Hooke s Law. Spring. Simple Harmonic Motion. Energy. 12/9/09 Physics 201, UWMadison 1


 Ethelbert Ramsey
 1 years ago
 Views:
Transcription
1 Hooke s Law Spring Simple Harmonic Motion Energy 12/9/09 Physics 201, UWMadison 1
2 relaxed position F X = kx > 0 F X = 0 x apple 0 x=0 x > 0 x=0 F X =  kx < 0 x 12/9/09 Physics 201, UWMadison 2
3 We know that if we stretch a spring with a mass on the end and let it go, the mass will oscillate back and forth (if there is no friction). k m This oscillation is called Simple Harmonic Motion, and is actually easy to understand... k k m m 12/9/09 Physics 201, UWMadison 3
4 k m m d 2 x dt 2 x d 2 x dt 2 = k m x a differential equation 12/9/09 Physics 201, UWMadison 4
5 d 2 x define dt = k 2 m x ω = k m d 2 x dt 2 = ω 2 x Where ω is the angular frequency of motion Try the solution x = A cos(ωt) dx dt d 2 x dt 2 = ω Asin ( ωt ) = ω 2 Acos( ωt) = ω 2 x This works, so it must be a solution! 12/9/09 Physics 201, UWMadison 5
6 d 2 x We just showed that (from F = ma) dt 2 = ω 2 x has the solution x = A cos(ωt) but x = A sin(ωt) is also a solution. The most general solution is a linear combination (sum) of these two solutions! x = Bsin( ωt) + C cos( ωt) = Acos( ωt + φ) dx dt d 2 x dt 2 = ω Bcos ωt ( ) ωc sin ( ωt ) = ω 2 Bsin( ωt) ω 2 C cos( ωt) = ω 2 x ok 12/9/09 Physics 201, UWMadison 6
7 x = Acos( ωt + ϕ) 3 But wait a minute...what does angular frequency ω have to do with moving back & forth in a straight line?? 4 2 y x The spring s motion with amplitude A is identical to the xcomponent of a particle in uniform circular motion with radius A. 2 3 π 2 4 π 5 6 θ 12/9/09 Physics 201, UWMadison 7
8 A mass m = 2 kg on a spring oscillates with amplitude A = 10 cm. At t = 0 its speed is maximum, and is v = +2 m/s. What is the angular frequency of oscillation ω? What is the spring constant k? v MAX = ωa ω = v MAX A = 2 m s 10cm = 20 s 1 ω = k Also: k = mω 2 m So k = (2 kg) x (20 s 1 ) 2 = 800 kg/s 2 = 800 N/m k m 12/9/09 Physics 201, UWMadison 8 x
9 Use initial conditions to determine phase φ! Suppose we are told x(0) = 0, and x is initially increasing (i.e. v(0) = positive): x(t) = A cos(ωt + φ) v(t) = ωa sin(ωt + φ) a(t) = ω 2 A cos(ωt + φ) x(0) = 0 = A cos(φ) v(0) > 0 = ωa sin(φ) φ < 0 φ = π/2 or π/2 So φ = π/2 π θ 2π k m cosθ sinθ 0 x 12/9/09 Physics 201, UWMadison 9
10 So we find φ= π/2 x(t) = A cos(ωt  π/2 ) v(t) = ωa sin(ωt  π/2 ) a(t) = ω 2 A cos(ωt  π/2 ) x(t) = A sin(ωt) v(t) = ωa cos(ωt) a(t) = ω 2 A sin(ωt) A x(t) k m A π 2π ωt 0 x 12/9/09 Physics 201, UWMadison 10
11 We already know that for a vertical spring if y is measured from the equilibrium position The force of the spring is the negative derivative of this function: U = 1 2 ky2 So this will be just like the horizontal case: k ky = ma = m d 2 y dt 2 y = 0 Which has solution y = A cos(ωt + φ) where ω = F = ky m 12/9/09 Physics 201, UWMadison 11 k m
12 Recall that the torque due to gravity about the rotation (z) axis is τ = mgd d = Lsin θ Lθ for small θ so τ = mg Lθ τ = I d 2 θ dt 2 z But τ = Iα, where I = ml 2 d 2 θ where dt 2 = ω 2 θ ω = g L Differential equation for simple harmonic motion! θ = θ 0 cos( ωt + ϕ) θ d L m mg 12/9/09 Physics 201, UWMadison 12
13 You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T 1. Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T 2. Which of the following is true: (a) T 1 = T 2 (b) T 1 > T 2 (c) T 1 < T 2 12/9/09 Physics 201, UWMadison 13
