Oscillations. Vern Lindberg. June 10, 2010

Save this PDF as:
Size: px
Start display at page:

Download "Oscillations. Vern Lindberg. June 10, 2010"

Transcription

1 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 Undamped Simple Harmonic Motion If a mass m is attached to an ideal spring of constant k, and the spring force is the only net force, then one dimensional simple harmonic motion (SHM) results. The equation of motion is The solution can be written is several forms. ẍ = k m x (1) x = A 0 sin(ωt + φ 0 ) (2) = A 1 cos ωt + A 2 sin ωt (3) = A 3 exp(iωt) + A 4 exp( iωt) (4) = A 5 exp(i(ωt + φ 5 )) (5) The angular frequency, ω = 2πf, is independent of amplitude and is k ω = m Since this is a second order differential equation, all solutions have two integration constants (such as amplitude and phase, A 0 and φ 0, or two amplitudes A 1 and A 2 ) that are determined by the initial conditions. Picking Eq. 2, the first solution, If at t = 0 we have x = x 0 and v = v 0 then (6) x(0) = A sin φ 0 = x 0 (7) ẋ(0) = ωa cos φ 0 = v 0 (8) 1

2 Solving these we get tan φ 0 = ωx 0 (9) v 0 A = x v2 0 ω 2 (10) We will choose the positive square root value for amplitude, but there remains an ambiguity in the inverse tangent that is resolved by deciding what quadrant for φ 0 is needed to produce the proper signs for initial position and velocity. We will spend some time on this in class. Example: Suppose the spring constant is 18 N/m and the mass is 2 kg. Then the angular frequency is ω = 3 rad/s. If x 0 = cm and v 0 = cm/s, then the amplitude is 5.0 cm and the inverse tangent gives phase angles of either or 2.80 radians. With a bit of thought you can decide that the correct phase is 2.80 radians. Example Using Eq. 3, find the values of A 1, A 2 Use the same values as in the previous example. The potential energy associated with this simple model SHM is V (x) = 1 2 kx2 (11) This is a parabolic potential energy, with the oscillations occurring around the minimum in the potential energy. The spring force is a restoring force, linear in displacement. 1.1 The Morse Potential for Diatomic Molecules: Approximations Any potential energy that has a minimum can have oscillations around that minimum. A natural question is to ask whether the oscillations are simple harmonic. If the system oscillates about x = x 0 with force F = k(x x 0 ) and potential energy V = 1 2 k(x x 0) 2 then the motion is SHM for any amplitude. What if we don t have forces of that form? For small oscillations about the minimum we can make suitable approximations to see if the small amplitude motion is SHM. Taylor Series and Approximations A vital skill in physics is making approximations. The Taylor series is a polynomial expansion of a function about some point. If we expand f(x) about x = a, the Taylor series is f(x) = f(a) + ( f x ) x=a (x a) + 1 2! ( 2 ) f x 2 (x x=a a)2 + (12) The terms are called the zeroeth order, first order, second order,.... We frequently want to find the first higher order term (first and above) that is non-zero. 2

3 Consider the Morse potential energy, a model of the potential energy for a vibrating diatomic molecule like O 2. [ V (x) = V 0 1 exp( x x 2 0 )] V 0 (13) δ First we find the location of the minimum by solving dv/dx = 0 and find that the minimum occurs at x = x 0. This is done as an example in Chapter 2 of the text. At this minimum we find V = V 0. Computational Hint Make appropriate substitutions of variables to make the calculus easier. For the Morse potential define z = (x x 0 )/δ. Then V = V 0 (1 exp z) 2 V 0, and we can do derivatives as dv/dx = dv/dz dz/dx. For small oscillations we expand the exponential in a Taylor series, V = V 0 [1 (1 z + z 2 /2 z 3 /6 + )] 2 V 0 = V 0 [z z 2 /2 + z 3 /6 + ] 2 V 0 = V 0 [z 2 z 3 + 7z 4 /12 + ] V 0 ( ) x 2 = V 0 z 2 x0 V 0 = V 0 V 0 (14) δ This is parabolic and so we expect SHM. The restoring force for small oscillations is linear, F (x) = dv dx = 2V 0 δ 2 (x x 0) (15) Thus for small oscillation we expect SHM with an angular frequency of 2V0 ω = mδ 2 (16) Example The following system is set up on a horizontal, frictionless table. A small block of mass m is attached to two identical springs of constant k. When the springs are unstretched they have identical lengths L 0. The equilibrium position is a straight line consisting of a wall, a spring, the block, the other spring, and a second wall. The block is pulled down a small distance y L 0 perpendicular to the equilibrium line. Determine whether or not the subsequent motion is simple harmonic. 1.2 Angular SHM Consider a simple pendulum of length l with point mass m attached at one end, in a uniform gravitational field g. The motion is along an arc of a circle, so it is natural to use 3

