Wave Phenomena Physics 15c

Size: px
Start display at page:

Download "Wave Phenomena Physics 15c"

Transcription

1 Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections , Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory " Lab Sections! (Tentative) Sections have been defined! Monday 6 PM and Tuesday 4 PM, starting next week! See Course Web " Sections! (Tentative) Office Hours! Masahiro: Wed :30-4:00 PM & Thu 3:30-5:00 PM! Antonio: Wed 4:00-6:00 PM

2 What We Did Last Time! Analyzed a simple harmonic oscillator! The equation of motion:! The general solution: x() t = acosωt+ bsinωt! Studied the solution! Frequency and period! Energy tossing m! Completeness of the solution d x t () = kx() t ω = k m Goals for Today! Just how common are harmonic oscillators?! Very Physics is filled with them! But why?! Introduce complex exponentials:! It replaces sine and cosine e ±ix! It makes the math easier once you get used to it! Attack a damped oscillator! Exercise of complex math! Tool of the day: Taylor expansion

3 Taylor Expansion! Any (smooth) function f(x) can be approximated around a given point x = a as: 1 1 n n f( x) f( a) + f ( a)( x a) + f ( a)( x a) +! + f ( a)( x a) +! n!! You know this already, right?! The approximation is better when x a is small.! Because the higher-order terms (x a) n shrinks faster. Ubiquity of Harmonic Oscillators! Harmonic oscillator s equation of motion: d x m = kx Hooke s Law force! The restoring force kx is linear with x! This is not exactly true in most cases! Springs do not follow Hooke s law beyond elastic limits! Still, the physical world is full of almost-harmonic oscillators! And for a good reason

4 Example: Pendulum! A pendulum swings because of the combined force of the gravity mg and the string tension T! Combined force is mgsinθ! Displacement from the equilibrium is Lθ Force is not linear with displacement Pendulum is not a harmonic oscillator T mg θ mg sinθ Small-Angle Approximation! Taylor-expand F = mgsinθ around θ = 0 1 sinθ = sin 0 + (sin θ) θ= 0θ + (sin θ) θ= 0θ +! = θ θ + θ +! 6 10! For small angle θ,! Equation of motion after approximation: d ( Lθ ) m = ml "" θ = mgθ This is linear! Solving this is pretty easy θ = acosωt+ bsinωt F = mgsinθ mgθ g ω = L

5 Potential Energy! Look at the same problem using the potential energy! At angle θ, the mass m is higher than the lowest position by h = L(1 cosθ)! The potential energy is E P = mgh = mgl(1 cosθ)! You get the force by differentiating E P with the distance x = Lθ de 1 h P dep F = = = mgsinθ dx L dθ Remember this? L x θ Small-Angle Approximation! We could Taylor-expand the potential energy E = mgl(1 cos θ ) mglθ P! The force is dep 1 d 1 F = = mglθ = mgθ dx L dθ! OK, we got the linear force again 1 cosθ = 1 θ + θ! Why are we doing this?

6 Linearizing Equation of Motion! We can often linearize the equation of motion for small oscillation E around a stable point (equilibrium)! Why?! Anything that is stable is at a minimum of the potential energy E! Let s call it x = 0 0! Taylor expansion of E near x = 0 is Ex ( ) = E(0) + E (0) x+ E (0) x + E (0) x = 0 > 0 x Linearizing Equation of Motion 1 1 Ex = E + E x + E x+ 6 3 ( ) (0) (0) (0)...! If the oscillation is small enough, the terms x 3, x 4, can be ignored.! E(x) looks like a parabola.! This gives a force de F = = E (0) x dx! This works for just about any stable physical system Linear E A harmonic oscillator! 0 x

7 Ubiquity of Harmonic Oscillators! Every physically stable object can make harmonic oscillation! Stable object sits where the potential energy is minimum! The potential near the minimum looks like a parabola! Its derivative gives a linear restoring force! This is true for small oscillation! How small depend on how the potential looks like! We observe oscillation only when small is large enough! Invisibly small oscillation will become important with quantum mechanics Complex Numbers! I assume you are familiar with complex numbers! A few reminders to make sure you got the key concepts i = 1 Complex plane ib z = a+ ib Real part Imaginary part Re( z) = a Im( z) = b a ib z* = a ib Complex conjugate Im( z*) = Im( z) z+ z* z z* Re( z) = Im( z) = i

