Mh1: Simple Harmonic Motion. Chapter 15. Motion of a Spring-Mass System. Periodic Motion. Oscillatory Motion

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Mh1: Simple Harmonic Motion. Chapter 15. Motion of a Spring-Mass System. Periodic Motion. Oscillatory Motion"

Transcription

1 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 object that regularly repeats! The object returns to a given position after a fixed tie interval! Siple Haronic Motion! When the force acting on the object is proportional to the position of the object relative to soe equilibriu position, and is directed towards it! F = - k. x Motion of a Spring-Mass Syste! Active Figure 15.0! Block of ass is attached to a spring, and free to ove on a frictionless horizontal surface! When the spring is neither stretched nor copressed, the block is at the equilibriu position! x = 0

2 Hooke s Law! Active Figure 15.01! Hooke s Law for the stretched spring F s = - kx! F s is the restoring force! It is always directed toward the equilibriu position! Thus it is always opposite the displaceent fro equilibriu! k is the force (spring) constant! x is the displaceent fro the equilibriu position Acceleration! The force described by Hooke s Law is the net force in Newton s Second Law F Hooke x = F! kx = a a Newton x k =! x The Acceleration is! Proportional to the displaceent of the block! Has direction opposite the direction of the displaceent fro equilibriu! Thus, this is Siple Haronic Motion! It is not constant a = " k x! Thus, the kineatic equations (i.e. v=u+at etc.) for constant acceleration cannot be applied A block on the end of a spring is pulled to position x = A and released fro rest. In one full cycle of its otion, through what total distance does it travel? 1. A/. A 3. A 4. 4A

3 PHYSCLIPS : Waves and Sound 1.1 Oscillations Vertical Orientation! When the block is hung fro a vertical spring, its weight will cause the spring to stretch! If the resting position of the spring is defined as y = 0, the sae analysis as was done for the horizontal spring will apply to the vertical spring-block syste! Here y = x - x equilibriu where g = k x equilibriu! See Movie HMM03AN1! SHM with vertical spring! 19sec Siple Haronic Motion Matheatical Representation! Model the block as a particle! Choose x as the axis along which the oscillation occurs! Apply Newton s nd Law:! Acceleration d x k a = =! x! We let k! =! Then a = -! x dt Siple Haronic Motion Graphical Representation! A solution is x(t) = A cos (!t + ")! A,!, " are all constants! Differentiate, to yield: v = -!A sin (!t + ")! a =-! A cos (!t + ")!! So that a = -! x as for SHM! Siilarly x(t) = A sin (!t + ") is also a solution

4 Siple Haronic Motion the constants of otion! A is the aplitude of the otion! This is the axiu position of the particle in either the positive or negative direction!! is called the angular frequency! Units are radian/s! " is the phase constant or the initial phase angle! Siple Haronic Motion, cont! Since x(t) = A cos (!t + "), then A and " are deterined uniquely by the position and velocity of the particle at t = 0! e.g. If the particle is at x = A at t = 0, then " = 0! The phase of the otion is (!t + ")! x (t) is periodic and its value is the sae each tie!t increases by " radians! PHYSCLIPS : Waves and Oscillations 1. The Equations Mh11: Siple Haronic Motion air track & springs No friction!

5 Consider a graphical representation of siple haronic otion. When the object is at point A on the graph, what can you say about its position and velocity? 1. The position and velocity are both positive.. The position and velocity are both negative. 3. The position is positive, and its velocity is zero. 4. The position is negative, and its velocity is zero. 5. The position is positive, and its velocity is negative. 6. The position is negative, and its velocity is positive. The figure shows two curves representing objects undergoing siple haronic otion. The correct description of these two otions is that the siple haronic otion of object B is: 1. of larger angular frequency and larger aplitude than that of object A.. of larger angular frequency and saller aplitude than that of object A. 3. of saller angular frequency and larger aplitude than that of object A. 4. of saller angular frequency and saller aplitude than that of object A. Period and Frequency! The period, T, is the tie interval required for the particle to go through one full cycle of its otion! i.e. x(t)=x(t+t) and v(t)=v(t+t) with T="/#! The inverse of the period is called the frequency! It represents the nuber of oscillations the particle undergoes per unit tie interval! f = 1/T = #/"! Units are cycles per second = hertz (Hz)! Since! = then k 1 T =! ƒ = k! k Period and Frequency: II! Frequency and period depend only on the ass of the particle and the force constant of the spring! See Active Figure 15.06! Frequency is larger for a stiffer spring (large values of k) and decreases with increasing ass of the particle f = 1 " k