4 an angular variable θ for the displacement. Using the angular form of Newton s Second Law, in scalar form since the angular motion is along a fixed direction, N = I θ (17) mgl sin θ = (ml 2 ) θ (18) Providing the angle (in radians) is small, we can use the Taylor Series expansion sin θ = θ θ 3 /3! + and keep it to first order to get the equation of motion. θ + g l θ = 0 (19) This is the same form as the SHM equation of motion, Equation 1, so the solution is SHM with angular frequency ω = g/l. We will solve the general equation of motion (large angles) in Mechanics 2 using Elliptic Integrals. 1.3 Energy and SHM For ideal SHM the energy is a constant of the motion and is E = 1 2 mẋ kx2 (20) 2 Damped Simple Harmonic Motion Linear Damping Suppose we add a linear damping term to the equation of motion to get ẍ = c mẋ k m x (21) It is convenient to define a damping factor γ = so that the equation of motion becomes ẍ + 2γẋ + ω0 2 = 0 (22) Next let s introduce a differential operator, D = d/dt. The operator has no independent meaning, but once it is applied to a function its meaning is clear. The equation of motion can be written [ D 2 + 2γD + ω0 2 ] x = 0 (23) [ ] [ ] D + γ γ 2 ω0 2 D + γ + γ 2 ω0 2 x = 0 (24) There are two solutions, one for the first bracket operating on x equalling 0 and one for the second bracket operating on x being zero. c 2m 4

5 These are first order differential equations and are easy to solve resulting in a general solution x(t) = A 1 exp [ (γ q)t] + A 2 exp [ (γ + q)t] (25) where q = γ 2 ω 2 0. Case 1 When γ 2 ω0 2 > 0, q is real and positive and we have the overdamped case, where a displaced mass slowly returns to equilibrium with no oscillations. Note that (γ q) > 0. Case 3 When γ 2 ω0 2 < 0, q is imaginary and we have the underdamped case where the mass oscillates about equilibrium with a decreasing amplitude. Defining = ω0 2 γ2 (26) and doing some work (shown in the text) we can write the solution can as where A and θ 0 are determined by initial conditions. x(t) = e γt A cos( t + θ 0 ) (27) Case 2 When γ 2 ω0 2 = 0 we have critical damping. The mass quickly returns to equilibrium without overshooting. If we start with Equation 25, we run into a problem: we have only one arbitrary integration constant. To find a solution with two integration constants we must return to the differential equation, Equation 24, that becomes [D + γ] [D + γ] x = 0 (28) This is a second order differential equation and we need two arbitrary constants. Here is the trick needed to get the general solution. Define u = (D + γ)x. Then (D + γ)u = 0 and this has the solution u = Ae γt. The solution for x is found from Ae γt = (D + γ)x A = e γt (D + γ)x = D(xe γt ) xe γt = At + B x = Ate γt + Be γt (29) Example Suppose we have a mass of 3.0 kg attached to a spring of constant 12.0 N/m, and with initial conditions that at t = 0, position is 1.0 m and velocity is 0.0 m/s. (a) For the underdamped case with c = 4.0 kg/s find the values of the integration constants. [A = 1.06 m, θ 0 = rad] (b) For the critically damped case with c = 12 kg/s find the values of the integration constants. [A = 2.0 m/s, B = 1.0 m] (c) For the overdamped case of c = 20 kg/s, find the values of the integration constants. [A 1 = m, A 2 = m] 5