8 Argument! For a complex number z,! The distance z from 0 is the absolute value: z = a + b! The angle θ is the argument, or phase: θ = arg( z)! z may be expressed as: z = z (cosθ + isin θ) = z i e θ What s this? θ z z = a + ib Euler s Identity e iθ = cosθ + isinθ Euler s Identity! This is a natural extension of the real exponential! Check this with Taylor expansion e x = 1+ x + x + x + x + x e ix 1 i i = 1+ ix x x + x + x cos( x) = 1 x + x sin( x) = x x + x

9 Complex Plane! e iθ goes around the unit circle on the complex plane. Im e iπ / = i e iθ = cosθ + isinθ iπ e = 1 θ 0 e =1 Re e i3 π / = i Complex Solutions! Let s try a complex exponential on a simple harmonic oscillator: d x = ω x Xt! Try x = e Xt de Solutions Xt = X e = ω e Xt xt () = i t e ± ω! We got complex solutions to a harmonic oscillator! But we need a real solution for a physical system! We need an interpretation of the complex solution X = ω X = ± iω

10 Complex " Real Solutions i t! The two complex solutions x() t = e ± ω are complex conjugates of each other.! Generally, if you have a complex solution z(t), the complex conjugate z*(t) must also be a solution d z( t) d z* ( t) = ω z( t) = ω z*( t)! Any linear combination of z(t) and z * (t) is a solution.! In particular: zt () + z*() t Re( zt ( )) = is a solution Linearity! Generality! Because of linearity, we can multiply the solution by any complex number iωt ( a ib) e = ( a ib)(cosωt+ isin ωt) = acosωt+ bsin ωt+ i( asinωt bcos ωt) Real part gives the general solution

11 Recipe! Instead of cosωt and sinωt, use e Xt where X is a complex number! This simplifies the differential math: d d ( cos t) ω = ω sin ωt ( sin t) ω = ω cosωt! When you get the final answer, take the real part! Let s look at an example d e Xt = X e Xt Damped Oscillator! An ideal LC oscillator is a capacitor C and an inductor L connected in parallel! It s a simple harmonic oscillator! Real-world LC oscillators have resistance R, or damping! Let s take such an oscillator, give charge q 0 to the capacitor, and throw the switch on at t = 0! What happens to q = q(t)? q C q R I I = dq Charge conservation L

12 Damped Oscillator! The voltages across C, R, and L add up to zero VC + VR + VL = 0! Define voltages clockwise q VC = Capacitance C = RI Ohm s Law VR V L di = L Inductance V C q q I V R I = dq V L Equation of Motion q C + dq R d q + L = 0 Note if R = 0, d q q L = C Damped Oscillator q C + dq d q R + L = 0! Now try q( t) = q0e 1 + RX + LX q0e C X + R L Xt = 0! Assume R is small! Oscillator is weakly-damped Xt 1 X + LC = 0! X is a complex number X = R L C q q ± R L R I 1 LC L This is small

13 Weakly-Damped Oscillator R X = ± L = γ ± γ R L ω! Suppose γ << ω X γ ± iω! The complex solution becomes ( i ) t t i t qt () qe γ ± ω qe γ ± = = e ω! Take the real part: γ t qt () = qe cosωt 1 LC γ R, ω 1 L LC very weak damping Weakly-Damped Oscillator t q t) = q e γ cosωt ( 0! Oscillation shrinks exponentially due to the damping term.! How long it keeps ringing depends on the ratio ω = γ R L C q 0 q γt 0e q 0

14 Complex vs. Real! We could solve the damped oscillator problem without complex exponential. (H&L Section 1.6)! The calculation is more difficult and less elegant! One must assume that the answer is e γt cosωt! It comes out naturally with complex math! Different assumption must be made if the damping is strong! We will deal with more and more complex examples in the future! Get used to it now! Strongly-Damped Oscillator! What if the resistor R is large?! The solution is still the same e Xt where X is X = γ ± γ ω γ =, ω =! With large R, γ ω >> ( γ ) R 1 L LC ω ω X γ ± γ γ γ! The two solutions are both exponentially decreasing! No oscillation. Just slowly going toward zero, ω t γ t γ 1 qt () = qe + qe q 1 and q must be fixed by the initial conditions.

