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Kinematics in 2-D (II)

Kinematics in 2-D (II). Uniform circular motion Tangential and radial components of Relative velocity and acceleration Serway and Jweett : 4.4 to 4.6. center. Uniform Circular Motion. “uniform” means constant speed velocity changes (direction changes)

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Kinematics in 2-D (II)

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  1. Kinematics in 2-D (II) • Uniform circular motion • Tangential and radial components of • Relative velocity and acceleration Serway and Jweett : 4.4 to 4.6 Physics 1D03 - Lecture 5

  2. center Uniform Circular Motion • “uniform” means constant speed • velocity changes (direction changes) • acceleration : find by subtracting vectors, then What is the value of Dv/Dt, as Dt0 ? Physics 1D03 - Lecture 5

  3. Similar triangles, Note is perpendicular to Subtract velocities: Compare with displacements: Physics 1D03 - Lecture 5

  4. as Dt 0, From previous slide: and so Direction: SinceDv is perpendicular toDr, ais perpendicular tov So,a is towards the centre of the circle (“centripetal”). Physics 1D03 - Lecture 5

  5. has components parallel and perpendicular to the motion ; and In general, direction and speed both change: Physics 1D03 - Lecture 5

  6. at The radial (centripetal) component is due to the change in direction. ac a , perpendicular to path The tangential component (tangent to the path) is equal to the rate of change of speed: Physics 1D03 - Lecture 5

  7. Application:What accelerations does a plane pilot feel at the top and bottom of a loop? g ac g Where does he feel the heaviest ? Physics 1D03 - Lecture 5

  8. bicycle wasp Relative Motion Example: A wasp is flying from north to south at 30 km/h. You are riding your bicycle northeast at 20 km/h. What is the velocity of the wasp relative to you (vwb) ? There are two reference frames (coordinate axes) for measuring from: the ground, and the bicycle. At left are the velocities relative to the ground. How do these look relative to the bicycle? Physics 1D03 - Lecture 5

  9. N 20 km/h  135° You can use your favorite method to solve for and angle  or .  30 km/h Using relative vectors: or Answers : 46 km/h at  = 27,  = 18 Physics 1D03 - Lecture 5

  10. Suppose we have a stationary reference frame (axes x,y), and another reference frame , moving at velocity v0 . The relative displacement of the two sets of axes is r0 (which changes with time). y' x' The positions are related by y y' v0 Differentiate to get velocities: particle Differentiate again. If v0 is constant, x' x The acceleration of a particle appears the same in both reference frames! Physics 1D03 - Lecture 5

  11. Inertial Frames A reference frame in which Newton’s Laws are true is called an Inertial Frame. Since Newton’s mechanics is based on acceleration (and on relative positions), any frame moving at constant velocity is an inertial frame Physics looks the same in all inertial frames. Physics 1D03 - Lecture 5

  12. cube moves away Dropping a sugar cube in a train going around a curve. cube goes straight View from inertial frame (top view) cup follows curve (away from cube) Example cube falls vertically Dropping a sugar cube into a coffee cup in an airplane traveling 600 km/h. But: Physics 1D03 - Lecture 5

  13. Summary • Acceleration has a tangential component (parallel to motion) and a radial component (perpendicular to the motion) • at = rate of increase of speed • towards the center of the circle or arc • Inertial reference frames move at constant velocity relative to each other • Acceleration is the same in all inertial frames, velocities obey : Physics 1D03 - Lecture 5

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