6 2.1 Energy and Linear Damping The instantaneous energy is still E = mẋ 2 /2 + kx 2 /2 so we can look at how this changes with time. de = mẋẍ + kẋx = (mẍ + kx)ẋ (30) dt But from Equation 21, mẍ + kx = cẋ, hence de dt = cẋ2 (31) 2.2 Quality Factor for Damped SHM For the underdamped case we can characterize the system by its Q-factor, Q = 2π E E (32) where E is the energy stored in the oscillator (think in terms of the average energy) and E is the energy lost in one period of the oscillation. Q is calculated usually for weakly damped systems where the energy does not change quickly. In this case the energy at time t can be written Using Equation 27 for x we differentiate to get Then, changing variables to z = t + θ 0 E = E = 1 2 ka2 e 2γt = 1 2 mω2 0A 2 e 2γt (33) ẋ = Ae γt (γ cos( t + θ 0 ) + sin( t + θ 0 )) (34) Td 0 = ca2 cẋ 2 dt (35) 2π+θ0 θ 0 e 2γt [ γ 2 cos 2 z + 2γ cos z sin z + ω 2 d sin2 z ] dz (36) But for weakly decaying systems the exponential changes very little during the integration, so we remove it from the integral. The integration limits can start anywhere as long as they extend over 2π. So 2π E = ca2 e 2γt 0 [ γ 2 cos 2 z + 2γ cos z sin z + ω 2 d sin2 z ] dz (37) 6

7 The integrals for cos 2 and sin 2 each yield π, and the cross term becomes 0, giving Then E = πca2 e 2γt [ γ 2 + ω 2 ] πca 2 d = e 2γt ω0 2 (38) Q = 2π 1 2 mω2 0 A2 e 2γt πca 2 e 2γt ω 2 0 = 2γ Table in the text lists some Q s for underdamped systems. The earth (responding to earthquakes) has Q = 250 to 1400 while a piano string has Q = (39) 2.3 An Example Suppose a mass of 500 kg is attached to an ideal spring of force constant 2000 N/m and a linear damping shock absorber of constant 20 N s/m. Find (a) The natural (undamped) angular frequency and frequency in Hz. (b) The frequency with damping applied. (c) The Q of the system (d) The time for the amplitude to reduce to 1/10 of its original value. (e) The energy when the amplitude has reduced to 1/10 its original value. We have m = 500 kg, k = 2000 N/m, and c = 20 N s/m. (a) Use ω 0 = k/m = rad/s. This gives a frequency of f = ω 0 /2π = Hz (and a period of s). (b) Compute γ = c/2m = 0.02 rad/s. Then = ω 2 0 γ2 = rad/s. (c) The quality factor is Q = /2γ = (d) The envelope of the oscillation varies as A(t) = Ae γt, so solve 0.1 = e 0.02t to get t = s. (37 oscillations during this time.) (e) Since E A 2, when the amplitude is reduced to 1/10 of its original value, the energy is reduced to 1/100 of its original value. 3 Phase Space Plots of position versus time and velocity versus time are useful, but often a plot of velocity (or momentum) versus position is drawn. A plot such as this is called a phase plot. 7

8 For undamped SHM, x = A sin(ω 0 t + φ 0 ) and ẋ = Aω 0 cos(ω 0 t + φ 0 ). We combine these so as to eliminate time and get the parametric equation x 2 A 2 + ẋ2 A 2 ω 2 0 = 1 (40) which is the equation of an ellipse. Figure shows several trajectories in phase space corresponding to different initial conditions. All trajectories in this situation form closed paths. For underdamped oscillations the phase plot spirals in toward a final value of v = 0. The text does some very clever changes of variables to make the phase plot more obvious: I will just demonstrate using Easy JAVA Simulation, and extend to the case of critical damping and over damping. Non linear oscillators can lead to chaotic motion, a topic left for other courses. 4 Forced Damped Harmonic Motion Consider the damped harmonic oscillator subject to a periodic driving force F 0 cos ωt. The equation of motion is mẍ = kx cẋ + F 0 cos ωt (41) First consider undamped motion and guess the solution to be x = A cos(ωt φ) with two variables, A, φ to be determined. The equation of motion becomes maω 2 cos(ωt φ) + ka cos(ωt φ) = F 0 cos ωt (42) This only works for the two values φ = 0, π, implying A = F 0/m ω 2 0 ω2 φ = 0 Low frequency (43) = F 0/m ω 2 ω 2 0 This gives a resonance at ω = ω 0. φ = π High frequency (44) When we add damping and the velocity term we have difficulty when starting with a cosine solution. Instead we assume a driving force as F 0 e iωt and consider the real part of 8