15 Critically-Damped Oscillator! What if the resistor R is chosen just so that γ = ω?! The two solutions for X becomes identical X = γ ± γ ω = γ! The second solution is qt () te γ t = Check this yourself.! This condition is called critical damping! Let s compare weak/strong/critical damping for a same ω and same initial condition Damping Strength q 0 Critically-damped Strongly-damped ωt Weakly-damped q 0

16 Critical Damping! Critically damped oscillator stops most quickly! The energy is dissipated in R as fast as possible! This is useful when you are trying to control the movement! Shock absorbers on automobiles! Feedback control systems (e.g. thermostats)! Underdamped oscillation can cause disasters! Overloaded tracks. Broken shock absorbers Summary! Studied how the equation of motion can be linearized for small oscillations! Taylor expansion of the potential near the minimum! This makes harmonic oscillators very common! Learned to deal with complex exponentials! Makes it easy to solve linear differential equations! Analyzed a damped oscillator! Tried using complex exponential! Behavior depends on the damping strength! Next target: forced oscillation

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory

More information

Wave Phenomena Physics 15c. Lecture 2 Damped Oscillators Driven Oscillators

Wave Phenomena Physics 15c. Lecture 2 Damped Oscillators Driven Oscillators Wave Phenomena Physics 15c Lecture Damped Oscillators Driven Oscillators What We Did Last Time Analyzed a simple harmonic oscillator The equation of motion: The general solution: Studied the solution m

More information

Wave Phenomena Physics 15c. Masahiro Morii

Wave Phenomena Physics 15c. Masahiro Morii Wave Phenomena Physics 15c Masahiro Morii Teaching Staff! Masahiro Morii gives the lectures.! Tuesday and Thursday @ 1:00 :30.! Thomas Hayes supervises the lab.! 3 hours/week. Date & time to be decided.!

More information

The Harmonic Oscillator

The Harmonic Oscillator The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can

More information

Oscillations. Tacoma Narrow Bridge: Example of Torsional Oscillation

Oscillations. Tacoma Narrow Bridge: Example of Torsional Oscillation Oscillations Mechanical Mass-spring system nd order differential eq. Energy tossing between mass (kinetic energy) and spring (potential energy) Effect of friction, critical damping (shock absorber) Simple

More information

Forced Oscillation and Resonance

Forced Oscillation and Resonance Chapter Forced Oscillation and Resonance The forced oscillation problem will be crucial to our understanding of wave phenomena Complex exponentials are even more useful for the discussion of damping and

More information

Topic 1: Simple harmonic motion

Topic 1: Simple harmonic motion Topic 1: Simple harmonic motion Introduction Why do we need to know about waves 1. Ubiquitous in science nature likes wave solutions to equations 2. They are an exemplar for some essential Physics skills:

More information

Chapter 16: Oscillations

Chapter 16: Oscillations Chapter 16: Oscillations Brent Royuk Phys-111 Concordia University Periodic Motion Periodic Motion is any motion that repeats itself. The Period (T) is the time it takes for one complete cycle of motion.

More information

APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS

APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration

More information

Chapter 2: Complex numbers

Chapter 2: Complex numbers Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We

More information

Chap. 15: Simple Harmonic Motion

Chap. 15: Simple Harmonic Motion Chap. 15: Simple Harmonic Motion Announcements: CAPA is due next Tuesday and next Friday. Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/ Examples of periodic motion vibrating guitar

More information

Physics 11b Lecture #15

Physics 11b Lecture #15 Physics 11b ecture #15 and ircuits A ircuits S&J hapter 3 & 33 Administravia Midterm # is Thursday If you can t take midterm, you MUST let us (me, arol and Shaun) know in writing before Wednesday noon

More information

2.4 Models of Oscillation

2.4 Models of Oscillation 2.4 Models of Oscillation In this section we give three examples of oscillating physical systems that can be modeled by the harmonic oscillator equation. Such models are ubiquitous in physics, but are

More information

2.4 Harmonic Oscillator Models

2.4 Harmonic Oscillator Models 2.4 Harmonic Oscillator Models In this section we give three important examples from physics of harmonic oscillator models. Such models are ubiquitous in physics, but are also used in chemistry, biology,

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion (FIZ 101E - Summer 2018) July 29, 2018 Contents 1 Introduction 2 2 The Spring-Mass System 2 3 The Energy in SHM 5 4 The Simple Pendulum 6 5 The Physical Pendulum 8 6 The Damped Oscillations

More information

Physics 115. AC: RL vs RC circuits Phase relationships RLC circuits. General Physics II. Session 33

Physics 115. AC: RL vs RC circuits Phase relationships RLC circuits. General Physics II. Session 33 Session 33 Physics 115 General Physics II AC: RL vs RC circuits Phase relationships RLC circuits R. J. Wilkes Email: phy115a@u.washington.edu Home page: http://courses.washington.edu/phy115a/ 6/2/14 1

More information

Handout 11: AC circuit. AC generator

Handout 11: AC circuit. AC generator Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For

More information

Vibrations and Waves Physics Year 1. Handout 1: Course Details

Vibrations and Waves Physics Year 1. Handout 1: Course Details Vibrations and Waves Jan-Feb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office

More information

Linear Second-Order Differential Equations LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS

Linear Second-Order Differential Equations LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS 11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS A Spring with Friction: Damped Oscillations The differential equation, which we used to describe the motion of a spring, disregards friction. But there

More information

The object of this experiment is to study systems undergoing simple harmonic motion.