6 Suary: Motion Equations for Siple Haronic Motion x( t) = A cos (! t + ") dx v = = #! Asin (! t + ") dt d x a = = #! A cos(! t + ") dt! Reeber, siple haronic otion is not uniforly accelerated otion Maxiu Values of v and a! Because the sine and cosine functions oscillate between ±1, then the axiu values of velocity and acceleration for an object in SHM are k vax =! A = A k aax =! A = A All three together HMA03AN3: SHM for x, v, a Phase of x, v, a! The graphs show:! (a) displaceent as a function of tie! (b) velocity as a function of tie! (c ) acceleration as a function of tie! Velocity is 90 o out of phase with the displaceent! Acceleration is 180 o out of phase with the displaceent 3 seconds

7 PHYSCLIPS : Waves and Oscillations 1.3 Forces & Energy An object of ass is hung fro a spring and set into oscillation. The period of the oscillation is easured and recorded as T. The object of ass is reoved and replaced with an object of ass. When this object is set into oscillation, what is the period of the otion? 1. T. 3. T T 4. T / 5. T/ SHM Exaple 1 SHM Exaple! x(t) = A cos (!t + ")! Initial conditions at t = 0:! x (0)= A! v (0) = 0! Thus " = 0! The acceleration reaches extrees of ±! A! The velocity reaches extrees of ±!A! x(t) = A cos (!t + ")! Initial conditions at t = 0:! x (0)=0! v (0) = v ax! Thus " = #/! i.e. x(t) = A sin (!t)! Graph is shifted one-quarter cycle to the right copared to graph when x(0) = A$

8 Energy of the SHM Oscillator: I! Assue a spring-ass syste is oving on a frictionless surface! This tells us the total energy is constant! The kinetic energy can be found by! K =! v =!! A sin (!t + ")! i.e. K =! k A sin (!t + ") as # =k/! The elastic potential energy can be found by! U = " Fdx = " kxdx = 1 kx! i.e. U =! kx =! ka cos (!t + ")! Thus, the total energy is K + U =! ka Energy of the SHM Oscillator: II! See Active Figure 15.09! The total echanical energy is constant =! ka! The total echanical energy is proportional to the square of the aplitude! Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block Suary of Energy in SHM Active Figure Mh7: Transfer of Energy Coupled pendula

9 Molecular Model of SHM! If the atos in the olecule do not ove too far, the force between the can be odeled as if there were springs between the atos! The potential energy acts siilar to that of the SHM oscillator Mh3: Siple Haronic Motion and Circular Motion Connection between circular otion and SHM can be seen on the overhead projector SHM and Circular Motion: I! See Active Figure 15.13! An overhead view which shows the relationship between SHM and circular otion! As the ball rotates with constant angular velocity, its shadow oves back and forth in siple haronic otion SHM and Circular Motion: II! The particle oves along the circle with constant angular velocity!! OP akes an angle % with the x axis! At tie t the angle between OP and the x axis will be % =!t + "$! So x(t) = A cos (!t + ") and v = -!A sin (!t + "), a =! A cos (!t + ")! Thus a =! x i.e. SHM!

10 SHM and Circular Motion: III! Siple Haronic Motion along a straight line can be represented by the projection of unifor circular otion along the diaeter of a reference circle! Unifor circular otion can be considered a cobination of two siple haronic otions! One along the x-axis! The other along the y-axis! The two differ in phase by 90 o The figure shows the position of an object in unifor circular otion at t = 0. A light shines fro above and projects a shadow of the object on a screen below the circular otion. What are the correct values for the aplitude and phase constant (relative to an x axis to the right) of the siple haronic otion of the shadow? and and and " and " Mh5: Siple Haronic Motion the siple pendulu Siple, but a classic! The Siple Pendulu! See Active Figure 15.16! Exhibiting periodic otion! The otion occurs in the vertical plane and is driven by the gravitational force! The otion is very close to that of the SHM oscillator for sall angles

11 Siple Pendulu II! The forces acting on the bob are T and g! T is the force exerted on the bob by the string! g is the gravitational force! The tangential coponent of the gravitational force is a restoring force! This is -g sin(%)! The arc length is s = L %! So ds dt = L d" and d s dt dt = L d " dt Siple Pendulu III! In the tangential direction, F t = "gsin# = d s dt = L d # dt! The length, L, of the pendulu is constant, and so, for sall values of % d! g g = " sin! = "! dt L L! Thus the otion is SHM with # =g/l Siple Pendulu IV! The function % can be written as % = % ax cos (!t + ")! The angular frequency is! = g L! The period is Mh1: The Pendulu period and length Half class tie top pendulu, half class tie botto pendulu. Is T $%L? T! = =! " L g