9 the complex quantity to describe the physical situation. We guess a solution x = Ae iωt φ so that the equation of motion becomes mω 2 Ae iωt φ + kae iωt φ + icωae iωt φ = F 0 e iωt (45) mω 2 A + ka + icωa = F 0 e iφ = F 0 (cos φ + i sin φ) (46) The real parts must be equal, and the imaginary parts must be equal giving These can be solved simultaneously and give A(k mω 2 ) = F 0 cos φ (47) c ω A = F 0 sin φ (48) tan φ = A = 2γω ω0 2 ω2 (49) F 0 /m (ω 2 0 ω 2 ) 2 + 4γ 2 ω 2 (50) Further details should have been discussed in Vibrations and Waves. You will probably see resonance in an electrical circuit when you take Electronic Measurements. When the resonance is fairly sharp, meaning relatively small damping, the resonant frequency is very close to the natural frequency. For these cases we define the sharpness of the resonance with another Q factor same symbol and definition as before, but a different meaning. If the resonance has an amplitude A max at resonance, ω 0, then there are frequencies above and below the resonant frequency at which A = A max / 2. These two frequencies are called the half-energy (or half-power) frequencies, and the difference in frequencies is ω = 2γ = c (51) m and Q resonance = ω 0 ω ω 0 2γ (52) 9

Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations

Spring 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 information

Mechanical Vibrations

Mechanical Vibrations Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

More information

Physics 1120: Simple Harmonic Motion Solutions

Physics 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 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

226 Chapter 15: OSCILLATIONS

226 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 information

Physics 41 HW Set 1 Chapter 15

Physics 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 information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

2.2 Magic with complex exponentials

2.2 Magic with complex exponentials 2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or

More information

Oscillations. Chapter 1. 1.1 Simple harmonic motion. 1.1.1 Hooke s law and small oscillations

Oscillations. Chapter 1. 1.1 Simple harmonic motion. 1.1.1 Hooke s law and small oscillations Chapter 1 Oscillations David Morin, morin@physics.harvard.edu A wave is a correlated collection of oscillations. For example, in a transverse wave traveling along a string, each point in the string oscillates

More information

Simple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines

Simple 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 information

HOOKE S LAW AND SIMPLE HARMONIC MOTION

HOOKE S LAW AND SIMPLE HARMONIC MOTION HOOKE S LAW AND SIMPLE HARMONIC MOTION Alexander Sapozhnikov, Brooklyn College CUNY, New York, alexs@brooklyn.cuny.edu Objectives Study Hooke s Law and measure the spring constant. Study Simple Harmonic

More information

Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k

Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k Physics 1C Midterm 1 Summer Session II, 2011 Solutions 1. If F = kx, then k m is (a) A (b) ω (c) ω 2 (d) Aω (e) A 2 ω Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of

More information

2.6 The driven oscillator

2.6 The driven oscillator 2.6. THE DRIVEN OSCILLATOR 131 2.6 The driven oscillator We would like to understand what happens when we apply forces to the harmonic oscillator. That is, we want to solve the equation M d2 x(t) 2 + γ

More information

AP Physics C. Oscillations/SHM Review Packet

AP 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

Introduction to Complex Numbers in Physics/Engineering

Introduction to Complex Numbers in Physics/Engineering Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The

More information

Physics 231 Lecture 15

Physics 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 information

Understanding Poles and Zeros

Understanding Poles and Zeros MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

More information

Practice Test SHM with Answers

Practice 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 information

Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD. q j

Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD. q j Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD Introduction Let take Lagrange s equations in the form that follows from D Alembert s principle, ) d T T = Q j, 1) dt q j q j suppose that the generalized

More information

Second Order Linear Differential Equations

Second 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 information

Unit - 6 Vibrations of Two Degree of Freedom Systems

Unit - 6 Vibrations of Two Degree of Freedom Systems Unit - 6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two

More information

AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?

AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

More information

Mechanics 1: Conservation of Energy and Momentum

Mechanics 1: Conservation of Energy and Momentum Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

More information

Chapter 15, example problems:

Chapter 15, example problems: Chapter, example problems: (.0) Ultrasound imaging. (Frequenc > 0,000 Hz) v = 00 m/s. λ 00 m/s /.0 mm =.0 0 6 Hz. (Smaller wave length implies larger frequenc, since their product,

More information

How do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation

How do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation 1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two

More information

19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.

