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S imple H armonic M otion ( S.H.M.)

S imple H armonic M otion ( S.H.M.). S.H.M. Definition Properties Forced Oscillation Resonance. Definition. So...?. Simple Harmonic Motion is a linear motion such that :. 1. its acceleration is directly proportional to its displacement from a fixed point

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S imple H armonic M otion ( S.H.M.)

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  1. Simple Harmonic Motion (S.H.M.)

  2. S.H.M. • Definition • Properties • Forced Oscillation • Resonance

  3. Definition So...? Simple Harmonic Motion is a linear motion such that : 1. its acceleration is directly proportional to its displacement from a fixed point (the equilibrium position), 2. its acceleration always points towards the fixed point.

  4. Equil. position a a a a Definition acceleration a µ -x displacement 0

  5. Mathematical Expression a µ -x i.e. a = - w2 x where w2 is a +ve const.

  6. a a a a Equil. position Example 1 Mass-Spring System

  7. a a a a Equil. position Example 2 Simple Pendulum

  8. a Equil. position a a a Example 3 Floating Cylinder

  9. Notes 1. The acceleration is due to the resultant force acting. 2. The system will oscillate when disturbed. The maximum displacement is called the amplitude (A).

  10. Mathematical Derivations Definition : a = -w2x where w2is a constant ……... integrating……… ……... integrating ……… We obtain another four equations of motion involving a , v , x and t .

  11. Equations of Motion (SHM) x = Acoswt v = - wA sinwt a = -w2A coswt v = ± w (A2- x2 )0.5 a = -w2x [the definition]

  12. Displacement-Time Graph x x = Acoswt A t 0 -A

  13. Velocity-Time Graph v v = - wA sinwt wA t 0 - wA

  14. Acceleration-Time Graph a a = -w2A coswt w2A t 0 -w2A

  15. Velocity-Displacement Graph v = ± w (A2- x2 )0.5 v wA t 0 -A A - wA

  16. Acceleration-Displacement Graph a = -w2x [the definition] a w2A 0 x -A A -w2A

  17. x v a Phase Relationship t 0

  18. Properties 1. S.H.M. is an oscillatory and periodic motion. 2. The time required for one complete oscillation is called the period. 3. The period is independentof the amplitude for a given system.

  19. Natural Frequency When a system is disturbed, it will oscillate with a frequency which is called the natural frequency ( fo ) of the system. e.g. for a mass-spring system :

  20. Forced Oscillation When a system is disturbed by a periodic driving force and then oscillate, this is called forced oscillation. Note : The system will oscillate with its natural frequency ( fo ) which is independent of the frequency of the driving force.

  21. Example(Mass-Spring System) Periodic driving force of freq. f Oscillating with natural freq. fo

  22. Resonance When a system is disturbed by a periodic driving force which frequency is equal to the natural frequency ( fo ) of the system, the system will oscillate with LARGE amplitude. Resonance is said to occur.

  23. Example 1 Breaking Glass System : glass Driving Force : sound wave

  24. Example 2 Collapse of the Tacoma Narrows suspension bridge in America in 1940 System : bridge Driving Force : strong wind

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