12 A grandfather clock depends on the period of a pendulu to keep correct tie. Suppose a grandfather clock is calibrated correctly and then a ischievous child slides the bob of the pendulu downward on the oscillating rod. The grandfather clock will run: A grandfather clock depends on the period of a pendulu to keep correct tie. Suppose a grandfather clock is calibrated correctly at sea level and is then taken to the top of a very tall ountain. The grandfather clock will now run: 1. slow.. fast. 3. correctly. 1. slow.. fast. 3. correctly. Daped Oscillations: I! See Active Figure 15.1! In any real systes, non-conservative forces are present! Friction is a coon non-conservative force! In this case, the echanical energy of the syste diinishes in tie, the otion is said to be daped Daped Oscillations: II! A graph for a daped oscillation! The aplitude decreases with tie! The blue dashed lines represent the envelope of the otion

13 Daped Oscillation, Exaple! One exaple of daped otion occurs when an object is attached to a spring and suberged in a viscous liquid! The retarding force can be expressed as R = - b v where b is a positive constant! b is called the daping coefficient! Fro Newton s Second Law the equation of otion becoes F x = -k x bv x = a x Types of Daping Not for exaination " b # = 0 $ % &!! ' ( k!! is also called the natural 0 = frequency of the syste! If R ax = bv ax < ka, the syste is said to be underdaped! When b reaches a critical value b c such that b c / =! 0, the syste will not oscillate! The syste is said to be critically daped! If R ax = bv ax > ka and b/ >! 0, the syste is said to be overdaped Types of Daping: II PHYSCLIPS : Sound and Waves 1.4 Non-linearity and Daping! Graphs of position versus tie for! (a) an underdaped oscillator! (b) a critically daped oscillator! (c) an overdaped oscillator! For critically daped and overdaped there is no angular frequency

14 Mh13: Double Pendulu Forced Oscillations Forced Oscillations: I! It is possible to copensate for the loss of energy in a daped syste by applying an external force! e.g. pushing child on a swing at angular frequency!! In general F = F 0 sin(!t) bv kx! The aplitude of the otion reains constant if the energy input per cycle exactly equals the decrease in echanical energy in each cycle that results fro resistive forces Forced Oscillations: II! Motion is x (t) = A cos (!t + ") where the aplitude of the driven oscillation is A = F 0 " b! # $ + % & ' ( (!! 0 ) Mh: Resonance Pendulu Large ball will cause all balls to oscillate but only the ball of the sae natural frequency will oscillate at large aplitude. Try and predict beforehand.!! 0 is the natural frequency of the undaped oscillator

15 Resonance: I! When the frequency of the driving force is near the natural frequency (! &! 0 ) an increase in aplitude occurs! This draatic increase in the aplitude is called resonance! The natural frequency! 0 is also called the resonance frequency of the syste Resonance: II! Resonance (axiu peak) occurs when the driving frequency equals the natural frequency! The aplitude increases with decreased daping! The curve broadens as the daping increases! The shape of the resonance curve depends on b PHYSCLIPS : Waves and Sound 1.5 Resonance Resonance: Tacoa Narrows Bridge Noveber 7, 1940, Washington State, USA

16 PHYSCLIPS : Waves and Sound Diensions

Chapter 13 Simple Harmonic Motion

Chapter 13 Simple Harmonic Motion We are to adit no ore causes of natural things than such as are both true and sufficient to explain their appearances. Isaac Newton 13.1 Introduction to Periodic Motion Periodic otion is any otion that

More information

Lesson 44: Acceleration, Velocity, and Period in SHM

Lesson 44: Acceleration, Velocity, and Period in SHM Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain

More information

Physics 211: Lab Oscillations. Simple Harmonic Motion.

Physics 211: Lab Oscillations. Simple Harmonic Motion. Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.

More information

Simple Harmonic Motion Concepts

Simple 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 information

Simple Harmonic Motion

Simple Harmonic Motion SHM-1 Siple Haronic Motion A pendulu, a ass on a spring, and any other kinds of oscillators ehibit a special kind of oscillatory otion called Siple Haronic Motion (SHM). SHM occurs whenever : i. there

More information

Simple Harmonic Motion. Simple Harmonic Motion

Simple 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 information

Work, Energy, Conservation of Energy

Work, Energy, Conservation of Energy This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and non-conservative forces, with soe

More information

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

Hooke s Law. Spring. Simple Harmonic Motion. Energy. 12/9/09 Physics 201, UW-Madison 1 Hooke s Law Spring Simple Harmonic Motion Energy 12/9/09 Physics 201, UW-Madison 1 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, UW-Madison 2 We know

More information

Lecture L9 - Linear Impulse and Momentum. Collisions

Lecture L9 - Linear Impulse and Momentum. Collisions J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9 - Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,

More information

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

PH 221-3A Fall Waves - I. Lectures Chapter 16 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)