The object of this experiment is to study systems undergoing simple harmonic motion. Chapter 9 Simple Harmonic Motion 9.1 Purpose The object of this experiment is to study systems undergoing simple harmonic motion. 9.2 Introduction This experiment will develop your ability to perform calculations

More information

Thursday, August 4, 2011

Thursday, August 4, 2011 Chapter 16 Thursday, August 4, 2011 16.1 Springs in Motion: Hooke s Law and the Second-Order ODE We have seen alrealdy that differential equations are powerful tools for understanding mechanics and electro-magnetism.

More information

EM Oscillations. David J. Starling Penn State Hazleton PHYS 212

EM Oscillations. David J. Starling Penn State Hazleton PHYS 212 I ve got an oscillating fan at my house. The fan goes back and forth. It looks like the fan is saying No. So I like to ask it questions that a fan would say no to. Do you keep my hair in place? Do you

More information

Chapter 7 Hooke s Force law and Simple Harmonic Oscillations

Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Hooke s Law An empirically derived relationship that approximately works for many materials over a limited range. Exactly true for a massless,

More information

Physics 141, Lecture 7. Outline. Course Information. Course information: Homework set # 3 Exam # 1. Quiz. Continuation of the discussion of Chapter 4.

Physics 141, Lecture 7. Outline. Course Information. Course information: Homework set # 3 Exam # 1. Quiz. Continuation of the discussion of Chapter 4. Physics 141, Lecture 7. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 07, Page 1 Outline. Course information: Homework set # 3 Exam # 1 Quiz. Continuation of the

More information

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying

More information

Supplemental Notes on Complex Numbers, Complex Impedance, RLC Circuits, and Resonance

Supplemental Notes on Complex Numbers, Complex Impedance, RLC Circuits, and Resonance Supplemental Notes on Complex Numbers, Complex Impedance, RLC Circuits, and Resonance Complex numbers Complex numbers are expressions of the form z = a + ib, where both a and b are real numbers, and i

More information

Chapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion:

Chapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion: Chapter 14 Oscillations Oscillations Introductory Terminology Simple Harmonic Motion: Kinematics Energy Examples of Simple Harmonic Oscillators Damped and Forced Oscillations. Resonance. Periodic Motion

More information

Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations Applications of Second-Order Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition

More information

In the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as

In the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as 2 MODELING Once the control target is identified, which includes the state variable to be controlled (ex. speed, position, temperature, flow rate, etc), and once the system drives are identified (ex. force,

More information

Oscillations and Waves

Oscillations and Waves Oscillations and Waves Somnath Bharadwaj and S. Pratik Khastgir Department of Physics and Meteorology IIT Kharagpur Module : Oscillations Lecture : Oscillations Oscillations are ubiquitous. It would be

More information

Thursday March 30 Topics for this Lecture: Simple Harmonic Motion Kinetic & Potential Energy Pendulum systems Resonances & Damping.

Thursday March 30 Topics for this Lecture: Simple Harmonic Motion Kinetic & Potential Energy Pendulum systems Resonances & Damping. Thursday March 30 Topics for this Lecture: Simple Harmonic Motion Kinetic & Potential Energy Pendulum systems Resonances & Damping Assignment 11 due Friday Pre-class due 15min before class Help Room: Here,

More information

Selected Topics in Physics a lecture course for 1st year students by W.B. von Schlippe Spring Semester 2007

Selected Topics in Physics a lecture course for 1st year students by W.B. von Schlippe Spring Semester 2007 Selected Topics in Physics a lecture course for st year students by W.B. von Schlippe Spring Semester 7 Lecture : Oscillations simple harmonic oscillations; coupled oscillations; beats; damped oscillations;

More information

Updated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum

Updated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are

More information

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring

More information

MATH 135: COMPLEX NUMBERS

MATH 135: COMPLEX NUMBERS MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex

More information

Physics 9 Friday, April 18, 2014

Physics 9 Friday, April 18, 2014 Physics 9 Friday, April 18, 2014 Turn in HW12. I ll put HW13 online tomorrow. For Monday: read all of Ch33 (optics) For Wednesday: skim Ch34 (wave optics) I ll hand out your take-home practice final exam

More information

1 Simple Harmonic Oscillator

1 Simple Harmonic Oscillator Physics 1a Waves Lecture 3 Caltech, 10/09/18 1 Simple Harmonic Oscillator 1.4 General properties of Simple Harmonic Oscillator 1.4.4 Superposition of two independent SHO Suppose we have two SHOs described

More information

Oscillations Simple Harmonic Motion

Oscillations Simple Harmonic Motion Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and

More information

3 + 4i 2 + 3i. 3 4i Fig 1b

3 + 4i 2 + 3i. 3 4i Fig 1b The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of

More information

z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1)

z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) 11 Complex numbers Read: Boas Ch. Represent an arb. complex number z C in one of two ways: z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) Here i is 1, engineers call

More information

Electric Circuit Theory

Electric Circuit Theory Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 8 Natural and Step Responses of RLC Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 8.1 Introduction to the Natural Response

More information

Mathematical Review for AC Circuits: Complex Number

Mathematical Review for AC Circuits: Complex Number Mathematical Review for AC Circuits: Complex Number 1 Notation When a number x is real, we write x R. When a number z is complex, we write z C. Complex conjugate of z is written as z here. Some books use

More information

Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits

Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits Module 24: Undriven RLC Circuits 1 Module 24: Outline Undriven RLC Circuits Expt. 8: Part 2:Undriven RLC Circuits 2 Circuits that Oscillate (LRC) 3 Mass on a Spring: Simple Harmonic Motion (Demonstration)

More information

Physics 132 3/31/17. March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman. Swing, Swing, Swing. Cartoon: Bill Watterson

Physics 132 3/31/17. March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman. Swing, Swing, Swing. Cartoon: Bill Watterson March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman Swing, Swing, Swing Cartoon: Bill Watterson Calvin & Hobbes 1 Outline The makeup exam Recap: the math of the harmonic oscillator

More information

The Pendulum Plain and Puzzling

The Pendulum Plain and Puzzling The Pendulum Plain and Puzzling Chris Sangwin School of Mathematics University of Edinburgh April 2017 Chris Sangwin (University of Edinburgh) Pendulum April 2017 1 / 38 Outline 1 Introduction and motivation

More information

Physics GRE Practice

Physics GRE Practice Physics GRE Practice Chapter 3: Harmonic Motion in 1-D Harmonic Motion occurs when the acceleration of a system is proportional to the negative of its displacement. or a x (1) ẍ x (2) Examples of Harmonic

More information

Vibrations and waves: revision. Martin Dove Queen Mary University of London

Vibrations and waves: revision. Martin Dove Queen Mary University of London Vibrations and waves: revision Martin Dove Queen Mary University of London Form of the examination Part A = 50%, 10 short questions, no options Part B = 50%, Answer questions from a choice of 4 Total exam

More information

P441 Analytical Mechanics - I. RLC Circuits. c Alex R. Dzierba. In this note we discuss electrical oscillating circuits: undamped, damped and driven.

P441 Analytical Mechanics - I. RLC Circuits. c Alex R. Dzierba. In this note we discuss electrical oscillating circuits: undamped, damped and driven. Lecture 10 Monday - September 19, 005 Written or last updated: September 19, 005 P441 Analytical Mechanics - I RLC Circuits c Alex R. Dzierba Introduction In this note we discuss electrical oscillating

More information

Chapter 14 (Oscillations) Key concept: Downloaded from

Chapter 14 (Oscillations) Key concept: Downloaded from Chapter 14 (Oscillations) Multiple Choice Questions Single Correct Answer Type Q1. The displacement of a particle is represented by the equation. The motion of the particle is (a) simple harmonic with

More information

C R. Consider from point of view of energy! Consider the RC and LC series circuits shown:

C R. Consider from point of view of energy! Consider the RC and LC series circuits shown: ircuits onsider the R and series circuits shown: ++++ ---- R ++++ ---- Suppose that the circuits are formed at t with the capacitor charged to value. There is a qualitative difference in the time development

More information

Work sheet / Things to know. Chapter 3

Work sheet / Things to know. Chapter 3 MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients

More information

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered

More information

Physics 4 Spring 1989 Lab 5 - AC Circuits

Physics 4 Spring 1989 Lab 5 - AC Circuits Physics 4 Spring 1989 Lab 5 - AC Circuits Theory Consider the series inductor-resistor-capacitor circuit shown in figure 1. When an alternating voltage is applied to this circuit, the current and voltage

More information

WPI Physics Dept. Intermediate Lab 2651 Free and Damped Electrical Oscillations

WPI Physics Dept. Intermediate Lab 2651 Free and Damped Electrical Oscillations WPI Physics Dept. Intermediate Lab 2651 Free and Damped Electrical Oscillations Qi Wen March 11, 2017 Index: 1. Background Material, p. 2 2. Pre-lab Exercises, p. 4 3. Report Checklist, p. 6 4. Appendix,

More information

Lecture 18. In other words, if you double the stress, you double the resulting strain.

