Section 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).


 Vincent Parrish
 1 years ago
 Views:
Transcription
1 Difference Equations to Differential Equations Section 8.5 Applications: Pendulums MassSpring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider the motion of a pendulum, a prolem originally mentioned in Section. in connection with the trigonometric functions. Second, we will discuss the motion of an oject virating at the end of a spring. The motion of a pendulum Consider a pendulum consisting of a o of mass m at the end of a rigid rod of length. We will assume that the mass of the rod is negligile in comparison with the mass of the o. Let x(t e the angle etween the rod the vertical at time t, with x(t > 0 for angles measured in the counterclockwise direction x(t < 0 for angles measured in the clockwise direction. See Figure Suppose the o is pulled through an angle α then released. That is, suppose our initial conditions are x(0 = α ẋ(0 = 0. If we view the motion of the pendulum in the complex plane, with the real axis vertical, positive direction downward, the imaginary axis horizontal, positive direction to the right, then the position of the o at time t is given y z(t = e ix(t. (8.5.1 x(t Figure A pendulum Then we have ż = iẋe ix (8.5. z = ẋ e ix + iẍe ix = ẋ (cos(x + i sin(x + iẍ(cos(x + i sin(x (8.5.3 = ( ẋ cos(x ẍ sin(x + i( ẋ sin(x + ẍ cos(x. 1 Copyright c y Dan Sloughter 000
2 Applications: Pendulums MassSpring Systems Section 8.5 Now z is the acceleration of the pendulum, so m z must e equal to the force of gravity acting on the o, namely, a force of magnitude mg acting in the downward direction, the direction of the positive real axis. Hence we must have g = z, that is, g = ( ẋ cos(x ẍ sin(x + i( ẋ sin(x + ẍ cos(x. (8.5.4 Equating the real imaginary parts of the two sides of (8.5.4 gives us g = ẋ cos(x ẍ sin(x ( = ẋ sin(x + ẍ cos(x. (8.5.6 Multiplying (8.5.5 y sin(x (8.5.6 y cos(x gives us g sin(x = ẋ cos(x sin(x + ẍ sin (x ( = ẋ sin(x cos(x + ẍ cos (x. (8.5.8 Adding (8.5.7 (8.5.8 together yields g sin(x = ẍ(sin (x + cos (x = ẍ. (8.5.9 Thus ẍ = g sin(x. ( So we have reduced the prolem of descriing the motion of the pendulum to the prolem of solving the second order differential equation ( suject to the initial conditions x(0 = α ẋ(0 = 0. Unfortunately, this equation is not linear. In fact, it is not possile to find a closed form solution for this equation. In Section 8.6 we will discuss how to study this equation using numerical approximations, ut for now we will take a different approach to finding an approximate solution. Since we know sin(x = x + o(x ( from our work on est affine approximations in Chapter, it is reasonale to replace sin(x y x for small values of x. Hence, if we restrict to the case where α is small, we may replace ( y the linear equation ẍ = g x. (8.5.1 Since this equation is homogeneous with constant coefficients, we may solve it using the techniques of Section 8.4. Specifically, the characteristic equation for this equation is k + g = 0, (8.5.13
3 Section 8.5 Applications: Pendulums MassSpring Systems Figure 8.5. Motion of a pendulum which has roots Hence the general solution is Then g k 1 = i g k = i ( ( ( g ( x = c 1 cos t g + c sin t. ( ẋ = c 1 g sin ( g t + c g cos ( g t, ( so x(0 = c 1 ẋ(0 = c g. Hence the initial conditions x(0 = α ẋ(0 = 0 imply c 1 = α c = 0. Thus ( g x = α cos t ( The graph of x for the case = 1 meter α = 0.1 radians, in which case we use g = 9.8 meters per second per second, is shown in Figure One consequence of ( is that the period of the motion, that is, the time it takes the o to make one complete oscillation, is π g = π g, ( independent of the value of α. Of course, we are working under the approximation sin(x x, so ( is actually only an approximation of the period. Nevertheless,
