1 Curvilinear Motion Chapter 11 pag.84

2 Curvilinear Motion: Velocity O R1R1 R2R2 RR SS The arc  S is longer than the chord  R What velocity has the body? v =  S /  t this difference is reduced by bringing two closer points together   shorter  time interval R3R3 SS R1R1 R3R3 RR  If we reduce  t to almost zero (infinitesimal) we have:  R =  S  R tangent to arc  

3 O R1R1 R3R3 Curvilinear Motion: Velocity SS RR If we take two points infinitely close on a curvilinear trajectory we have:  R =  S  R tangent to arc  The Velocity is a vector that is tangent at the trajectory instant after instant  

4  Velocity Vector: Direction:tangent to circumference therefore change over time  an Acceleration exists v2v2 v1v1 Acceleration Vector: Direction:parallel to radius Way:towards centre.  CENTRIPETAL ACCELERATION Circular Motion: Acceleration vv 

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

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

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

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

9  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 2  v  if we use the expressions of angular and linear velocity we have: v 

10 Curvilinear Varied Motion aNaN aTaT a For Curvilinear Varied motion the velocity changes in direction intensity Variation of velocity in direction  Centripetal component of acceleration Normal on trajectory Variation of velocity in intensity  Tangential component of acceleration Tangent on trajectory   

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

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

13  SHM: Period O X Y A B The Harmonic Motion is a Periodic motion The shortest time interval after there the motion have again the same propriety is known as PERIOD is the duration of a complete oscillation The period of SHM is the same as the period of CUM The Angular Velocity 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 point F  -x  F = - k x

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

16 SHM: Equation O X Y tt x t O R -R //  /   /2   /2  x = R cos(  t)  

17 SHM: Velocity O X Y tt v t O RR -R-R //  /   /2   /2  v x = -  R sin(  t) MAXIMUM in centre of oscillation ZERO at extremes of oscillation Negative for ‘going’ Positive for ‘return'  

18  SHM: Acceleration a t O 2R2R -2R-2R //  /   /2   /2  a = -  2 R cos(  t) = -  2 x NEGATIVE for positive elongations POSITIVE for negative elongations MASSIMUM (absolute value) at extremes of oscillation ZERO in centre of oscillation 

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

20 Chapter 11 pag.84 The End 