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Unformatted text preview: Chapter 15 Oscillations In this chapter we will cover the following topics: Displacement, velocity and acceleration of a simple harmonic oscillator Energy of a simple harmonic oscillator Examples of simple harmonic oscillators: springmass system, simple pendulum, physical pendulum, torsion pendulum Damped harmonic oscillator Forced oscillations/Resonance (151) In fig.a we show snapshots of a s Simple Harmon imple oscillat ic ing Motio syst n ( em SHM) . The motion is periodic i.e. it repeats in time. The time needed to complete one repetition is known as the The number of repetitions per u (symbol , units: s ). nit time is called the f perio c d rquen T ( 29 1 ( ) cos y (symbol The displacement of the particle is given by the equation: Fig.b is a plot of ( ) versus . The quantity is called the of the motion. ampli , unit hertz) I e tud m m x f T t t x f x t x t = = + t gives the maximum possible displacement of the oscillating object The quantity is called the of the oscillator. It is given by angular fre the equatio qu n: enc y (152) ( 29 ( ) cos m x t x t = + 2 2 f T = = ( 29 ( ) cos m x t x t = + The quantity is called the phase angle of the oscillator. The value of is determined from the displacement (0) and the velocity (0) at 0. In fig.a ( ) is plotted versus for 0. ( ) x v t x t t x t x = = = ( 29 ( 29 Velocity of SHM velocity cos ( ) ( ) cos sin The quantity is called the It expresses the maximum possible value of ( ) In fig.b the am velocity ( ) is plo pl tt itude m m m m m v t dx t d v t x t x t dt dt x v t v t = = + =  + ed versus for 0. ( ) sin m t v t x t = =  ( 29 2 2 2 ( ) ( ) sin cos The quantity is called the Acceleration of SHM: acceleration ampli .It expresses the maximum possible value of a( ). In fig.c the accele t r d at u e a m m m m dv t d a t x t x t x dt dt x t = = + =  =  2 ion a( ) is plotted versus for 0. ( ) cos m t t a t x t = =  (153) ( 29 2 2 2 We saw that the acceleration of an object undergoing SHM is: If we apply Newton's second law we get: The Force Law for Simple Harmonic Motion Simple harmonic motion occurs whe a x F ma m x m x =  = =  =  The force can be wr n the force acting on itten as: where is a constant. If we compare the two expressions for F we have an object is paroportional to the disaplacement but opposite in sign.to the disaplacement but opposite in sign....
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 Spring '11
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 Physics, Acceleration, Energy, Mass

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