4 4 Applications: Pendulums MassSpring Systems Section 8.5 the approximation is very good for small oscillations is the reason pendulums were used to measure time in early clocks. Virations in mechanical systems: massspring systems In this example we consider the motion of an oject of mass m suspended on a spring, as shown in Figure We will measure the position of the oject along a vertical axis, with the equilirium position at 0 the positive direction downward. Let x(t denote the position of the oject at time t suppose the oject is released from rest at position x 0. That is, we suppose that x(0 = x 0 ẋ(0 = 0. If we ignore any damping forces, such as resistance to the motion due to the surrounding medium, such as air or oil, then the only forces acting on the oject are the force of gravity, contriuting a term of mg, the restorative force of the spring, given, according to Hooke s law, y kl for some constant k > 0, where l is the amount the spring is stretched or compressed from its natural length. If we let l e the amount the spring is stretched when the oject is at the equilirium position, that is, when x = 0, then at any time the spring is stretched or compressed y x + l. Thus at any time t the force acting on the oject is F = mg k(x + l. (8.5.0 x = 0 Figure Mass on a spring at equilirium In particular, if the oject is at rest at its equilirium position, then oth x = 0 F = 0. Hence 0 = mg k l, (8.5.1 so mg = k l. (8.5. Thus (8.5.0 simplifies to F = kx. Applying Newton s second law of motion, we have mẍ = kx, (8.5.3 from which we otain ẍ = k x. (8.5.4 m
5 Section 8.5 Applications: Pendulums MassSpring Systems Figure Motion of a massspring system without damping This equation is of the same form as the equation derived aove for approximating the motion of a pendulum. Hence, using the same reasoning, the solution is ( k x = x 0 cos m t. (8.5.5 The graph of x for k = 10, m = 5, x 0 = is shown in Figure Notice that the period of the motion is T = π k m m = π k. (8.5.6 The frequency of the motion, that is, the numer of complete oscillations in one unit of time, is f = 1 T = 1 k π m. (8.5.7 Hence for a fixed mass, increasing the spring constant, that is, increasing the stiffness of the spring, decreases the period increases the frequency; for a fixed spring constant, increasing the mass increases the period decreases the frequency. Now suppose there is a damping force, a force resisting the motion of the oject, which is proportional to the velocity. This adds an additional term of cẋ, where c is a positive constant, to the force acting on the oject, giving us F = kx cẋ. Thus mẍ = kx cẋ, (8.5.8 so ẍ + c mẋ + k m x = 0 (8.5.9
6 6 Applications: Pendulums MassSpring Systems Section 8.5 replaces (8.5.4 as the equation descriing the motion of the oject. notation, we will let = c m Then our differential equation ecomes a = k m. To simplify the ẍ + ẋ + a x = 0, ( with characteristic equation (using s for the variale s + s + a = 0. ( Hence the roots of the characteristic equation are s 1 = 4 4a = a (8.5.3 s = + 4 4a = + a. ( Thus the ehavior of the system depends on whether a > 0, a = 0, or a < 0. Equivalently, since a = c 4m k m, the ehavior of the system depends on whether c > 4mk, c = 4mk, or c < 4mk. In the first case the system is said to e overdamped, in the second it is critically damped, in the third it is underdamped. First consider the overdamped case a > 0. In this case the characteristic equation has distinct real roots, so the general solution is x = c 1 e s 1t + c e s t. ( Now ẋ = c 1 s 1 e s 1t + c s e s t, ( so x(0 = c 1 + c ẋ(0 = c 1 s 1 + c s. Hence the initial conditions, x(0 = x 0 ẋ(0 = 0, give us x 0 = c 1 + c 0 = c 1 s 1 + c s.
7 Section 8.5 Applications: Pendulums MassSpring Systems Figure Motion of an overdamped massspring system Multiplying the first equation y s 1 sutracting from the second gives us Hence Thus Now > 0 > a, so Hence x 0 s 1 = c (s s 1. c = x 0s 1 s s 1 c 1 = x 0 c = x 0(s s 1 s s 1 + x 0s 1 s s 1 = x 0s s s 1. It follows that e s t > e s 1t, s s 1 > 0, x = x 0 s s 1 (s e s 1t s 1 e s t. ( s = + a < 0. s 1 < s < 0. ( s e s 1t s 1 e s t > s e s t s 1 e s t = e s t (s s 1 > 0 for all t 0. Hence if x 0 < 0, then x(t < 0 for all t 0, if x 0 > 0, then x(t > 0 for all t > 0. Comining this with lim x(t = 0, ( t we see that in this case the system does not oscillate at all. After release, the oject simply returns to the equilirium position. Figure shows this ehavior for k = 10, m = 5, c = 0, x 0 =.
