ball
  88. 88 15.4 Momentum of a System of Point Particles
  89. 89 15.5 Force on a System of Particles
  90. 90 16.1 Cases of Constant Momentum
  91. 91 16.2 Momentum Diagrams
  92. 92 17.1 Definition of the Center of Mass
  93. 93 17.2 Worked Example - Center of Mass of 3 Objects
  94. 94 17.3 Center of Mass of a Continuous System
  95. 95 17.5 Worked Example - Center of Mass of a Uniform Rod
  96. 96 17.6 Velocity and Acceleration of the Center of Mass
  97. 97 17.7 Reduction of a System to a Point Particle
  98. 98 18.0 Week 6 Introduction
  99. 99 18.1 Relative Velocity
  100. 100 18.2 Set up a Recoil Problem
  101. 101 18.3 Solve for Velocity in the Ground Frame
  102. 102 18.4 Solve for Velocity in the Moving Frame
  103. 103 19.1 Rocket Problem 1 - Set up the Problem
  104. 104 19.2 Rocket Problem 2 - Momentum Diagrams
  105. 105 19.3 Rocket Problem 3 - Mass Relations
  106. 106 19.4 Rocket Problem 4 - Solution
  107. 107 19.5 Rocket Problem 5 - Thrust and External Forces
  108. 108 19.6 Rocket Problem 6 - Solution for No External Forces
  109. 109 19.7 Rocket Problem 7 - Solution with External Forces
  110. 110 PS.6.1 Rocket Sled - Differential Equation
  111. 111 PS.6.1 Rocket Sled - Integrate the Rocket Equation
  112. 112 PS.6.1 Rocket Sled - Solve for Initial Velocity
  113. 113 PS.6.2 Snowplow Problem
  114. 114 20.0 Week 7 Introduction
  115. 115 20.1 Kinetic Energy
  116. 116 20.2 Work by a Constant Force
  117. 117 20.3 Work by a Non-Constant Force
  118. 118 20.4 Integrate adt and adx
  119. 119 20.5 Work-Kinetic Energy Theorem
  120. 120 20.6 Power
  121. 121 21.1 Scalar Product Properties
  122. 122 21.2 Scalar Product in Cartesian Coordinates
  123. 123 21.3 Kinetic Energy as a Scalar Product
  124. 124 21.4 Work in 2D and 3D
  125. 125 21.5 Work-Kinetic Energy Theorem in 2D and 3D
  126. 126 21.6 Worked Example: Block Going Down a Ramp
  127. 127 22.1 Path Independence - Gravity
  128. 128 22.2 Path Dependence - Friction
  129. 129 22.3 Conservative Forces
  130. 130 22.4 Non-conservative Forces
  131. 131 22.5 Summary of Work and Kinetic Energy
  132. 132 PS.7.1 Worked Example - Collision and Sliding on a Rough Surface
  133. 133 23.0 Week 8 Introduction
  134. 134 23.1 Introduction to Potential Energy
  135. 135 23.2 Potential Energy of Gravity near the Surface of the Earth
  136. 136 23.3 Potential Energy Reference State
  137. 137 23.4 Potential Energy of a Spring
  138. 138 23.5 Potential Energy of Gravitation
  139. 139 24.1 Mechanical Energy and Energy Conservation
  140. 140 24.2 Energy State Diagrams
  141. 141 24.3 Worked Example - Block Sliding Down Circular Slope
  142. 142 24.4 Newton's 2nd Law and Energy Conservation
  143. 143 25.1 Force is the Derivative of Potential
  144. 144 25.2 Stable and Unstable Equilibrium Points
  145. 145 25.3 Reading Potential Energy Diagrams
  146. 146 26.0 Week 9 Introduction
  147. 147 26.1 Momentum in Collisions
  148. 148 26.2 Kinetic Energy in Collisions
  149. 149 26.3 Totally Inelastic Collisions
  150. 150 27.1 Worked Example: Elastic 1D Collision
  151. 151 27.2 Relative Velocity in 1D
  152. 152 27.3 Kinetic Energy and Momentum Equation
  153. 153 27.4 Worked Example: Elastic 1D Collision Again
  154. 154 27.5 Worked Example: Gravitational Slingshot
  155. 155 27.6 2D Collisions
  156. 156 DD.2.1 Position in the CM Frame
  157. 157 DD.2.2 Relative Velocity is Independent of Reference Frame