More information

Determination of Acceleration due to Gravity

Determination of Acceleration due to Gravity Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

More information

Let s first see how precession works in quantitative detail. The system is illustrated below: ...

Let s first see how precession works in quantitative detail. The system is illustrated below: ... lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

More information

Simple Harmonic Motion Experiment. 1 f

Simple Harmonic Motion Experiment. 1 f Simple Harmonic Motion Experiment In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring

More information

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

More information

Lecture L5 - Other Coordinate Systems

Lecture L5 - Other Coordinate Systems S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

More information

Simple Harmonic Motion

Simple 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 information

Lesson 11. Luis Anchordoqui. Physics 168. Tuesday, December 8, 15

Lesson 11. Luis Anchordoqui. Physics 168. Tuesday, December 8, 15 Lesson 11 Physics 168 1 Oscillations and Waves 2 Simple harmonic motion If an object vibrates or oscillates back and forth over same path each cycle taking same amount of time motion is called periodic

More information

1.2 Second-order systems

1.2 Second-order systems 1.2. SECOND-ORDER SYSTEMS 25 if the initial fluid height is defined as h() = h, then the fluid height as a function of time varies as h(t) = h e tρg/ra [m]. (1.31) 1.2 Second-order systems In the previous

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT 4 Understand single-phase alternating current (ac) theory Single phase AC

More information

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

More information

Module P5.3 Forced vibrations and resonance

Module P5.3 Forced vibrations and resonance F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P5.3 Forced vibrations and resonance 1 Opening items 1.1 Module introduction 1.2 Fast track questions 1.3 Ready to study? 2 Driven

More information

SOLUTIONS TO CONCEPTS CHAPTER 15

SOLUTIONS TO CONCEPTS CHAPTER 15 SOLUTIONS TO CONCEPTS CHAPTER 15 1. v = 40 cm/sec As velocity of a wave is constant location of maximum after 5 sec = 40 5 = 00 cm along negative x-axis. [(x / a) (t / T)]. Given y = Ae a) [A] = [M 0 L

More information

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

More information

Experiment 9. The Pendulum

Experiment 9. The Pendulum Experiment 9 The Pendulum 9.1 Objectives Investigate the functional dependence of the period (τ) 1 of a pendulum on its length (L), the mass of its bob (m), and the starting angle (θ 0 ). Use a pendulum

More information

Differential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University

Differential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter. Linear second order ODEs 5.. Newton s second law 5.2. Springs

More information

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

More information

State Newton's second law of motion for a particle, defining carefully each term used.

State Newton's second law of motion for a particle, defining carefully each term used. 5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

More information

ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1

ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1 19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point

More information

Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.

Notice 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 information

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* + v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure

More information

Second Order Systems

Second Order Systems Second Order Systems Second Order Equations Standard Form G () s = τ s K + ζτs + 1 K = Gain τ = Natural Period of Oscillation ζ = Damping Factor (zeta) Note: this has to be 1.0!!! Corresponding Differential

More information

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Rotational Motion: 11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

More information

PENDULUM PERIODS. First Last. Partners: student1, student2, and student3

PENDULUM PERIODS. First Last. Partners: student1, student2, and student3 PENDULUM PERIODS First Last Partners: student1, student2, and student3 Governor s School for Science and Technology 520 Butler Farm Road, Hampton, VA 23666 April 13, 2011 ABSTRACT The effect of amplitude,

More information

HSC Mathematics - Extension 1. Workshop E4

HSC Mathematics - Extension 1. Workshop E4 HSC Mathematics - Extension 1 Workshop E4 Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong

More information

Chapter 6 Circular Motion

Chapter 6 Circular Motion Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

More information

BASIC VIBRATION THEORY

BASIC VIBRATION THEORY CHAPTER BASIC VIBRATION THEORY Ralph E. Blae INTRODUCTION This chapter presents the theory of free and forced steady-state vibration of single degree-of-freedom systems. Undamped systems and systems having

More information

Aim : To study how the time period of a simple pendulum changes when its amplitude is changed.

Aim : To study how the time period of a simple pendulum changes when its amplitude is changed. Aim : To study how the time period of a simple pendulum changes when its amplitude is changed. Teacher s Signature Name: Suvrat Raju Class: XIID Board Roll No.: Table of Contents Aim..................................................1

More information

Section 5.0 : Horn Physics. By Martin J. King, 6/29/08 Copyright 2008 by Martin J. King. All Rights Reserved.