8 8 Applications: Pendulums MassSpring Systems Section Figure Motion of a critically damped massspring system Next consider the case when a = 0. In this case the characteristic equation has only one real root, s 1 = s =, so the general solution is x = c 1 e t + c te t. ( Then ẋ = c 1 e t c te t + c e t, ( so x(0 = c 1 ẋ(0 = c 1 + c. Hence the initial conditions, x(0 = x 0 ẋ(0 = 0, give us c 1 = x 0 c = x 0. Thus Equivalently, since = Now for any t 0, x = x 0 e t + x 0 te t = x 0 e t (1 + t. ( c m, x = x 0 e c m t( c m t > 0. c m t. (8.5.4 Hence, as in the overdamped case, the system does not oscillate. Once released, the oject moves ack to the equilirium position without ever crossing it. Figure shows this ehavior for k = 10, m = 5, c = 10, x 0 =. This motion is said to e critically damped ecause any increase in c results in overdamped motion, while any decrease in c results in underdamped motion, which we consider next. Finally, consider the case when a < 0. The roots of the characteristic equation are now s 1 = a = i a ( s = + a = + i a (8.5.44
9 Section 8.5 Applications: Pendulums MassSpring Systems 9 If we let α = a, then the general solution is Then x = e t (c 1 cos(αt + c sin(αt. ( ẋ = e t ( αc 1 sin(αt + αc cos(αt e t (c 1 cos(αt + c sin(αt, ( so x(0 = c 1 ẋ(0 = αc c 1. Hence the initial conditions, x(0 = x 0 ẋ(0 = 0, imply that c 1 = x 0 Thus c = x 0 α. x = e t (x 0 cos(αt + x 0 α sin(αt = x 0 α e t (α cos(αt + sin(αt. ( This expression simplifies somewhat if we introduce the angle Then ( θ = tan 1. ( α cos(θ = α α + sin(θ = α +. Moreover, since α = a, Hence x = x 0 α + α α + = (a + = a = k m. ( e t α α + cos(αt + = x 0 k α m e t (cos(θ cos(αt + sin(θ sin(αt. α + sin(αt Using the angle sutraction formula for cosine, this ecomes x = x 0 k α m e t cos(αt θ. ( The presence of the cosine factor in this expression shows us that, even though we still have lim x(t = 0, t
10 10 Applications: Pendulums MassSpring Systems Section Figure Motion of an underdamped massspring system the underdamped massspring system will oscillate aout the equilirium position with a decreasing amplitude of x 0 k α m e t. ( Figure shows this ehavior for k = 10, m = 5, c = 5, x 0 =. Prolems 1. In an experiment to determine g, a pendulum of length 50 centimeters is oserved to have a period of oscillation of 1.4 seconds. Approximate g ased on this oservation.. The period of oscillation of a pendulum of length given in ( is, as mentioned, only an approximation of the true period. It can e shown that the true period of a pendulum released from an angle α is given y T = 4 g π 0 1 dφ, 1 k sin (φ where 0 < α < π k = sin ( α. (a Find the period of oscillation for a pendulum of length 50 centimeters for α = π 4, α = π 6, α = π 50, α = π 100. Compare these results with the approximation given in ( ( Graph T as a function of α for π 4 α π 4. For comparison, also plot the horizontal line T = π g. 3. Consider a massspring system with x 0 = 10, ẋ(0 = 0, k = 10, m = 10. Plot x(t for c = 0, c = 5, c = 10, c = 0, c = 5, c = 30. Identify each motion as overdamped, critically damped, underdamped, or undamped.
11 Section 8.5 Applications: Pendulums MassSpring Systems Consider a massspring system with x 0 = 10, ẋ(0 = 0, m = 10, c = 0. Plot x(t for k =, k = 5, k = 10, k = 15. Identify each motion as overdamped, critically damped, underdamped, or undamped. 5. Consider the underdamped motion of a massspring system expressed in ( (a Show that the maximum values of x(t occur at t = 0, T, T,..., where T = π. k m c 4m Note that when c = 0, T reduces to the period of the motion for the massspring system without damping. ( Show that if x 1 x are two successive maximum values of x(t, then x 1 x = e ct m. 6. Inside the earth, the force of gravity acting on an oject is proportional to the distance etween the oject the center of the earth. (a Suppose a hole is drilled through the earth from pole to pole a rock is dropped into the hole. If x(t is the distance from the oject to the center of the earth at time t, show that, ignoring any resistive forces, where R is the radius of the earth. ( g x = R cos R t, ( How long, in minutes, does it take for the rock to make one complete trip from pole to pole ack? Use R = 3950 miles. (c What is the velocity of the rock, in miles per hour, when it reaches the center of the earth?