14 We have shown that for a simple pendulum ω = g L Recall the pendulum depended on the torque from gravity. Now the moment arm is from the center of mass and the oscillation point so L is shorter Since T = 2π ω T = 2π L g If we make a pendulum shorter, it oscillates faster (shorter period) 12/9/09 Physics 201, UWMadison 14
15 Force: d 2 s dt 2 = ω 2 s ω = k k m k m m s 0 0 s Solution: s = A cos(ωt + φ) ω = g L s L 12/10/09 Physics 201, UWMadison 15
16 U t x = Acos( ωt) U = 1 2 kx2 = 1 2 ka2 cos 2 ( ωt) K t v = dx dt = ω Asin ( ωt ) K = 1 2 mv2 = 1 2 mω 2 A 2 sin 2 ( ωt) t E = K +U = K Max = 1 2 mω2 A 2 = U Max = 1 2 k 2 A 2 12/10/09 16
17 U = 1 2 kx2 U = 1 2 kx2 K = 1 2 mv2 E = U + K = 1 2 ka2 = 1 2 mv2 max 12/10/09 Physics 201, UWMadison 17
18 For both the spring and the pendulum, we can derive the SHM solution by using energy conservation. The total energy (K + U) of a system undergoing SHM will always be constant! This is not surprising since there are only conservative forces present, hence K+U energy is conserved. U E K U A 0 A s 12/10/09 Physics 201, UWMadison 18
19 m x=0 x E total = 1/2 Mv 2 + 1/2 kx 2 = constant KE PE KE max = Mv 2 max /2 = Mω2 A 2 /2 =ka 2 /2 PE max = ka 2 /2 E total = ka 2 /2 12/10/09 Physics 201, UWMadison 19
20 U 1 2 kx2 for small x 12/10/09 Physics 201, UWMadison 20
21 Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, and that we know where the CM is located and what the moment of inertia I about the axis is. The torque about the rotation (z) axis for small θ is (using sin θ θ) τ = Mgd MgRθ MgRθ = I d 2 θ dt 2 d 2 θ dt = ω 2 θ 2 where ω = MgR I ( ) θ = θ 0 cos ωt + ϕ τ α zaxis R θ x CM d Mg 12/10/09 Physics 201, UWMadison 21
22 Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis, and the moment of inertia I about this axis is known. The wire acts like a rotational spring. When the object is rotated, the wire is twisted. This produces a torque that opposes the rotation. In analogy with a spring, the torque produced is proportional to the displacement: τ = kθ τ wire θ I 12/10/09 Physics 201, UWMadison 22
23 Since τ = kθ, τ = Iα becomes kθ = I d 2 θ dt 2 d 2 θ dt = ω 2 θ where ω = k 2 I τ I wire θ This is similar to the mass on spring except I has taken the place of m (no surprise). 12/10/09 Physics 201, UWMadison 23
24 In many real systems, nonconservative forces are present The system is no longer ideal Friction is a common nonconservative force In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped The amplitude decreases with time The blue dashed lines on the graph represent the envelope of the motion 12/10/09 Physics 201, UWMadison 24
25 One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid The retarding force can be expressed as R =  b v where b is a constant b is called the damping coefficient The restoring force is kx From Newton s Second Law ΣF x = k x bv x = ma x or, m d 2 x dt 2 + b dx dt + kx = 0 12/10/09 Physics 201, UWMadison 25
26 m d 2 x dt 2 + b dx dt + kx = 0 When b is small enough, the solution to this equation is: with x = A 0 e (b/2m)t cos( ω t + δ ) ω = ω 0 (This solution applies when b<2mω 0.) b 1 2mω 0 2, ω 0 = k m 12/10/09 Physics 201, UWMadison 26
27 The position can be described by x = A e (b/2m)t cos( ω t + δ ) 0 The angular frequency is ω = k m b 2m 2 When the retarding force is small, the oscillatory character of the motion is preserved, but the amplitude decreases exponentially with time The motion ultimately ceases 12/10/09 Physics 201, UWMadison 27
