lecture-ch15 - Contents 15 Oscillations 1 15.1 Simple...

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Unformatted text preview: Contents 15 Oscillations 1 15.1 Simple Harmonic Motion (SHM) . . . . . . . . . . . . . . . . . 1 15.1.1 Velocity of SHM: . . . . . . . . . . . . . . . . . . . . . 3 15.1.2 Acceleration of SHM: . . . . . . . . . . . . . . . . . . . 3 15.2 The Force Law for Simple Harmonic Motion . . . . . . . . . . 3 15.3 An Angular Simple Harmonic Oscillator . . . . . . . . . . . . 4 15.3.1 Torsion Pendulum . . . . . . . . . . . . . . . . . . . . 4 15.3.2 The Simple Pendulum . . . . . . . . . . . . . . . . . . 5 15.3.3 Physical Pendulum . . . . . . . . . . . . . . . . . . . . 6 15.3.4 Simple Harmonic Motion and Uniform Circular Motion 7 15.4 Damped Simple Harmonic Motion . . . . . . . . . . . . . . . . 8 15.4.1 Over Damping: ( b 2 m ) 2 − k m > 0 . . . . . . . . . . . . . 9 15.4.2 Critical Damping: ( b 2 m ) 2 − k m = 0 . . . . . . . . . . . . 10 15.4.3 Under Damping: ( b 2 m ) 2 − k m < 0 . . . . . . . . . . . . . 10 15.5 Forced Oscillations and Resonance . . . . . . . . . . . . . . . 12 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: spring-mass system, simple pen- dulum, physical pendulum, torsion pendulum, Damped harmonic oscillator, Forced oscillations/Resonance. 15.1 Simple Harmonic Motion (SHM) In the following, we show snapshots of a simple oscillating system. 1 The motion is periodic i.e. it repeats in time. The time needed to complete one repetition is known as the period (symbol T , units: s ). The number of repetitions per unit time is called the frequency (symbol f , unit hertz) f = 1 T The displacement of the particle is given by the equation: x ( t ) = x m cos ( ωt + φ ) Fig. b is a plot of x ( t ) versus t . The quantity x m is called the amplitude of the motion. It gives the maximum possible displacement of the oscillating object. The quantity ω is called the angular frequency of the oscillator. It is given by the equation: ω = 2 πf = 2 π T The quantity φ is called the phase angle of the oscillator. The value of φ is determined from the displacement x (0) and the velocity v (0) at t = 0. In fig. a below, x ( t ) is plotted versus t for φ = 0. x ( t ) = x m cos ωt 2 15.1.1 Velocity of SHM: v ( t ) = dx ( t ) dt = d ( x m cos ( ωt + φ )) dt = − ωx m sin ( ωt + φ ) The quantity ωx m is called the velocity amplitude v m . It expresses the max- imum possible value of v ( t ). In fig. b the velocity v ( t ) is plotted versus t for φ = 0. v ( t ) = − ωx m sin ωt. 15.1.2 Acceleration of SHM: a ( t ) = dv ( t ) dt = d ( − ωx m sin ( ωt + φ )) dt = − ω 2 x m cos ( ωt + φ ) The quantity ω 2 x m is called the acceleration amplitude a m . It expresses the maximum possible value of a ( t ). In fig. c the acceleration a ( t ) is plotted versus t for φ = 0. a ( t ) = − ω 2 x m cos ( ωt ) 15.2 The Force Law for Simple Harmonic Motion We saw that the acceleration of an object undergoing SHM is: a = − ω 2 x 3 If we apply Newton’s second law we get:...
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lecture-ch15 - Contents 15 Oscillations 1 15.1 Simple...

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