Chapter_15

# Chapter_15 - Chapter 15 Oscillations In this chapter we...

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Unformatted text preview: Chapter 15 Oscillations In this chapter we will cover the following topics: Simple harmonic oscillator: Displacement, velocity, acceleration, kinetic and potential energy. Examples of simple harmonic oscillators: spring-mass system, simple pendulum, physical pendulum, torsion pendulum Damped harmonic oscillator Forced oscillations/Resonance (15-1) In fig.a we show snapshots of a simple oscillating system. Simple Harmonic Motion (SHM) The motion is periodic, i.e. it repeats in time. The time needed to complete one repetition is the The number of repetitions per u (symb nit t ol , unit ime is th s s). (symbol , unit e T f period frequency ( 29 The displacement of the particle is given by (see Fig.b). is called the of the SHM. It gives the maximum displacement of the oscillating object is cal h l er 1 ( ) c tz ed th . ) os = = + m m f T x x t x t ϖ ϖ φ amplitude e of the SHM. It is given by the equation: angular frequency (15-2) ( 29 ( ) cos m x t x t ϖ φ = + 2 2 f T π ϖ π = =-x m x m O ( 29 ( ) cos m x t x t ϖ φ = + ( 29 ( 29 ( 29 ( 29 The quantity is called the . The value of is determined from (0) cos and the velocity (0) sin at 0. ( ) ( ) cos sin The quantity = = - = = = + = - + = Velocity of SHM m m m m m x x v x t dx t d v t x t x t dt dt v φ φ φ ϖ φ ϖ φ ϖ ϖ φ phase angle is called the It expresses the maximum valu e of ( ) m x v t ϖ velocity amplitude ( 29 ( 29 2 2 2 ( ) ( ) sin cos The quantity is called the . It expresses the maximum value of a( ). = =- + = - + = - = Acceleration of SHM m m m m dv t d a t x t x t x dt dt a x t ϖ ϖ φ ϖ ϖ φ ϖ ϖ acceleration amplitude (15-3) 2 a x ϖ = - = φ ( 29 ( 29 = + φ m x t x cosωt Simple Harmonic Motion (cont’d) cos( ϖ t) cos( ϖ t- π /4) cos( ϖ t+ π /4) ? → m m x x' T T → ' π/4 ϖ 15.2.1. Object A is attached to ideal spring A and is moving in simple harmonic motion. Object B is attached to ideal spring B and is moving in simple harmonic motion. The period and the amplitude of object B are both two times the corresponding values for object A. How do the maximum speeds of the two objects compare? a) The maximum speed of A is one fourth that of object B. b) The maximum speed of A is one half that of object B. c) The maximum speed of A is the same as that of object B. d) The maximum speed of A is two times that of object B. e) The maximum speed of A is four times that of object B. 15.2.1. Object A is attached to ideal spring A and is moving in simple harmonic motion. Object B is attached to ideal spring B and is moving in simple harmonic motion. The period and the amplitude of object B are both two times the corresponding values for object A. How do the maximum speeds of the two objects compare?...
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Chapter_15 - Chapter 15 Oscillations In this chapter we...

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