Chapter 15

# Chapter 15 - Announcements Practice Exams 3 4 are posted on...

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Unformatted text preview: Announcements Practice Exams 3, 4 are posted on E-learning Exam #3, Wednesday, July 22 in class. It will cover ch. 12-15 Exam #4, Wednesday, August 5, ch 1-17 Chapter 15, oscillations Circular motion Simple Harmonic Motion (oscillations) Forces causing simple harmonic oscillations Circular Motion b Simple Harmonic Motion y(t) ϕ ο simple harmonic sin( ) 2 period 1 frequency 2 sin( ) phase initial phase (phase const) angular frequency motion ( 2 ) m m m t y y T f T y y t f y ϕ ϕ ϖ ϕ π ϖ ϖ π ϖ ϕ ϕ ϕ ϖ ϖ π = + = = → = = → = + → → → → = amplitude → y Circular Motion b Simple Harmonic Motion y(t) ϕ ο sin( ) 2 m y y t T ϖ ϕ π ϖ = + = time displacement initial phase = left-right shifts y Circular Motion b Simple Harmonic Motion x(t) ϕ ο simple harmonic motio cos( ) 2 period 1 frequency 2 cos( ) phase initial phase (phase const) angular frequency ( n 2 ) m m m t x x T f T x x t f x ϕ ϕ ϖ ϕ π ϖ ϖ π ϖ ϕ ϕ ϕ ϖ ϖ π = + = = → = = → = + → → → → = amplitude → x Simple Harmonic Motion x(t) ( 29 ( 29 2 2 2 2 2 ( ) cos( ), ( ) sin( ) ( ) ( ) cos( ) cos( ) ( ) ( ) ( ) m m m m x t x t T dx v t x t dt dv a t x t x t dt d x t a t x t dt π ϖ ϕ ϖ ϖ ϖ ϕ ϖ ϖ ϖ ϕ ϖ ϖ ϕ ϖ = + = = = - ⋅ + = = ⋅ - ⋅ + = - ⋅ + = = - ⋅ displacement velocity acceleration 2 nd order differential equation for x(t)! Simple Harmonic Motion x(t) 2 2 2 ( ) cos( ), ( ) ( ) ( ) m x t x t T a t x t F ma F m x t π ϖ ϕ ϖ ϖ ϖ = + = = - ⋅ = = - ⋅ 2 ( ) 2 = , 2 ( ) cos( ) m F k x t m k k m T m k x t x t ϖ π ϖ π ϖ ϖ ϕ = - ⋅ = = = = + (our friend the mass on a spring) x x Example 1 An object of mass m is attached to two springs with Hooke’s constants k 1 and k 2 . Find the period of oscillations. k 1 k 2 Example 2 An mass m= 1kg oscillates with SH motion according to x ( t )=2.0 cos[( π /2)t+ π /4] meters. 1. what are the displacement, velocity, acceleration at t =2s?...
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Chapter 15 - Announcements Practice Exams 3 4 are posted on...

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