PHY183-Lecture48 - April 14, 2006 Physics for...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 5

PHY183-Lecture48 - April 14, 2006 Physics for...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online