28 The energy of a weakly damped harmonic oscillator decays exponentially in time. The potential energy is ½kx 2 : x 2 = ( A e (b/2m)t cos( ω t + δ )) 2 0 = A 2 e (b/m)t cos 2 ( ω t + δ ) 0 Average over a period when (b/m)<<ω : Average of cos 2 = ½, so averaged over a period, x 2 A 0 2 e (b/m)t. 12/10/09 Physics 201, UWMadison 28
29 The energy of a weakly damped harmonic oscillator decays exponentially in time. The kinetic energy is ½mv 2 : v 2 = d dt ( A 0 e (b/2m)t cos( ω t + δ )) 2 ( ) 2 = (b / 2m)A 0 e (b/2m)t cos( ω t + δ ) A 0 e (b/2m)t sin( ω t + δ ) Average energy over a period when (b/m)<<ω : Average of cos 2 = average of sin 2 = ½, and average of sinx cosx=0, so averaged over a period, v 2 e (b/m)t. 12/10/09 Physics 201, UWMadison 29
30 The energy of a weakly damped harmonic oscillator decays exponentially in time. Both kinetic and potential energy (averaged over a period) decay as e (b/m)t. So total energy is proportional to e (b/m)t. 12/10/09 Physics 201, UWMadison 30
31 ω 0 is also called the natural frequency of the system If R max = bv max < ka, the system is said to be underdamped When b reaches a critical value b c such that b c / 2 m = ω 0, the system will not oscillate The system is said to be critically damped If R max = bv max > ka and b/2m > ω 0, the system is said to be overdamped For critically damped and overdamped there is no angular frequency 12/10/09 Physics 201, UWMadison 31
32 It is possible to compensate for the loss of energy in a damped system by applying an external sinusoidal force (with angular frequency ω) The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increase After a sufficiently long period of time, E driving = E lost to internal Then a steadystate condition is reached The oscillations will proceed with constant amplitude The amplitude of a driven oscillation is ω 0 is the natural frequency of the undamped oscillator 12/10/09 Physics 201, UWMadison 32
33 When ω ω 0 an increase in amplitude occurs This dramatic increase in the amplitude is called resonance The natural frequency 0 is also called the resonance frequency At resonance, the applied force is in phase with the velocity and the power transferred to the oscillator is a maximum The applied force and v are both proportional to sin (ωt + φ) The power delivered is F. v» This is a maximum when F and v are in phase Resonance (maximum peak) occurs when driving frequency equals the natural frequency The amplitude increases with decreased damping The curve broadens as the damping increases The shape of the resonance curve depends on b 12/10/09 Physics 201, UWMadison 33
34 The amplitude of a system moving with simple harmonic motion is doubled. The total energy will then be 4 times larger 2 times larger the same as it was half as much quarter as much U = 1 2 kx mv2 at x = A,v = 0 U = 1 2 ka2 12/10/09 Physics 201, UWMadison 34
35 A glider of mass m is attached to springs on both ends, which are attached to the ends of a frictionless track. The glider moved by 0.2 m to the right, and let go to oscillate. If m = 2 kg, and spring constants are k 1 = 800 N/m and k 2 = 500 N/m, the frequency of oscillation (in Hz) is approximately 6 Hz 2 Hz 4 Hz 8 Hz 10 Hz f = ω 2π = k / m 2π = π = 4 Hz 12/10/09 Physics 201, Fall 2006, UWMadison 35