Lecture 18. In other words, if you double the stress, you double the resulting strain. Lecture 18 Stress and Strain and Springs Simple Harmonic Motion Cutnell+Johnson: 10.1-10.4,10.7-10.8 Stress and Strain and Springs So far we ve dealt with rigid objects. A rigid object doesn t change shape

More information

CHAPTER 12 OSCILLATORY MOTION

CHAPTER 12 OSCILLATORY MOTION CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time

More information

!T = 2# T = 2! " The velocity and acceleration of the object are found by taking the first and second derivative of the position:

!T = 2# T = 2!  The velocity and acceleration of the object are found by taking the first and second derivative of the position: A pendulum swinging back and forth or a mass oscillating on a spring are two examples of (SHM.) SHM occurs any time the position of an object as a function of time can be represented by a sine wave. We

More information

Chapter 6. Second order differential equations

Chapter 6. Second order differential equations Chapter 6. Second order differential equations A second order differential equation is of the form y = f(t, y, y ) where y = y(t). We shall often think of t as parametrizing time, y position. In this case

More information

Mass on a Horizontal Spring

Mass on a Horizontal Spring Course- B.Sc. Applied Physical Science (Computer Science) Year- IInd, Sem- IVth Subject Physics Paper- XIVth, Electromagnetic Theory Lecture No. 22, Simple Harmonic Motion Introduction Hello friends in

More information

Math 211. Substitute Lecture. November 20, 2000

Math 211. Substitute Lecture. November 20, 2000 1 Math 211 Substitute Lecture November 20, 2000 2 Solutions to y + py + qy =0. Look for exponential solutions y(t) =e λt. Characteristic equation: λ 2 + pλ + q =0. Characteristic polynomial: λ 2 + pλ +

More information

AC Circuits III. Physics 2415 Lecture 24. Michael Fowler, UVa

AC Circuits III. Physics 2415 Lecture 24. Michael Fowler, UVa AC Circuits III Physics 415 Lecture 4 Michael Fowler, UVa Today s Topics LC circuits: analogy with mass on spring LCR circuits: damped oscillations LCR circuits with ac source: driven pendulum, resonance.

More information

MAT01B1: Separable Differential Equations

MAT01B1: Separable Differential Equations MAT01B1: Separable Differential Equations Dr Craig 3 October 2018 My details: acraig@uj.ac.za Consulting hours: Tomorrow 14h40 15h25 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/

More information

Physics 142 AC Circuits Page 1. AC Circuits. I ve had a perfectly lovely evening but this wasn t it. Groucho Marx

Physics 142 AC Circuits Page 1. AC Circuits. I ve had a perfectly lovely evening but this wasn t it. Groucho Marx Physics 142 A ircuits Page 1 A ircuits I ve had a perfectly lovely evening but this wasn t it. Groucho Marx Alternating current: generators and values It is relatively easy to devise a source (a generator

More information

Inductance, RL Circuits, LC Circuits, RLC Circuits

Inductance, RL Circuits, LC Circuits, RLC Circuits Inductance, R Circuits, C Circuits, RC Circuits Inductance What happens when we close the switch? The current flows What does the current look like as a function of time? Does it look like this? I t Inductance

More information

Resonance and response

Resonance and response Chapter 2 Resonance and response Last updated September 20, 2008 In this section of the course we begin with a very simple system a mass hanging from a spring and see how some remarkable ideas emerge.