Unforced Mechanical Vibrations
Unforced Mechanical Vibrations Today we begin to consider applications of second order ordinary differential equations. 1. SpringMass Systems 2. Unforced Systems: Damped Motion 1 SpringMass Systems We
More informationMath Assignment 5
Math 2280  Assignment 5 Dylan Zwick Fall 2013 Section 3.41, 5, 18, 21 Section 3.51, 11, 23, 28, 35, 47, 56 Section 3.61, 2, 9, 17, 24 1 Section 3.4  Mechanical Vibrations 3.4.1  Determine the period
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A
More information5.6 Unforced Mechanical Vibrations
5.6 Unforced Mechanical Vibrations 215 5.6 Unforced Mechanical Vibrations The study of vibrating mechanical systems begins here with examples for unforced systems with one degree of freedom. The main example
More informationragsdale (zdr82) HW7 ditmire (58335) 1 The magnetic force is
ragsdale (zdr8) HW7 ditmire (585) This printout should have 8 questions. Multiplechoice questions ma continue on the net column or page find all choices efore answering. 00 0.0 points A wire carring
More informationAPPLICATIONS OF SECONDORDER DIFFERENTIAL EQUATIONS
APPLICATIONS OF SECONDORDER DIFFERENTIAL EQUATIONS Secondorder linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationOscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance
Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum
More informationOscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Massspring system Energy in SHM Pendulums
PHYS 101 Previous Exam Problems CHAPTER 15 Oscillations Simple harmonic motion Massspring system Energy in SHM Pendulums 1. The displacement of a particle oscillating along the x axis is given as a function
More informationOscillations. 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 18. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationEquations of motion for the Pendulum and AugmentedReality Pendulum sketches
Equations of motion for the Pendulum and AugmentedReality Pendulum sketches Ludovico Carbone Issue: Date: February 3, 20 School of Physics and Astronomy University of Birmingham Birmingham, B5 2TT Introduction
More informationSection 7.3 Double Angle Identities
Section 7.3 Double Angle Identities 3 Section 7.3 Double Angle Identities Two special cases of the sum of angles identities arise often enough that we choose to state these identities separately. Identities
More informationOscillations 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 informationRobot Position from Wheel Odometry
Root Position from Wheel Odometry Christopher Marshall 26 Fe 2008 Astract This document develops equations of motion for root position as a function of the distance traveled y each wheel as a function
More informationChapter 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 informationVibrations: Second Order Systems with One Degree of Freedom, Free Response
Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single
More information3 Forces and pressure Answer all questions and show your working out for maximum credit Time allowed : 30 mins Total points available : 32
1 3 Forces and pressure Answer all questions and show your working out for maximum credit Time allowed : 30 mins Total points availale : 32 Core curriculum 1 A icycle pump has its outlet sealed with a
More informationChapter 1. Harmonic Oscillator. 1.1 Energy Analysis
Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the massspring system. A mass is attached to one end of a spring. The other end of the spring is attached to
More informationP = ρ{ g a } + µ 2 V II. FLUID STATICS
II. FLUID STATICS From a force analysis on a triangular fluid element at rest, the following three concepts are easily developed: For a continuous, hydrostatic, shear free fluid: 1. Pressure is constant
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx When the mass is released, the spring will pull
More informationLAST TIME: Simple Pendulum:
LAST TIME: Simple Pendulum: The displacement from equilibrium, x is the arclength s = L. s / L x / L Accelerating & Restoring Force in the tangential direction, taking cw as positive initial displacement
More informationUndamped Free Vibrations (Simple Harmonic Motion; SHM also called Simple Harmonic Oscillator)
Section 3. 7 MassSpring Systems (no damping) Key Terms/ Ideas: Hooke s Law of Springs Undamped Free Vibrations (Simple Harmonic Motion; SHM also called Simple Harmonic Oscillator) Amplitude Natural Frequency
More informationLab 14  Simple Harmonic Motion and Oscillations on an Incline
Lab 14  Simple Harmonic Motion and Oscillations on an Incline Name I. Introduction/Theory Partner s Name The purpose of this lab is to measure the period of oscillation of a spring and mass system on
More informationLab M5: Hooke s Law and the Simple Harmonic Oscillator
M5.1 Lab M5: Hooke s Law and the Simple Harmonic Oscillator Most springs obey Hooke s Law, which states that the force exerted by the spring is proportional to the extension or compression of the spring
More informationChapter 11 Vibrations and Waves
Chapter 11 Vibrations and Waves 111 Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic.
