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Unformatted text preview: April 14, 2006 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2006 Lecture 48 April 14, 2006 Physics for Scientists&Engineers 1 2 Review Review ! Pendulum. Differential equation: ! Solution: harmonic motion in ! ! Period and frequency of a pendulum of length l are f = 1 2 ! g l T = 2 ! l g ! ( t ) = ! cos g l t " # $ % & ’ d 2 ! dt 2 + g l ! = April 14, 2006 Physics for Scientists&Engineers 1 3 Quiz 48.1 Quiz 48.1 ! Which of the following will double the period of a pendulum? A) Mass -> 4 x Mass B) Length -> 2 x Length C) Length -> 4 x Length D) Length -> Length / 2 E) Length -> Length / 4 April 14, 2006 Physics for Scientists&Engineers 1 4 Damped Oscillations Damped Oscillations ! If you wait long enough, a pendulum will eventually stop oscillating and just hang straight down ! Why? ! Air resistance and other friction effects ! From 1-d motion chapter: air resistance depends on velocity and are directed opposite to velocity vector ! In general: need to introduce a damping force into our equations of motion: (valid for small v) F d = ! bv April 14, 2006 Physics for Scientists&Engineers 1 5 Equation of Motion Equation of Motion ! New equation of motion ! Divide by mass to get equation into standard form: ! Note: This equation contains the position vector, its derivative, and its second derivative m d 2 x dt 2 = ! b dx dt ! kx " d 2 x dt 2 + b m dx dt + k m x = April 14, 2006 Physics for Scientists&Engineers 1 6 Small Damping Small Damping ! For small values of the damping constant (we will define what “small” means), the solution of this equation is given as: ! The coefficients A and B are determined by the initial conditions, i.e. the position x and velocity v at time t = 0 ! The angular frequencies appearing in this solution are given by: x ( t ) = Ae ! " # t cos( " ' t ) + Be ! " # t sin( " ' t ) A = x and B = v + x ! " ! ' ! " = b 2 m ! ' = ! 2 # ! " 2 = k m # b 2 m $ % & ’ ( ) 2 ! = k m April 14, 2006 Physics for Scientists&Engineers 1 7 Damping Damping ! This solution is valid for all values of the damping constant b for which the argument of the square root for w’ remains positive ! This is the condition for “small” damping...
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This note was uploaded on 03/19/2008 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.
- Spring '08