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Unformatted text preview: 1 Chapter 10 Simple Harmonic Motion SHM Examples: MassSpring System, Simple Pendulum Energy Conservation Applied to SHM 2 10.5 Simple Harmonic Motion Condition for oscillations around a point: restoring force 3 10.5 Simple Harmonic Motion Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point ) is proportional to the displacement from equilibrium. 4 The motion of a mass on a spring is an example of SHM. The restoring force is F=kx. x Equilibrium position x y 5 Assuming the table is frictionless: ( 29 ( 29 t x m k t a ma kx F x x x = = = Also, ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 1 2 1 t kx t mv t U t K t E + = + = x Equilibr ium positio n x y 6 10.67 Representing Simple Harmonic Motion When a massspring system is oriented vertically, it will exhibit SHM with the same period and frequency as a horizontally placed system. 7 SHM demo http://www.ngsir.netfirms.com/englishhtm/Sprin http://www.phy.ntnu.edu.tw/ntnujava/index.php 8 At the equilibrium point x =0 so a =0 too. When the stretch is a maximum, a will be a maximum too. The velocity at the end points will be zero, and it is a maximum at the equilibrium point. 9 A simple harmonic oscillator can be described mathematically by: ( 29 ( 29 ( 29 t A t v t a t A t x t v t A t x cos sin cos 2 = = = = = Or by: ( 29 ( 29 ( 29 t A t v t a t A t x t v t A t x sin cos sin 2 = = = = = where A is the amplitude of the motion, the maximum displacement from equilibrium, A =v max , and A 2 =a max. 10 SHM graphically 11 The period of oscillation is . 2 = T where is the angular frequency of the oscillations, k is the spring constant and m is the mass of the block. m k = 12 Example (text problem 10.28): The period of oscillation of an object in an ideal massspring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point? At equilibrium x=0: 2 2 2 2 1 2 1 2 1 mv kx mv U K E = + = + = Since E=constant, at equilibrium (x = 0) the KE must be a maximum. Here v = v max = A . 13 ( 29 ( 29 cm/sec 8 . 62 rads/sec 6 . 12 cm 5.0 and rads/sec 6 . 12 s 50 . 2 2 = = = = = = A v T The amplitude A is given, but is not....
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This note was uploaded on 12/23/2011 for the course PHY 2130 taught by Professor Rehse during the Fall '08 term at Wayne State University.
 Fall '08
 REHSE
 Energy, Force, Mass, Simple Harmonic Motion

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