  158. 158 DD.2.3 1D Elastic Collision Velocities in CM Frame
  159. 159 DD.2.4 Worked Example: 1D Elastic Collision in CM Frame
  160. 160 DD.2.5 Kinetic Energy in Different Reference Frames
  161. 161 DD.2.6 Kinetic Energy in the CM Frame
  162. 162 DD.2.7 Change in the Kinetic Energy
  163. 163 28.0 Week 10 Introduction
  164. 164 28.1 Rigid Bodies
  165. 165 28.2 Introduction to Translation and Rotation
  166. 166 28.3 Review of Angular Velocity and Acceleration
  167. 167 29.1 Kinetic Energy of Rotation
  168. 168 29.2 Moment of Inertia of a Rod
  169. 169 29.3 Moment of Inertia of a Disc
  170. 170 29.4 Parallel Axis Theorem
  171. 171 29.5 Deep Dive - Moment of Inertia of a Sphere
  172. 172 29.6 Deep Dive - Derivation of the Parallel Axis Theorem
  173. 173 30.1 Introduction to Torque and Rotational Dynamics
  174. 174 30.2 Cross Product
  175. 175 30.3 Cross Product in Cartesian Coordinates
  176. 176 30.4 Torque
  177. 177 30.5 Torque from Gravity
  178. 178 31.1 Relationship between Torque and Angular Acceleration
  179. 179 31.2 Internal Torques Cancel in Pairs
  180. 180 31.3 Worked Example - Find the Moment of Inertia of a Disc from a Falling Mass
  181. 181 31.4 Worked Example - Atwood Machine
  182. 182 31.5 Massive Pulley Problems
  183. 183 31.7 Worked Example - Two Blocks and a Pulley Using Energy
  184. 184 PS.10.1 Worked Example - Blocks with Friction and Massive Pulley
  185. 185 32.0 Week 11 Introduction
  186. 186 32.1 Angular Momentum for a Point Particle
  187. 187 32.2 Calculating Angular Momentum
  188. 188 32.3 Worked Example - Angular Momentum About Different Points
  189. 189 32.4 Angular Momentum of Circular Motion
  190. 190 33.1 Worked Example - Angular Momentum of 2 Rotating Point Particles
  191. 191 33.2 Angular Momentum of a Symmetric Object
  192. 192 33.4 If Momentum is Zero then Angular Momentum is Independent of Origin
  193. 193 33.5 Kinetic Energy of a Symmetric Object
  194. 194 34.1 Torque Causes Angular Momentum to Change - Point Particle
  195. 195 34.2 Torque Causes Angular Momentum to Change - System of Particles
  196. 196 34.3 Angular Impulse
  197. 197 34.4 Demo: Bicycle Wheel Demo
  198. 198 34.5 Worked Example - Particle Hits Pivoted Ring
  199. 199 35.0 Week 12 Introduction
  200. 200 35.1 Translation and Rotation of a Wheel
  201. 201 35.2 Rolling Wheel in the Center of Mass Frame
  202. 202 35.3 Rolling Wheel in the Ground Frame
  203. 203 35.4 Rolling Without Slipping Slipping and Skidding
  204. 204 35.5 Contact Point of a Wheel Rolling Without Slipping
  205. 205 36.1 Friction on a Rolling Wheel
  206. 206 36.2 Worked Example - Wheel Rolling Without Slipping Down Inclined Plane - Torque Method
  207. 207 36.3 Demo: Spool Demo
  208. 208 36.4 Worked Example - Yoyo Pulled Along the Ground
  209. 209 36.5 Analyze Force and Torque in Translation and Rotation Problems
  210. 210 37.1 Kinetic Energy of Translation and Rotation
  211. 211 37.2 Worked Example - Wheel Rolling Without Slipping Down Inclined Plane
  212. 212 37.3 Angular Momentum of Translation and Rotation
  213. 213 DD.3.1 Deep Dive - Gyroscopes - Free Body Diagrams, Torque, and Rotating Vectors
  214. 214 DD.3.2 Deep Dive - Gyroscopes - Precessional Angular Velocity and Titled Gyroscopes
  215. 215 DD.3.3 Deep Dive - Gyroscopes - Nutation and Total Angular Momentum

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