HR15 - Chapter 15 Oscillations In this chapter we will...

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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: spring-mass system, simple pendulum, physical pendulum, torsion pendulum Damped harmonic oscillator Forced oscillations/Resonance (15-1)
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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 known as the The number of repetitions per u (symbol , units: s ). nit time is called the T period frequen () The displacement of the particle is given by the equation: Fig.b is a plot of ( ) versus . The quantity is called the of th (sym 1 co e motio bol , unit hertz) . n s m m f T xt x f t t x ω φ = = + cy amplitude It gives the maximum possible displacement of the oscillating object The quantity is called the of the oscillator. It is given by the equatio n: angular frequency (15-2) ( ) cos m x t = + 2 2 f T π ωπ ==
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() cos m xt x t ω φ =+ The quantity is called the 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 vt x t tx t x = == phase angle cos s in The quantity is called the It expresses the maximum possible value of ( ) In fig.b the velocity ( ) is plo t t m mm t dx t d x t x t dt dt xv ωφ + = + ⎡⎤ ⎣⎦ Velocity of SHM velocity amplitude ed versus for 0. s m t x t ωω = =− 22 2 s The quantity is called the a .It expresses the maximum possible value of a( ). In fig.c the accelerati dv t d at x t x t x dt dt x t + = + = Acceleration of SHM : acceleration amplitude 2 on a( ) is plotted versus for 0. m tt x t = (15-3) 2 ax
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() 22 We saw that the acceleration of an object undergoing SHM is: If we apply Newton's second law we get: Fm a mx m x ωω == = The Forc Simple h e Law for Sim armonic motio ple H n occ armonic urs whe n Motion the The force can be written as: where is a constant. If we compare the two expressions for F we have: F C Cx m =− force acting on an object is proportional to the disaplacement but opposite in sign. 2 2 an d2 Cm CT mC π ω =→ = = = Consider the motion of a mass attached to a spring of srping constant than moves on a frictionless horizontal floor as shown in the figure. m k The net force on is given by Hooke's law: . If we compare this equation with the expression we identify the constant C with the sping constant k.
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This note was uploaded on 10/31/2010 for the course EE 423 taught by Professor Mitin during the Spring '10 term at SUNY Buffalo.

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HR15 - Chapter 15 Oscillations In this chapter we will...

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