Harmonic MotionVector ComponentsVelocity ComponentsAcceleration ComponentsChanging Angle to PositionAcceleration and PositionSpring OscilationsSpringboardHarmonic MotionHarmonic MotionVector ComponentsVector ComponentsCircular motion can be Circular motion can be described by components.described by components.•xx = = rr cos cos •yy = = rr sin sin For uniform circular motion For uniform circular motion the angle is related to the the angle is related to the angular velocity.angular velocity.• = = ttThe motion can be described The motion can be described as a function of time.as a function of time.•xx = = rr cos cos tt•yy = = rr sin sin ttrr sinr cosVelocity ComponentsVelocity ComponentsThe velocity vector can also be The velocity vector can also be described by components.described by components.•vvxx = - = -vv sin sin •vvyy = = vv cos cos This is the derivative of the This is the derivative of the position.position.v-v sinv costrtrdtddtdyvtrtrdtddtdxvyxcos)sin(sin)cos(Acceleration ComponentsAcceleration ComponentsFor uniform circular motion the For uniform circular motion the acceleration vector points inward.acceleration vector points inward.•aaxx = - = -aa cos cos •aayy = = -a-a sin sin This is the derivative of the This is the derivative of the velocity.velocity.a -a sin-a costrtrdtddtdvatrtrdtddtdvayyxxsin)cos(cos)sin(22Changing Angle to PositionChanging Angle to PositionIf only one component is viewed the motion is If only one component is viewed the motion is sinusoidal in time.sinusoidal in time.This is called harmonic motion.This is called harmonic motion.Springs and pendulums also have harmonic motion.Springs and pendulums also have harmonic motion.xx = = AA cos cos tt1 periodAcceleration and PositionAcceleration and PositionIn uniform circular motion acceleration is opposite to In uniform circular motion acceleration is opposite to the position from the center .the position from the center .In harmonic motion the acceleration is also opposite In harmonic motion the acceleration is also opposite to the position.to the position.xtrax22cosThis is true for all small oscillationsSpring OscilationsSpring OscilationsFrom the law of action the From the law of action the force is proportional to the force is proportional to the acceleration.acceleration.Harmonic motion has a Harmonic motion has a position-dependent force.position-dependent force.•Force is negativeForce is negative•Restoring forceRestoring forcexmmaFx2mkxmkxF/2SpringboardSpringboardA diving board oscillates with a A diving board oscillates with a frequency of 5.0 cycles per frequency of 5.0 cycles per second with a person of mass second with a person of mass 70. kg. What is the spring 70. kg. What is the spring constant of the board?constant of the board?mfkfmkf2222244/2Find the spring constant Find the spring constant from the mass and from the mass and frequency.frequency.With values:With values:•kk = 4 = 422(5.0 /s)(5.0 /s)22(70. kg)(70. kg)•K = 6.9 x 10K = 6.9 x 1044 N/m
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