More informationChapter 14. Oscillations and Resonance RESONANCE
Chapter 14 Oscillations and Resonance CHAPTER 14 RESONANCE OSCILLATIONS AND Oscillations and vibrations play a more significant role in our lives than we realize. When you strike a bell, the metal vibrates,
More informationHonors Algebra 2 Chapter 14 Page 1
Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using tbar method. A. Sine and Cosecant. y = sinθ y y y y 0   80   30 0 0 300 5 35 5 35 60 50 0
More informationAP Physics. Harmonic Motion. Multiple Choice. Test E
AP Physics Harmonic Motion Multiple Choice Test E A 0.10Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.
More information1 Systems of Differential Equations
March, 20 7 Systems of Differential Equations Let U e an open suset of R n, I e an open interval in R and : I R n R n e a function from I R n to R n The equation ẋ = ft, x is called a first order ordinary
More informationSection Mass Spring Systems
Asst. Prof. Hottovy SM212Section 3.1. Section 5.12 Mass Spring Systems Name: Purpose: To investigate the mass spring systems in Chapter 5. Procedure: Work on the following activity with 23 other students
More informationLab 11. SpringMass Oscillations
Lab 11. SpringMass Oscillations Goals To determine experimentally whether the supplied spring obeys Hooke s law, and if so, to calculate its spring constant. To find a solution to the differential equation
More informationSolution Derivations for Capa #12
Solution Derivations for Capa #12 1) A hoop of radius 0.200 m and mass 0.460 kg, is suspended by a point on it s perimeter as shown in the figure. If the hoop is allowed to oscillate side to side as a
More informationInvestigating Springs (Simple Harmonic Motion)
Investigating Springs (Simple Harmonic Motion) INTRODUCTION The purpose of this lab is to study the wellknown force exerted by a spring The force, as given by Hooke s Law, is a function of the amount
More informationChapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion
Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular
More information5.1: Angles and Radian Measure Date: PreCalculus
5.1: Angles and Radian Measure Date: PreCalculus *Use Section 5.1 (beginning on pg. 482) to complete the following Trigonometry: measurement of triangles An angle is formed by two rays that have a common
More informationChapter 07: Kinetic Energy and Work
Chapter 07: Kinetic Energy and Work Conservation of Energy is one of Nature s fundamental laws that is not violated. Energy can take on different forms in a given system. This chapter we will discuss work
More informationSolutions to Exam 2, Math 10560
Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If
More informationModeling and Experimentation: MassSpringDamper System Dynamics
Modeling and Experimentation: MassSpringDamper System Dynamics Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin July 20, 2014 Overview 1 This lab is meant to
More informationTest, Lesson 7 Waves  Answer Key Page 1
Test, Lesson 7 Waves  Answer Key Page 1 1. Match the proper units with the following: W. wavelength 1. nm F. frequency 2. /sec V. velocity 3. m 4. ms 1 5. Hz 6. m/sec (A) W: 1, 3 F: 2, 4, 5 V: 6 (B)
More informationPhysics A  PHY 2048C
Physics A  PHY 2048C Newton s Laws & Equations of 09/27/2017 My Office Hours: Thursday 2:003:00 PM 212 Keen Building Warmup Questions 1 In uniform circular motion (constant speed), what is the direction
More informationCopyright 2009, August E. Evrard.