Section 5.0 : Horn Physics. By Martin J. King, 6/29/08 Copyright 2008 by Martin J. King. All Rights Reserved. Section 5. : Horn Physics Section 5. : Horn Physics By Martin J. King, 6/29/8 Copyright 28 by Martin J. King. All Rights Reserved. Before discussing the design of a horn loaded loudspeaker system, it is

More information

! n. Problems and Solutions Section 1.5 (1.66 through 1.74)

! n. Problems and Solutions Section 1.5 (1.66 through 1.74) Problems and Solutions Section.5 (.66 through.74).66 A helicopter landing gear consists of a metal framework rather than the coil spring based suspension system used in a fixed-wing aircraft. The vibration

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com

Copyright 2011 Casa Software Ltd. www.casaxps.com Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations

More information

AP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017

AP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017 AP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017 Dear Student: The AP physics course you have signed up for is designed to prepare you for a superior performance on the AP test. To complete material

More information

SOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM

SOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM SOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM This work covers elements of the syllabus for the Engineering Council Exam D5 Dynamics of Mechanical Systems, C05 Mechanical and

More information

Review D: Potential Energy and the Conservation of Mechanical Energy

Review D: Potential Energy and the Conservation of Mechanical Energy MSSCHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Fall 2005 Review D: Potential Energy and the Conservation of Mechanical Energy D.1 Conservative and Non-conservative Force... 2 D.1.1 Introduction...

More information

physics 1/12/2016 Chapter 20 Lecture Chapter 20 Traveling Waves

physics 1/12/2016 Chapter 20 Lecture Chapter 20 Traveling Waves Chapter 20 Lecture physics FOR SCIENTISTS AND ENGINEERS a strategic approach THIRD EDITION randall d. knight Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide

More information

Differentiation of vectors

Differentiation of vectors Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

More information

Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.

Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad. Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.

More information

PHYS 211 FINAL FALL 2004 Form A

PHYS 211 FINAL FALL 2004 Form A 1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each

More information

F = ma. F = G m 1m 2 R 2

F = ma. F = G m 1m 2 R 2 Newton s Laws The ideal models of a particle or point mass constrained to move along the x-axis, or the motion of a projectile or satellite, have been studied from Newton s second law (1) F = ma. In the

More information

A) F = k x B) F = k C) F = x k D) F = x + k E) None of these.

A) F = k x B) F = k C) F = x k D) F = x + k E) None of these. CT16-1 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 information

L and C connected together. To be able: To analyse some basic circuits.

L and C connected together. To be able: To analyse some basic circuits. circuits: Sinusoidal Voltages and urrents Aims: To appreciate: Similarities between oscillation in circuit and mechanical pendulum. Role of energy loss mechanisms in damping. Why we study sinusoidal signals

More information

3.2 Sources, Sinks, Saddles, and Spirals

3.2 Sources, Sinks, Saddles, and Spirals 3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients

More information

Orbital Mechanics. Angular Momentum

Orbital Mechanics. Angular Momentum Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

More information

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc. Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

More information

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v, 1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

More information

Periodic wave in spatial domain - length scale is wavelength Given symbol l y

Periodic wave in spatial domain - length scale is wavelength Given symbol l y 1.4 Periodic Waves Often have situations where wave repeats at regular intervals Electromagnetic wave in optical fibre Sound from a guitar string. These regularly repeating waves are known as periodic

More information

ELASTIC FORCES and HOOKE S LAW

ELASTIC FORCES and HOOKE S LAW PHYS-101 LAB-03 ELASTIC FORCES and HOOKE S LAW 1. Objective The objective of this lab is to show that the response of a spring when an external agent changes its equilibrium length by x can be described

More information

State Newton's second law of motion for a particle, defining carefully each term used.

State Newton's second law of motion for a particle, defining carefully each term used. 5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

More information

both double. A. T and v max B. T remains the same and v max doubles. both remain the same. C. T and v max

both double. A. T and v max B. T remains the same and v max doubles. both remain the same. C. T and v max Q13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object s maximum speed

More information

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

More information

PHYSICS HIGHER SECONDARY FIRST YEAR VOLUME - II. Revised based on the recommendation of the Textbook Development Committee. Untouchability is a sin

PHYSICS HIGHER SECONDARY FIRST YEAR VOLUME - II. Revised based on the recommendation of the Textbook Development Committee. Untouchability is a sin PHYSICS HIGHER SECONDARY FIRST YEAR VOLUME - II Revised based on the recommendation of the Textbook Development Committee Untouchability is a sin Untouchability is a crime Untouchability is inhuman TAMILNADU

More information

1) The time for one cycle of a periodic process is called the A) wavelength. B) period. C) frequency. D) amplitude.

