Physics 1D03 - Lecture 51 Kinematics in 2-D (II) Uniform circular motion Tangential and radial components of Relative velocity and acceleration Serway.

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Physics 1D03 - Lecture 51 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 52 Uniform Circular Motion “uniform” means constant speed velocity changes (direction changes) acceleration : find by subtracting vectors, then center What is the value of  v/  t, as  t  0 ?

Physics 1D03 - Lecture 53 Subtract velocities: Compare with displacements: Similar triangles, Note is perpendicular to

Physics 1D03 - Lecture 54 From previous slide: as  t 0, and so Direction: Since  v is perpendicular to  r, a is perpendicular to v So, a is towards the centre of the circle (“centripetal”).

Physics 1D03 - Lecture 55 In general, direction and speed both change: has components parallel and perpendicular to the motion ; and

Physics 1D03 - Lecture 56 The radial (centripetal) component is due to the change in direction., perpendicular to path The tangential component (tangent to the path) is equal to the rate of change of speed: atat acac a

Physics 1D03 - Lecture 57 Application: What accelerations does a plane pilot feel at the top and bottom of a loop? acac g g Where does he feel the heaviest ?

Physics 1D03 - Lecture 58 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 (v wb) ? There are two reference frames (coordinate axes) for measuring from: the ground, and the bicycle. wasp bicycle Relative Motion At left are the velocities relative to the ground. How do these look relative to the bicycle?

Physics 1D03 - Lecture 59 Using relative vectors: You can use your favorite method to solve for and angle  or . Answers : 46 km/h at  = 27 ,  = 18    20 km/h 30 km/h 135° N or

Physics 1D03 - Lecture 510 The acceleration of a particle appears the same in both reference frames! y y'y' x'x' x v0v0 Suppose we have a stationary reference frame (axes x,y), and another reference frame, moving at velocity v 0. The relative displacement of the two sets of axes is r 0 (which changes with time). x'x' y'y' particle The positions are related by Differentiate to get velocities: Differentiate again. If v 0 is constant,

Physics 1D03 - Lecture 511 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 512 Dropping a sugar cube into a coffee cup in an airplane traveling 600 km/h. cube falls vertically Dropping a sugar cube in a train going around a curve. cube moves away Example cup follows curve (away from cube) View from inertial frame (top view) cube goes straight But:

Physics 1D03 - Lecture 513 Summary Acceleration has a tangential component (parallel to motion) and a radial component (perpendicular to the motion) a t = 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 :

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