Unless otherwise noted, the content of this course material is licensed under a Creative Commons BY 3.0 License. http://creativecommons.org/licenses/by/3.0/ Copyright 2009, August E. Evrard. You assume
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationChapter 5 Notes. 5.1 Using Fundamental Identities
Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx
More informationKEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM
KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered
More informationDesign Principles of Seismic Isolation
Design Principles of Seismic Isolation 3 George C. Lee and Zach Liang Multidisciplinary Center for Earthquake Engineering Research, University at Buffalo, State University of New York USA 1. Introduction
More informationHalldorson Honors Pre Calculus Name 4.1: Angles and Their Measures
.: Angles and Their Measures. Approximate each angle in terms of decimal degrees to the nearest ten thousandth. a. θ = 5 '5" b. θ = 5 8'. Approximate each angle in terms of degrees, minutes, and seconds
More informationChapter 2: Linear Constant Coefficient Higher Order Equations
Chapter 2: Linear Constant Coefficient Higher Order Equations The wave equation is a linear partial differential equation, which means that sums of solutions are still solutions, just as for linear ordinary
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1  TRIGONOMETRICAL GRAPHS
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1  TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply
More informationFor a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is
Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For
More informationHalldorson Honors Pre Calculus Name 4.1: Angles and Their Measures
Halldorson Honors Pre Calculus Name 4.1: Angles and Their Measures 1. Approximate each angle in terms of decimal degrees to the nearest ten thousandth. a. θ = 56 34'53" b. θ = 35 48'. Approximate each
More informationPhysics 210: Worksheet 26 Name:
(1) A all is floating in. If the density of the all is 0.95x10 kg/m, what percentage of the all is aove the? (2) An with a density of 19.x10 kg/m and a mass m50 kg is placed into a cylinder which contains
More informationThroughout this chapter you will need: pencil ruler protractor. 7.1 Relationship Between Sides in Rightangled. 13 cm 10.5 cm
7. Trigonometry In this chapter you will learn aout: the relationship etween the ratio of the sides in a rightangled triangle solving prolems using the trigonometric ratios finding the lengths of unknown
More information4.2. The Normal Force, Apparent Weight and Hooke s Law
4.2. The Normal Force, Apparent Weight and Hooke s Law Weight The weight of an object on the Earth s surface is the gravitational force exerted on it by the Earth. When you weigh yourself, the scale gives
More informationThe Phasor Analysis Method For Harmonically Forced Linear Systems
The Phasor Analysis Method For Harmonically Forced Linear Systems Daniel S. Stutts, Ph.D. April 4, 1999 Revised: 1015010, 91011 1 Introduction One of the most common tasks in vibration analysis is
More information4.5. Applications of Trigonometry to Waves. Introduction. Prerequisites. Learning Outcomes
Applications of Trigonometry to Waves 4.5 Introduction Waves and vibrations occur in many contexts. The water waves on the sea and the vibrations of a stringed musical instrument are just two everyday
More informationPHYS 1401 General Physics I EXPERIMENT 14 SIMPLE HARMONIC MOTION. II. APPARATUS Spring, weights, strings, meter stick, photogate and a computer.
PHYS 1401 General Physics I EXPERIMENT 14 SIMPLE HARMONIC MOTION I. INTRODUCTION The objective of this experiment is the study of oscillatory motion. In particular the springmass system will be studied.
More informationConstruction of new Nonstandard Finite Difference Schemes for the. Solution of a free undamped Harmonic Oscillator Equation
Construction of new Nonstandard Finite Difference Schemes for the Abstract Solution of a free undamped Harmonic Oscillator Equation Adesoji A.Obayomi * Adetolaju Olu Sunday Department of Mathematical
More informationChapters 10 & 11: Energy
Chapters 10 & 11: Energy Power: Sources of Energy Tidal Power SF Bay Tidal Power Project Main Ideas (Encyclopedia of Physics) Energy is an abstract quantity that an object is said to possess. It is not
More information5.1: Graphing Sine and Cosine Functions
5.1: Graphing Sine and Cosine Functions Complete the table below ( we used increments of for the values of ) 4 0 sin 4 2 3 4 5 4 3 7 2 4 2 cos 1. Using the table, sketch the graph of y sin for 0 2 2. What
More informationChapter 2 Second Order Differential Equations
Chapter 2 Second Order Differential Equations Either mathematics is too big for the human mind or the human mind is more than a machine.  Kurt Gödel (19061978) 2.1 The Simple Harmonic Oscillator The
More informationPhysics 326 Lab 6 10/18/04 DAMPED SIMPLE HARMONIC MOTION
DAMPED SIMPLE HARMONIC MOTION PURPOSE To understand the relationships between force, acceleration, velocity, position, and period of a mass undergoing simple harmonic motion and to determine the effect
More informationLECTURE 12 FRICTION & SPRINGS. Instructor: Kazumi Tolich