36 A 810g block oscillates on the end of a spring whose force constant is 60 N/m. The mass moves in a fluid which offers a resistive force proportional to its speed the proportionality constant is N.s/m. Write the equation of motion and solution. F s + F R = ma m d 2 x dt 2 dx = kx b dt Solution: x(t) = Ae b 2m t cos(ωt + φ) What is the period of motion? s ω = k m b 2m 2 What is the fractional decrease in amplitude per cycle? Write the displacement as a function of time if at t = 0, x = 0 and at t = 1 s, x = m. Solution: x(t) = 0.182e 0.100t sin(8.61t + φ) 12/10/09 Physics 201, Fall 2006, UWMadison 36
Simple Harmonic Motion
Simple Harmonic Motion Restating Hooke s law The equation of motion Phase, frequency, amplitude Simple Pendulum Damped and Forced oscillations Resonance Harmonic Motion A lot of motion in the real world
More informationPeriodic Motion or Oscillations. Physics 232 Lecture 01 1
Periodic Motion or Oscillations Physics 3 Lecture 01 1 Periodic Motion Periodic Motion is motion that repeats about a point of stable equilibrium Stable Equilibrium Unstable Equilibrium A necessary requirement
More informationPhysics 53. Oscillations. You've got to be very careful if you don't know where you're going, because you might not get there.
Physics 53 Oscillations You've got to be very careful if you don't know where you're going, because you might not get there. Yogi Berra Overview Many natural phenomena exhibit motion in which particles
More informationMECHANICS IV  SIMPLE HARMONIC MOTION
MIVp.1 A. OSCILLATIONS B. SIMPLE PENDULUM C. KINEMATICS OF SIMPLE HARMONIC MOTION D. SPRINGANDMASS SYSTEM E. ENERGY OF SHM F. DAMPED HARMONIC MOTION G. FORCED VIBRATION A. OSCILLATIONS A toandfro
More informationChapter 13, example problems: x (cm) 10.0
Chapter 13, example problems: (13.04) Reading Fig. 1330 (reproduced on the right): (a) Frequency f = 1/ T = 1/ (16s) = 0.0625 Hz. (since the figure shows that T/2 is 8 s.) (b) The amplitude is 10 cm.
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationChapter 1. Oscillations. Oscillations
Oscillations 1. A mass m hanging on a spring with a spring constant k has simple harmonic motion with a period T. If the mass is doubled to 2m, the period of oscillation A) increases by a factor of 2.
More informationChapter 24 Physical Pendulum
Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...
More informationPhysics 211 Week 12. Simple Harmonic Motion: Equation of Motion
Physics 11 Week 1 Simple Harmonic Motion: Equation of Motion A mass M rests on a frictionless table and is connected to a spring of spring constant k. The other end of the spring is fixed to a vertical
More informationPHYS2020: General Physics II Course Lecture Notes Section VII
PHYS2020: General Physics II Course Lecture Notes Section VII Dr. Donald G. Luttermoser East Tennessee State University Edition 4.0 Abstract These class notes are designed for use of the instructor and
More informationPhysics 1022: Chapter 14 Waves
Phys 10: Introduction, Pg 1 Physics 10: Chapter 14 Waves Anatomy of a wave Simple harmonic motion Energy and simple harmonic motion Phys 10: Introduction, Pg Page 1 1 Waves New Topic Phys 10: Introduction,
More informationAdvanced Higher Physics: MECHANICS. Simple Harmonic Motion
Advanced Higher Physics: MECHANICS Simple Harmonic Motion At the end of this section, you should be able to: Describe examples of simple harmonic motion (SHM). State that in SHM the unbalanced force is
More informationSIMPLE HARMONIC MOTION
PERIODIC MOTION SIMPLE HARMONIC MOTION If a particle moves such that it repeats its path regularly after equal intervals of time, its motion is said to be periodic. The interval of time required to complete
More informationSimple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines
Simple Harmonic Motion(SHM) Vibration (oscillation) Equilibrium position position of the natural length of a spring Amplitude maximum displacement Period and Frequency Period (T) Time for one complete
More informationName: Lab Partner: Section:
Chapter 10 Simple Harmonic Motion Name: Lab Partner: Section: 10.1 Purpose Simple harmonic motion will be examined in this experiment. 10.2 Introduction A periodic motion is one that repeats itself in
More informationA B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product
1 Dot Product and Cross Products For two vectors, the dot product is a number A B = AB cos(θ) = A B = AB (1) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More informationPeople s Physics book 3e Ch 251
The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate
More informationNotes on the Periodically Forced Harmonic Oscillator
Notes on the Periodically orced Harmonic Oscillator Warren Weckesser Math 38  Differential Equations 1 The Periodically orced Harmonic Oscillator. By periodically forced harmonic oscillator, we mean the
More informationLecture Presentation Chapter 14 Oscillations
Lecture Presentation Chapter 14 Oscillations Suggested Videos for Chapter 14 Prelecture Videos Describing Simple Harmonic Motion Details of SHM Damping and Resonance Class Videos Oscillations Basic Oscillation
More informationRecitation 2 Chapters 12 and 13
Recitation 2 Chapters 12 and 13 Problem 12.20. 65.0 kg bungee jumper steps off a bridge with a light bungee cord tied to her and the bridge (Figure P12.20. The unstretched length of the cord is 11.0 m.
More informationLab M1: The Simple Pendulum
Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of
More informationType: Double Date: Simple Harmonic Motion III. Homework: Read 10.3, Do CONCEPT QUEST #(7) Do PROBLEMS #(5, 19, 28) Ch. 10
Type: Double Date: Objective: Simple Harmonic Motion II Simple Harmonic Motion III Homework: Read 10.3, Do CONCEPT QUEST #(7) Do PROBLEMS #(5, 19, 28) Ch. 10 AP Physics B Mr. Mirro Simple Harmonic Motion
More informationExperiment Type: OpenEnded
Simple Harmonic Oscillation Overview Experiment Type: OpenEnded In this experiment, students will look at three kinds of oscillators and determine whether or not they can be approximated as simple harmonic
More informationAP Physics C. Oscillations/SHM Review Packet
AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 3)
Chapter 2. Small Oscillations Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 3) 2.1 Introduction If a particle, originally in a position of equilibrium we limit
More informationSimple Harmonic Motion Concepts
Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called
More informationSIMPLE HARMONIC MOTION Ken Cheney
SIMPLE HARMONIC MOTION Ken Cheney INTRODUCTION GENERAL Probably no tools that you will learn in Physics are more widely used than those that deal with simple harmonic motion. Here we will be investigating
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More information8 SIMPLE HARMONIC MOTION
8 SIMPLE HARMONIC MOTION Chapter 8 Simple Harmonic Motion Objectives After studying this chapter you should be able to model oscillations; be able to derive laws to describe oscillations; be able to use
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 informationSIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE
MISN026 SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE by Kirby Morgan 1. Dynamics of Harmonic Motion a. Force Varies in Magnitude and Direction................