1) The time for one cycle of a periodic process is called the A) wavelength. B) period. C) frequency. D) amplitude. practice wave test.. Name Use the text to make use of any equations you might need (e.g., to determine the velocity of waves in a given material) MULTIPLE CHOICE. Choose the one alternative that best completes

More information

Fourier Analysis. u m, a n u n = am um, u m

Fourier Analysis. u m, a n u n = am um, u m Fourier Analysis Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That s the subject of this chapter.

More information

7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )

7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( ) 34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using

More information

Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

More information

Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay

Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay Module - 2 Vibration Theory Lecture - 8 Forced Vibrations, Dynamic Magnification Factor Let

More information

Alternating-Current Circuits

Alternating-Current Circuits hapter 1 Alternating-urrent ircuits 1.1 A Sources... 1-1. Simple A circuits... 1-3 1..1 Purely esistive load... 1-3 1.. Purely Inductive oad... 1-5 1..3 Purely apacitive oad... 1-7 1.3 The Series ircuit...

More information

In order to describe motion you need to describe the following properties.

In order to describe motion you need to describe the following properties. Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties.

More information

Engineering Feasibility Study: Vehicle Shock Absorption System

Engineering Feasibility Study: Vehicle Shock Absorption System Engineering Feasibility Study: Vehicle Shock Absorption System Neil R. Kennedy AME40463 Senior Design February 28, 2008 1 Abstract The purpose of this study is to explore the possibilities for the springs

More information

Chapter 18 Static Equilibrium

Chapter 18 Static Equilibrium Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example

More information

PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION

PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION I. INTRODUCTION The objective of this experiment is the study of oscillatory motion. In particular the springmass system and the simple

More information

Tennessee State University

Tennessee State University Tennessee State University Dept. of Physics & Mathematics PHYS 2010 CF SU 2009 Name 30% Time is 2 hours. Cheating will give you an F-grade. Other instructions will be given in the Hall. MULTIPLE CHOICE.

More information

Lecture L6 - Intrinsic Coordinates

Lecture L6 - Intrinsic Coordinates S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6 - Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed

More information

HOOKE S LAW AND OSCILLATIONS

HOOKE S LAW AND OSCILLATIONS 9 HOOKE S LAW AND OSCILLATIONS OBJECTIVE To measure the effect of amplitude, mass, and spring constant on the period of a spring-mass oscillator. INTRODUCTION The force which restores a spring to its equilibrium

More information

The Math Circle, Spring 2004

The Math Circle, Spring 2004 The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

More information

Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE

Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE 1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object

More information

Difference between a vector and a scalar quantity. N or 90 o. S or 270 o

Difference between a vector and a scalar quantity. N or 90 o. S or 270 o Vectors Vectors and Scalars Distinguish between vector and scalar quantities, and give examples of each. method. A vector is represented in print by a bold italicized symbol, for example, F. A vector has

More information

VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

More information

Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

More information

Chapter 8: Potential Energy and Conservation of Energy. Work and kinetic energy are energies of motion.

Chapter 8: Potential Energy and Conservation of Energy. Work and kinetic energy are energies of motion. Chapter 8: Potential Energy and Conservation of Energy Work and kinetic energy are energies of motion. Consider a vertical spring oscillating with mass m attached to one end. At the extreme ends of travel

More information

APPLICATIONS. are symmetric, but. are not.

APPLICATIONS. are symmetric, but. are not. CHAPTER III APPLICATIONS Real Symmetric Matrices The most common matrices we meet in applications are symmetric, that is, they are square matrices which are equal to their transposes In symbols, A t =

More information

How to Graph Trigonometric Functions

How to Graph Trigonometric Functions How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

More information

16.2 Periodic Waves Example:

16.2 Periodic Waves Example: 16.2 Periodic Waves Example: A wave traveling in the positive x direction has a frequency of 25.0 Hz, as in the figure. Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of the wave. 1

More information