More information

4. Complex Oscillations

4. Complex Oscillations 4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic

More information

1 Oscillations MEI Conference 2009

1 Oscillations MEI Conference 2009 1 Oscillations MEI Conference 2009 Some Background Information There is a film clip you can get from Youtube of the Tacoma Narrows Bridge called Galloping Gertie. This shows vibrations in the bridge increasing

More information

Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2

Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2 Module 25: Driven RLC Circuits 1 Module 25: Outline Resonance & Driven LRC Circuits 2 Driven Oscillations: Resonance 3 Mass on a Spring: Simple Harmonic Motion A Second Look 4 Mass on a Spring (1) (2)

More information

Complex Numbers and the Complex Exponential

Complex Numbers and the Complex Exponential Complex Numbers and the Complex Exponential φ (2+i) i 2 θ φ 2+i θ 1 2 1. Complex numbers The equation x 2 + 1 0 has no solutions, because for any real number x the square x 2 is nonnegative, and so x 2

More information

dx n a 1(x) dy

dx n a 1(x) dy HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)

More information

Physics 2112 Unit 19

Physics 2112 Unit 19 Physics 11 Unit 19 Today s oncepts: A) L circuits and Oscillation Frequency B) Energy ) RL circuits and Damping Electricity & Magnetism Lecture 19, Slide 1 Your omments differential equations killing me.

More information

Physics 1C. Lecture 12B

Physics 1C. Lecture 12B Physics 1C Lecture 12B SHM: Mathematical Model! Equations of motion for SHM:! Remember, simple harmonic motion is not uniformly accelerated motion SHM: Mathematical Model! The maximum values of velocity

More information

T 2. Key take-aways: k m. Tuesday March 28

T 2. Key take-aways: k m. Tuesday March 28 Tuesday March 28 Topics for this Lecture: Simple Harmonic Motion Periodic (a.k.a. repetitive) motion Hooke s Law Mass & spring system Assignment 11 due Friday Pre-class due 15min before class Help Room:

More information

HOMEWORK ANSWERS. Lesson 4.1: Simple Harmonic Motion

HOMEWORK ANSWERS. Lesson 4.1: Simple Harmonic Motion DEVIL PHYSICS HOMEWORK ANSWERS Tsokos, Chapter 3 Test Lesson 4.1: Simple Harmonic Motion 1. Objectives. By the end of this class you should be able to: a) Understand that in simple harmonic motion there

More information

Differential Equations

Differential Equations Differential Equations A differential equation (DE) is an equation which involves an unknown function f (x) as well as some of its derivatives. To solve a differential equation means to find the unknown

More information

PHYSICS - CLUTCH CH 15: PERIODIC MOTION (NEW)

PHYSICS - CLUTCH CH 15: PERIODIC MOTION (NEW) !! www.clutchprep.com CONCEPT: Hooke s Law & Springs When you push/pull against a spring (FA), spring pushes back in the direction. (Action-Reaction!) Fs = FA = Ex. 1: You push on a spring with a force

More information

1 (2n)! (-1)n (θ) 2n

1 (2n)! (-1)n (θ) 2n Complex Numbers and Algebra The real numbers are complete for the operations addition, subtraction, multiplication, and division, or more suggestively, for the operations of addition and multiplication

More information

Parametric Resonance and Elastic Pendulums

Parametric Resonance and Elastic Pendulums Parametric Resonance and Elastic Pendulums Ravitej Uppu Abstract In this I try to extend the theoretical conception of Elastic Pendulum that can be explained by the Driven Pendulums that I presented during

More information

Inductance, RL and RLC Circuits

Inductance, RL and RLC Circuits Inductance, RL and RLC Circuits Inductance Temporarily storage of energy by the magnetic field When the switch is closed, the current does not immediately reach its maximum value. Faraday s law of electromagnetic

More information

PHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc. PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 15 Lecture RANDALL D. KNIGHT Chapter 15 Oscillations IN THIS CHAPTER, you will learn about systems that oscillate in simple harmonic

More information

1 Pushing your Friend on a Swing

1 Pushing your Friend on a Swing Massachusetts Institute of Technology MITES 017 Physics III Lecture 05: Driven Oscillations In these notes, we derive the properties of both an undamped and damped harmonic oscillator under the influence

More information

L = 1 2 a(q) q2 V (q).

L = 1 2 a(q) q2 V (q). Physics 3550 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6, 11.1 11.3 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion

More information

Chapter 15 - Oscillations

Chapter 15 - Oscillations The pendulum of the mind oscillates between sense and nonsense, not between right and wrong. -Carl Gustav Jung David J. Starling Penn State Hazleton PHYS 211 Oscillatory motion is motion that is periodic

More information

Introductory Physics. Week 2015/05/29

Introductory Physics. Week 2015/05/29 2015/05/29 Part I Summary of week 6 Summary of week 6 We studied the motion of a projectile under uniform gravity, and constrained rectilinear motion, introducing the concept of constraint force. Then

More information

Physics 294H. lectures will be posted frequently, mostly! every day if I can remember to do so