More informationHomework #7 Solutions
MAT 0 Spring 201 Problems Homework #7 Solutions Section.: 4, 18, 22, 24, 4, 40 Section.4: 4, abc, 16, 18, 22. Omit the graphing part on problems 16 and 18...4. Find the general solution to the differential
More informationSIMPLE HARMONIC MOTION
SIMPLE HARMONIC MOTION PURPOSE The purpose of this experiment is to investigate one of the fundamental types of motion that exists in nature  simple harmonic motion. The importance of this kind of motion
More informationExperiment 4: Harmonic Motion Analysis
Experiment 4: Harmonic Motion Analysis Background In this experiment you will investigate the influence of damping on a driven harmonic oscillator and study resonant conditions. The following theoretical
More informationSimple Harmonic Motion
Periodic motion Earth around the sun Elastic ball bouncing up an down Quartz in your watch, computer clock, ipod clock, etc. Heart beat, and many more In taking your pulse, you count 70.0 heartbeats in
More information2.3 Cantilever linear oscillations
.3 Cantilever linear oscillations Study of a cantilever oscillation is a rather science  intensive problem. In many cases the general solution to the cantilever equation of motion can not be obtained
More informationHooke s Law and Simple Harmonic Motion
Hooke s Law and Simple Harmonic Motion OBJECTIVE to measure the spring constant of the springs using Hooke s Law to explore the static properties of springy objects and springs, connected in series and
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationSHM Simple Harmonic Motion revised June 16, 2012
SHM Simple Harmonic Motion revised June 16, 01 Learning Objectives: During this lab, you will 1. communicate scientific results in writing.. estimate the uncertainty in a quantity that is calculated from
More informationSolutions 2.4Page 140
Solutions.4Page 4 Problem 3 A mass of 3 kg is attached to the end of a spring that is stretched cm by a force of 5N. It is set in motion with initial position = and initial velocity v = m/s. Find the
More informationLABORATORY 9. Simple Harmonic Motion
LABORATORY 9 Simple Harmonic Motion Purpose In this experiment we will investigate two examples of simple harmonic motion: the massspring system and the simple pendulum. For the massspring system we
More information1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!
1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating
More informationMh1: Simple Harmonic Motion. Chapter 15. Motion of a SpringMass System. Periodic Motion. Oscillatory Motion
Mh1: Siple Haronic Motion Chapter 15 Siple block and spring Oscillatory Motion Exaple: the tides, a swing Professor Michael Burton School of Physics, UNSW Periodic Motion! Periodic otion is otion of an
More information1 of 10 11/23/2009 6:37 PM
hapter 14 Homework Due: 9:00am on Thursday November 19 2009 Note: To understand how points are awarded read your instructor's Grading Policy. [Return to Standard Assignment View] Good Vibes: Introduction
More informationPhysics 231 Lecture 15
Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)
More informationSimple Harmonic Motion
Simple Harmonic Motion Objective: In this exercise you will investigate the simple harmonic motion of mass suspended from a helical (coiled) spring. Apparatus: Spring 1 Table Post 1 Short Rod 1 Rightangled
More informationturntable in terms of SHM and UCM: be plotted as a sine wave. n Think about spinning a ball on a string or a ball on a
RECALL: Angular Displacement & Angular Velocity Think about spinning a ball on a string or a ball on a turntable in terms of SHM and UCM: If you look at the ball from the side, its motion could be plotted
More informationSimple Harmonic Motion
Simple Harmonic Motion Theory Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is eecuted by any quantity obeying the Differential
More informationPhysics Exam 2 Formulas
INSTRUCTIONS: Write your NAME on the front of the blue exam booklet. The exam is closed book, and you may have only pens/pencils and a calculator (no stored equations or programs and no graphing). Show
More informationSolving the Harmonic Oscillator Equation. Morgan Root NCSU Department of Math
Solving the Harmonic Oscillator Equation Morgan Root NCSU Department of Math SpringMass System Consider a mass attached to a wall by means of a spring. Define y to be the equilibrium position of the block.