LECTURE 12 FRICTION & SPRINGS Instructor: Kazumi Tolich Lecture 12 2 Reading chapter 61 to 62 Friction n Static friction n Kinetic friction Springs Origin of friction 3 The origin of friction is electromagnetic
More informationME 323 Examination #2
ME 33 Eamination # SOUTION Novemer 14, 17 ROEM NO. 1 3 points ma. The cantilever eam D of the ending stiffness is sujected to a concentrated moment M at C. The eam is also supported y a roller at. Using
More informationChapter 15 SIMPLE HARMONIC MOTION
Physics Including Human Applications 309 Chapter 15 SIMPLE HARMONIC MOTION GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define
More informationExamination paper for TMA4195 Mathematical Modeling
Department of Mathematical Sciences Examination paper for TMA4195 Mathematical Modeling Academic contact during examination: Elena Celledoni Phone: 48238584, 73593541 Examination date: 11th of December
More informationVibrations and Waves Physics Year 1. Handout 1: Course Details
Vibrations and Waves JanFeb 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 informationMeasuring the Universal Gravitational Constant, G
Measuring the Universal Gravitational Constant, G Introduction: The universal law of gravitation states that everything in the universe is attracted to everything else. It seems reasonable that everything
More informationImaginary. Axis. Real. Axis
Name ME6 Final. I certify that I upheld the Stanford Honor code during this exam Monday December 2, 25 3:36:3 p.m. ffl Print your name and sign the honor code statement ffl You may use your course notes,
More informationELASTIC STRINGS & SPRINGS
ELASTIC STRINGS & SPRINGS Question 1 (**) A particle of mass m is attached to one end of a light elastic string of natural length l and modulus of elasticity 25 8 mg. The other end of the string is attached
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
More informationPhysics 202 Homework 1
Physics 202 Homework Apr 3, 203. A person who weighs 670 newtons steps onto a spring scale in the bathroom, (a) 85 kn/m (b) 290 newtons and the spring compresses by 0.79 cm. (a) What is the spring constant?
More informationChem(bio) Week 2: Determining the Equilibrium Constant of Bromothymol Blue. Keywords: Equilibrium Constant, ph, indicator, spectroscopy
Ojectives: Keywords: Equilirium Constant, ph, indicator, spectroscopy Prepare all solutions for measurement of the equilirium constant for romothymol Make a series of spectroscopic measurements on the
More informationAP Calculus Summer Homework MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
AP Calculus Summer Homework 20152016 Part 2 Name Score MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the distance d(p1, P2) between the points
More informationSTRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula
UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving
More informationModeling with Differential Equations
Modeling with Differential Equations 1. Exponential Growth and Decay models. Definition. A quantity y(t) is said to have an exponential growth model if it increases at a rate proportional to the amount
More informationSimple Harmonic Motion
3/5/07 Simple Harmonic Motion 0. The Ideal Spring and Simple Harmonic Motion HOOKE S AW: RESTORING FORCE OF AN IDEA SPRING The restoring force on an ideal spring is F x k x spring constant Units: N/m 3/5/07
More informationC. points X and Y only. D. points O, X and Y only. (Total 1 mark)
Grade 11 Physics  Homework 16  Answers on a separate sheet of paper, please 1. A cart, connected to two identical springs, is oscillating with simple harmonic motion between two points X and Y that
More informationLab 12. SpringMass Oscillations
Lab 12. SpringMass Oscillations Goals To determine experimentally whether the supplied spring obeys Hooke s law, and if so, to calculate its spring constant. To determine the spring constant by another
More informationThe content contained in all sections of chapter 6 of the textbook is included on the AP Physics B exam.
WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system is always
More informationRotational Kinetic Energy
Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body
More informationEquations. A body executing simple harmonic motion has maximum acceleration ) At the mean positions ) At the two extreme position 3) At any position 4) he question is irrelevant. A particle moves on the
More informationDetermining the Acceleration Due to Gravity with a Simple Pendulum
Determining the Acceleration Due to Gravity with a Simple Pendulum Quintin T. Nethercott and M. Evelynn Walton Department of Physics, University of Utah, Salt Lake City, 84112, UT, USA (Dated: March 6,
More informationName: AP Homework 9.1. Simple Harmonic Motion. Date: Class Period:
AP Homework 9.1 Simple Harmonic Motion (1) If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced 0.120 m from its
More informationVARIABLE COEFFICIENT OF FRICTION: AN EFFECTIVE VFPI PARAMETER TO CONTROL NEARFAULT GROUND MOTIONS
ISET Journal of Earthquake Technology, Paper No. 5, Vol. 49, No. 34, Sept.Dec. 0, pp. 73 87 VARIABLE COEFFICIENT OF FRICTION: AN EFFECTIVE VFPI PARAMETER TO CONTROL NEARFAULT GROUND MOTIONS Girish Malu*
More informationThis module requires you to read a textbook such as Fowles and Cassiday on material relevant to the following topics.
