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Curvilinear Motion

Curvilinear Motion. Chapter 11 pag.84. D S. D R. R 1. R 1. R 3. R 3. R 2. O. D S. D R. Curvilinear Motion: Velocity. What velocity has the body?. v = D S / D t. . . . . . . . . The arc D S is longer than the chord D R.

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Curvilinear Motion

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  1. CurvilinearMotion Chapter 11 pag.84

  2. DS DR R1 R1 R3 R3 R2 O DS DR Curvilinear Motion: Velocity What velocity has the body? v = DS / Dt         The arc DS is longer than the chordDR this difference is reduced by bringing two closer points together Þshortertime interval DR =DS DR tangent to arc If we reduce Dt to almost zero (infinitesimal) we have: Þ 

  3. R1 DS DR R3 O Curvilinear Motion: Velocity If we take two points infinitely close on a curvilinear trajectory we have: DR =DS DR tangent to arc  ß The Velocity is a vector that is tangent at the trajectory instant after instant 

  4. Dv v2 v1 Circular Motion: Acceleration Velocity Vector: Direction: tangent to circumference therefore change over time ß an Acceleration exists     Acceleration Vector: Direction: parallel to radius Way: towards centre. ß CENTRIPETAL ACCELERATION 

  5. Circular Uniform Motion: Definition Velocityintensity remains constant we define PERIOD (T) the time taken to complete one revolution unity of measurement: second  we define frequency (n) the reciprocal of period n =1/T unity of measurement: Hz (1 Hz = 1 revolution/second) 

  6. R Circular Uniform Motion: Velocity What is the expression of velocity? the SPACE is the length of circumference the TIME is the period ß If we utilize the frequency 

  7. R r C.U.M. : linearand angular Velocity V v The radius describes an angle that changes instant after instant therefore we can talk about a velocity connected to angle: ANGULAR VELOCITY(w) a     The body covers 2p radians during one period, therefore the expressions of w is: 

  8. C.U.M. : Linearand Angular Velocity Angolar Velocity rad/s Linear Velocity m/s V=wR For Circular Uniform Motion the Angolarvelocity and the Linear velocity are CONSTANTS 

  9. v C.U.M: Acceleration So we draw a circumference with the length of radius as the intensity of velocity!    If the body covers one revolution, the velocity vector changes by about 2pv ß if we use the expressions of angular and linear velocity we have: 

  10. Curvilinear Varied Motion For Curvilinear Varied motion the velocity changes in direction intensity aT a aN      Centripetal component of acceleration Normalon trajectory Þ Variation of velocity in direction Tangential component of acceleration Tangenton trajectory Þ Variation of velocity in intensity 

  11. Observation Linear Uniform Motion Acceleration NONE Velocity CONSTANT  Velocity changes in intensity Acceleration tangential Linear Varied Motion  Velocity changes in direction Acceleration centripetal Curvilinear Uniform Motion  Acceleration centripetal tangential Velocity changes in direction intensity Curvilinear Varied Motion  

  12. Y O X Simple Harmonic Motion (SHM): Definitions It is the projection of circular uniform motion onto a diameter Oscillation Complete: Motion from A to B and back to A: ABA Extremes of Oscillation: Points A and B A B   Centre of Oscillation: Points O Elongation: distance between the point and centre of oscillation Amplitude of motion: Maximum elongation 

  13. Y O X SHM: Period The Harmonic Motion is a Periodic motion The shortesttime interval after there the motion have again the same propriety is known as PERIOD is the duration of a complete oscillation A B   The period of SHM is the same as the period of CUM The AngularVelocity of CUM is known as pulsation of SHM 

  14. SHM: Dynamics Definition Simple Harmonic Motion is the motion of a body that is being acted on by a force, whose magnitude is proportional to the displacement of the body from fixed point (centre of oscillation) Fx whose direction is always towards that pointF-x F = - k x 

  15. 0 Max 0 Max 0 Max F=-kx Max 0 F=-kx Max Velocity Potential Energy Max 0 Potential Energy Max Acceleration Velocity Acceleration Kinetic Energy Kinetic Energy 0 Max 0 SHM: Summary B O A X  

  16. R Y p/2w 3p/2w O 2p/w t p/w wt O X -R SHM: Equation x      x = R cos(wt) 

  17. v wR Y p/2w 3p/2w O 2p/w t p/w wt O X -wR SHM: Velocity         vx = - wR sin(wt) Negative for ‘going’ Positive for ‘return' MAXIMUM in centre of oscillation ZERO at extremes of oscillation 

  18. a w2R p/2w 3p/2w O 2p/w t p/w -w2R SHM: Acceleration  a = - w2R cos(wt) = - w2x NEGATIVE for positive elongations POSITIVE for negative elongations MASSIMUM (absolute value) at extremes of oscillation ZERO in centre of oscillation 

  19. Y O X Composition to 2 SHM x = R cos(wt) y = R sin(wt) ß A B Parametric Equation of a circumference drawn by a body that moves with Angular Velocity w  TWO PERPENDICULAR SHMs whit the SAME PULSATION w, generate a CIRCULAR UNIFORM MOTION 

  20. The End  Chapter 11 pag.84

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