More informationp = F net t (2) But, what is the net force acting on the object? Here s a little help in identifying the net force on an object:
Harmonic Oscillator Objective: Describe the position as a function of time of a harmonic oscillator. Apply the momentum principle to a harmonic oscillator. Sketch (and interpret) a graph of position as
More informationIMPORTANT NOTE ABOUT WEBASSIGN:
Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationSimple Harmonic Motion
Simple Harmonic Motion 9M Object: Apparatus: To determine the force constant of a spring and then study the harmonic motion of that spring when it is loaded with a mass m. Force sensor, motion sensor,
More informationLecture L19  Vibration, Normal Modes, Natural Frequencies, Instability
S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19  Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free
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 informationLab 5: Conservation of Energy
Lab 5: Conservation of Energy Equipment SWS, 1meter stick, 2meter stick, heavy duty bench clamp, 90cm rod, 40cm rod, 2 double clamps, brass spring, 100g mass, 500g mass with 5cm cardboard square
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 informationNotice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW1 Possible Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 14.P.003 An object attached to a spring has simple
More informationHW 7 Q 14,20,20,23 P 3,4,8,6,8. Chapter 7. Rotational Motion of the Object. Dr. Armen Kocharian
HW 7 Q 14,20,20,23 P 3,4,8,6,8 Chapter 7 Rotational Motion of the Object Dr. Armen Kocharian Axis of Rotation The radian is a unit of angular measure The radian can be defined as the arc length s along
More informationSolution Derivations for Capa #10
Solution Derivations for Capa #10 1) The flywheel of a steam engine runs with a constant angular speed of 172 rev/min. When steam is shut off, the friction of the bearings and the air brings the wheel
More informationSAT Subject Physics Formula Reference
This guide is a compilation of about fifty of the most important physics formulas to know for the SAT Subject test in physics. (Note that formulas are not given on the test.) Each formula row contains
More informationForced Oscillations in a Linear System
Forced Oscillations in a Linear System Manual Eugene Butikov Annotation. The manual includes a description of the simulated physical system and a summary of the relevant theoretical material for students
More information16 OSCILLATORY MOTION AND WAVES
CHAPTER 16 OSCILLATORY MOTION AND WAVES 549 16 OSCILLATORY MOTION AND WAVES Figure 16.1 There are at least four types of waves in this picture only the water waves are evident. There are also sound waves,
More informationLecture L13  Conservative Internal Forces and Potential Energy
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L13  Conservative Internal Forces and Potential Energy The forces internal to a system are of two types. Conservative forces, such as gravity;
More information7. Kinetic Energy and Work
Kinetic Energy: 7. Kinetic Energy and Work The kinetic energy of a moving object: k = 1 2 mv 2 Kinetic energy is proportional to the square of the velocity. If the velocity of an object doubles, the kinetic
More information14 Mass on a spring and other systems described by linear ODE
14 Mass on a spring and other systems described by linear ODE 14.1 Mass on a spring Consider a mass hanging on a spring (see the figure). The position of the mass in uniquely defined by one coordinate
More informationPhysics 1120: Waves Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Waves Solutions 1. A wire of length 4.35 m and mass 137 g is under a tension of 125 N. What is the speed of a wave in this wire? If the tension
More informationSIMPLE HARMONIC MOTION
SIMPLE HRMONIC MOION PREVIOUS EMCE QUESIONS ENGINEERING. he displacement of a particle executin SHM is iven by :y = 5 sin 4t +. If is the time period and 3 the mass of the particle is ms, the kinetic enery
More informationSimple Harmonic Motion
Simple Harmonic Motion Simple harmonic motion is one of the most common motions found in nature and can be observed from the microscopic vibration of atoms in a solid to rocking of a supertanker on the
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationA C O U S T I C S of W O O D Lecture 3
Jan Tippner, Dep. of Wood Science, FFWT MU Brno jan. tippner@mendelu. cz Content of lecture 3: 1. Damping 2. Internal friction in the wood Content of lecture 3: 1. Damping 2. Internal friction in the wood
More informationF mg (10.1 kg)(9.80 m/s ) m
Week 9 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationSimple Harmonic Motion
5 Simple Harmonic Motion Note: this section is not part of the syllabus for PHYS26. You should already be familiar with simple harmonic motion from your first year course PH115 Oscillations and Waves.