Module M2 Lagrangian Mechanics and Oscillations Prerequisite: Module C1 This module requires you to read a textbook such as Fowles and Cassiday on material relevant to the following topics. Topics: Hamilton
More informationDesign Variable Concepts 19 Mar 09 Lab 7 Lecture Notes
Design Variale Concepts 19 Mar 09 La 7 Lecture Notes Nomenclature W total weight (= W wing + W fuse + W pay ) reference area (wing area) wing aspect ratio c r root wing chord c t tip wing chord λ taper
More information5. Forces and FreeBody Diagrams
5. Forces and FreeBody Diagrams A) Overview We will begin by introducing the bulk of the new forces we will use in this course. We will start with the weight of an object, the gravitational force near
More information= y(x, t) =A cos (!t + kx)
A harmonic wave propagates horizontally along a taut string of length L = 8.0 m and mass M = 0.23 kg. The vertical displacement of the string along its length is given by y(x, t) = 0. m cos(.5 t + 0.8
More informationDistance travelled time taken and if the particle is a distance s(t) along the xaxis, then its instantaneous speed is:
Chapter 1 Kinematics 1.1 Basic ideas r(t) is the position of a particle; r = r is the distance to the origin. If r = x i + y j + z k = (x, y, z), then r = r = x 2 + y 2 + z 2. v(t) is the velocity; v =
More information7. Vibrations DE2EA 2.1: M4DE. Dr Connor Myant
DE2EA 2.1: M4DE Dr Connor Myant 7. Vibrations Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Introduction...
More informationTHE SCREW GAUGE. AIM: To learn to use a Screw Gauge and hence use it to find the dimensions of various regular materials given.
EXPERIMENT NO: DATE: / / 0 THE SCREW GAUGE AIM: To learn to use a Screw Gauge and hence use it to find the dimensions of various regular materials given. APPARUTUS: Given a Screw Gauge, cylindrical glass
More informationWork Done by a Constant Force
Work and Energy Work Done by a Constant Force In physics, work is described by what is accomplished when a force acts on an object, and the object moves through a distance. The work done by a constant
More informationPREMED COURSE, 14/08/2015 OSCILLATIONS
PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS
More informationab is shifted horizontally by h units. ab is shifted vertically by k units.
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an
More information_CH01_p qxd 1/20/10 8:35 PM Page 1 PURPOSE
9460218_CH01_p001010.qxd 1/20/10 8:35 PM Page 1 1 GRAPHING AND ANALYSIS PURPOSE The purpose of this lab is to investigate the relationship between displacement and force in springs and to practice acquiring
More information4. Sinusoidal solutions
16 4. Sinusoidal solutions Many things in nature are periodic, even sinusoidal. We will begin by reviewing terms surrounding periodic functions. If an LTI system is fed a periodic input signal, we have
More informationBasic Theory of Differential Equations
page 104 104 CHAPTER 1 FirstOrder Differential Equations 16. The following initialvalue problem arises in the analysis of a cable suspended between two fixed points y = 1 a 1 + (y ) 2, y(0) = a, y (0)
More informationPH211 Chapter 10 Solutions
PH Chapter 0 Solutions 0.. Model: We will use the particle model for the bullet (B) and the running student (S). Solve: For the bullet, K B = m v = B B (0.00 kg)(500 m/s) = 50 J For the running student,
More informationGood Vibes: Introduction to Oscillations
Good Vibes: Introduction to Oscillations Description: Several conceptual and qualitative questions related to main characteristics of simple harmonic motion: amplitude, displacement, period, frequency,
More informationMATH 32 FALL 2012 FINAL EXAM  PRACTICE EXAM SOLUTIONS
MATH 3 FALL 0 FINAL EXAM  PRACTICE EXAM SOLUTIONS () You cut a slice from a circular pizza (centered at the origin) with radius 6 along radii at angles 4 and 3 with the positive horizontal axis. (a) (3
More informationExperiment 12 Damped Harmonic Motion
Physics Department LAB A  120 Experiment 12 Damped Harmonic Motion References: Daniel Kleppner and Robert Kolenkow, An Introduction to Mechanics, McGraw Hill 1973 pp. 414418. Equipment: Air track, glider,
More information