Physics 294H. lectures will be posted frequently, mostly! every day if I can remember to do so Physics 294H l Professor: Joey Huston l email:huston@msu.edu l office: BPS3230 l Homework will be with Mastering Physics (and an average of 1 hand-written problem per week) Help-room hours: 12:40-2:40

More information

Chapter 10: Sinusoidal Steady-State Analysis

Chapter 10: Sinusoidal Steady-State Analysis Chapter 10: Sinusoidal Steady-State Analysis 1 Objectives : sinusoidal functions Impedance use phasors to determine the forced response of a circuit subjected to sinusoidal excitation Apply techniques

More information

Vibrations and Waves MP205, Assignment 4 Solutions

Vibrations and Waves MP205, Assignment 4 Solutions Vibrations and Waves MP205, Assignment Solutions 1. Verify that x = Ae αt cos ωt is a possible solution of the equation and find α and ω in terms of γ and ω 0. [20] dt 2 + γ dx dt + ω2 0x = 0, Given x

More information

Physics Mechanics. Lecture 32 Oscillations II

Physics Mechanics. Lecture 32 Oscillations II Physics 170 - Mechanics Lecture 32 Oscillations II Gravitational Potential Energy A plot of the gravitational potential energy U g looks like this: Energy Conservation Total mechanical energy of an object

More information

11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion

11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion 11/17/10 Chapter 14. Oscillations This striking computergenerated image demonstrates an important type of motion: oscillatory motion. Examples of oscillatory motion include a car bouncing up and down,

More information

APPPHYS 217 Tuesday 6 April 2010

APPPHYS 217 Tuesday 6 April 2010 APPPHYS 7 Tuesday 6 April Stability and input-output performance: second-order systems Here we present a detailed example to draw connections between today s topics and our prior review of linear algebra

More information

Exam Question 6/8 (HL/OL): Circular and Simple Harmonic Motion. February 1, Applied Mathematics: Lecture 7. Brendan Williamson.

Exam Question 6/8 (HL/OL): Circular and Simple Harmonic Motion. February 1, Applied Mathematics: Lecture 7. Brendan Williamson. in a : Exam Question 6/8 (HL/OL): Circular and February 1, 2017 in a This lecture pertains to material relevant to question 6 of the paper, and question 8 of the Ordinary Level paper, commonly referred

More information

Lecture XXVI. Morris Swartz Dept. of Physics and Astronomy Johns Hopkins University November 5, 2003

Lecture XXVI. Morris Swartz Dept. of Physics and Astronomy Johns Hopkins University November 5, 2003 Lecture XXVI Morris Swartz Dept. of Physics and Astronomy Johns Hopins University morris@jhu.edu November 5, 2003 Lecture XXVI: Oscillations Oscillations are periodic motions. There are many examples of

More information

Wave Phenomena Physics 15c. Lecture 11 Dispersion

Wave Phenomena Physics 15c. Lecture 11 Dispersion Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed

More information

Complex Numbers Review

Complex Numbers Review Complex Numbers view ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The real numbers (denoted R) are incomplete

More information

Homework #5 Solutions

Homework #5 Solutions Homework #5 Solutions Math 123: Mathematical Modeling, Spring 2019 Instructor: Dr. Doreen De Leon 1. Exercise 7.2.5. Stefan-Boltzmann s Law of Radiation states that the temperature change dt/ of a body

More information

Announcements: Today: more AC circuits

Announcements: Today: more AC circuits Announcements: Today: more AC circuits I 0 I rms Current through a light bulb I 0 I rms I t = I 0 cos ωt I 0 Current through a LED I t = I 0 cos ωt Θ(cos ωt ) Theta function (is zero for a negative argument)

More information

L = 1 2 a(q) q2 V (q).

L = 1 2 a(q) q2 V (q). Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion

More information

LAB 10: HARMONIC MOTION AND THE PENDULUM

LAB 10: HARMONIC MOTION AND THE PENDULUM 163 Name Date Partners LAB 10: HARMONIC MOION AND HE PENDULUM Galileo reportedly began his study of the pendulum in 1581 while watching this chandelier swing in Pisa, Italy OVERVIEW A body is said to be

More information

Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis

Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass is attached to one end of a spring. The other end of the spring is attached to

More information

11. The Series RLC Resonance Circuit

11. The Series RLC Resonance Circuit Electronicsab.nb. The Series RC Resonance Circuit Introduction Thus far we have studied a circuit involving a () series resistor R and capacitor C circuit as well as a () series resistor R and inductor

More information