More informationphysics 111N rotational motion
physics 111N rotational motion rotations of a rigid body! suppose we have a body which rotates about some axis! we can define its orientation at any moment by an angle, θ (any point P will do) θ P physics
More informationRotational Dynamics. Luis Anchordoqui
Rotational Dynamics Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ). The radius of the circle is r. All points on a straight line
More information5.2 Rotational Kinematics, Moment of Inertia
5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position
More informationPhysics 271 FINAL EXAMSOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath
Physics 271 FINAL EXAMSOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath 1. The exam will last from 8:00 am to 11:00 am. Use a # 2 pencil to make entries on the answer sheet. Enter the following id information
More informationCenter of Mass/Momentum
Center of Mass/Momentum 1. 2. An Lshaped piece, represented by the shaded area on the figure, is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the Lshaped
More informationChapter 8 Rotational Motion
Chapter 8 Rotational Motion Textbook (Giancoli, 6 th edition): Assignment 9 Due on Thursday, November 26. 1. On page 131 of Giancoli, problem 18. 2. On page 220 of Giancoli, problem 24. 3. On page 221
More informationSimple harmonic motion
PH122 Dynamics Page 1 Simple harmonic motion 02 February 2011 10:10 Force opposes the displacement in A We assume the spring is linear k is the spring constant. Sometimes called stiffness constant Newton's
More informationVersion PREVIEW Practice 8 carroll (11108) 1
Version PREVIEW Practice 8 carroll 11108 1 This printout should have 12 questions. Multiplechoice questions may continue on the net column or page find all choices before answering. Inertia of Solids
More informationBead moving along a thin, rigid, wire.
Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing
More informationSimple Harmonic Motion. Simple Harmonic Motion
Siple Haronic Motion Basic oscillations are siple haronic otion (SHM) where the position as a function of tie is given by a sign or cosine Eaples of actual otion: asses on springs vibration of atos in
More informationPhysics 207 Lecture 25. Lecture 25. For Thursday, read through all of Chapter 18. Angular Momentum Exercise
Lecture 5 Today Review: Exam covers Chapters 1417 17 plus angular momentum, statics Assignment For Thursday, read through all of Chapter 18 Physics 07: Lecture 5, Pg 1 Angular Momentum Exercise A mass
More informationUNIT 14: HARMONIC MOTION
Name St.No.  Date(YY/MM/DD) / / Section UNIT 14: HARMONIC MOTION Approximate Time three 100minute sessions Back and Forth and Back and Forth... Cameo OBJECTIVES 1. To learn directly about some of the
More informationMathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.
Chapter 3 Mechanical Systems Dr. A.Aziz. Bazoune 3.1 INTRODUCTION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENTS Any mechanical
More informationPractice Test SHM with Answers
Practice Test SHM with Answers MPC 1) If we double the frequency of a system undergoing simple harmonic motion, which of the following statements about that system are true? (There could be more than one
More informationChapter 23 Simple Harmonic Motion
Chapter 3 Simple Harmonic Motion 3.1 Introduction: Periodic Motion... 1 3.1.1 Simple Harmonic Motion: Quantitative... 1 3. Simple Harmonic Motion: Analytic... 3 3..1 General Solution of Simple Harmonic
More informationA) F = k x B) F = k C) F = x k D) F = x + k E) None of these.
CT161 Which of the following is necessary to make an object oscillate? i. a stable equilibrium ii. little or no friction iii. a disturbance A: i only B: ii only C: iii only D: i and iii E: All three Answer:
More informationLab 5: Harmonic Oscillations and Damping
3 Lab 5: Harmonic Oscillations and Damping I. Introduction A. Objectives for this lab: 1. Learn how to quantitatively model a real harmonic oscillator 2. Learn how damping affects simple harmonic motion
More informationWork and Kinetic Energy
Chapter 6 Work and Kinetic Energy PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 6 To understand and calculate
More informationPhysics in the Laundromat
Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized
More informationPrelab Exercises: Hooke's Law and the Behavior of Springs
59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically
More informationL = 1 2 a(q) q2 V (q).
Physics 3550, Fall 2012 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
More informationChapter 10: Elasticity and Oscillations
Chapter 10: Elasticity and Oscillations Elastic Deformations Hooke s Law Stress and Strain Shear Deformations Volume Deformations Simple Harmonic Motion The Pendulum Damped Oscillations, orced Oscillations,
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
More informationUpdated 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