PH 221-3A Fall Waves - I. Lectures Chapter 16 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) PH 1-3A Fall 010 Waves - I Lectures 5-6 Chapter 16 (Halliday/Resnick/Walker, Fundaentals of Physics 8 th edition) 1 Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We

More information

1 of 10 11/23/2009 6:37 PM

1 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 information

Chapter 1. Oscillations. Oscillations

Chapter 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 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

Mechanics 1: Motion of a Particle in 1 Dimension

Mechanics 1: Motion of a Particle in 1 Dimension Mechanics 1: Motion of a Particle in 1 Diension Motion of a Particle in One Diension eans that the particle is restricted to lie on a curve. If you think hard about this stateent you ight think that the

More information

Periodic Motion or Oscillations. Physics 232 Lecture 01 1

Periodic 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 information

Simple Harmonic Motion

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 information

Answer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Hoework 7 David McIntyre 453 Mar 5, 004 This print-out should have 4 questions. Multiple-choice questions ay continue on the next colun or page find all choices before aking your selection.

More information

ConcepTest 4.6 Force and Two Masses

ConcepTest 4.6 Force and Two Masses ConcepTest 4.6 Force and Two Masses A force F acts on ass 1 giving acceleration a 1. The sae force acts on a different ass 2 giving acceleration a 2 = 2a 1. If 1 and 2 are glued together and the sae force

More information

SIMPLE HARMONIC MOTION

SIMPLE 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 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

LABORATORY 9. Simple Harmonic Motion

LABORATORY 9. Simple Harmonic Motion LABORATORY 9 Simple Harmonic Motion Purpose In this experiment we will investigate two examples of simple harmonic motion: the mass-spring system and the simple pendulum. For the mass-spring system we

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

If E is doubled, A will be increased by a factor of 2. That is ANew = 2 AOld. Old Old New New

If E is doubled, A will be increased by a factor of 2. That is ANew = 2 AOld. Old Old New New OSCILLAIONS 4 Q4.. Reason: here are any exaples in daily life, such as a ass hanging fro a spring, a tennis ball being volleyed back and forth, washboard road bups, a beating heart, AC electric voltage,

More information

Phys 207. Announcements. Hwk 6 is posted online; submission deadline = April 4 Exam 2 on Friday, April 8th. Today s Agenda

Phys 207. Announcements. Hwk 6 is posted online; submission deadline = April 4 Exam 2 on Friday, April 8th. Today s Agenda Phys 07 Announceents Hwk 6 is posted online; subission deadline = April 4 Ea on Friday, April 8th Today s Agenda Review of Work & Energy (Chapter 7) Work of ultiple constant forces Work done by gravity

More information

PH 221-3A Fall Waves - I. Lectures Chapter 16 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)

PH 221-3A Fall Waves - I. Lectures Chapter 16 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) PH 1-3A Fall 009 Waves - I Lectures 5-6 Chapter 16 (Halliday/Resnick/Walker, Fundaentals of Physics 8 th edition) 1 Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We

More information

Hooke s Law and Simple Harmonic Motion

Hooke 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 information

Center of Mass/Momentum

Center of Mass/Momentum Center of Mass/Momentum 1. 2. An L-shaped 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 L-shaped

More information

Lecture Presentation Chapter 14 Oscillations

Lecture 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 information

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.

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. 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 information

Experiment Type: Open-Ended

Experiment Type: Open-Ended Simple Harmonic Oscillation Overview Experiment Type: Open-Ended In this experiment, students will look at three kinds of oscillators and determine whether or not they can be approximated as simple harmonic

More information

2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position.

2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position. CHAPTER : Vibrations and Waes Answers to Questions The acceleration o a siple haronic oscillator is zero wheneer the oscillating object is at the equilibriu position 5 The iu speed is gien by = A k Various

More information

PHYS-2020: General Physics II Course Lecture Notes Section VII

PHYS-2020: General Physics II Course Lecture Notes Section VII PHYS-2020: 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 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

TOPIC E: OSCILLATIONS SPRING 2017

TOPIC E: OSCILLATIONS SPRING 2017 TOPIC E: OSCILLATIONS SPRING 017 1. Introduction 1.1 Overview 1. Degrees of freedo 1.3 Siple haronic otion. Undaped free oscillation.1 Generalised ass-spring syste: siple haronic otion. Natural frequency

More information

A Gas Law And Absolute Zero

A Gas Law And Absolute Zero A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very

More information

AP Physics. Chapter 9 Review. Momentum et. al.

AP Physics. Chapter 9 Review. Momentum et. al. AP Physics Chapter 9 Review Moentu et. al. 1. A 2000-kg truck traveling at a speed of 3.0 akes a 90 turn in a tie of 4.0 onds and eerges fro this turn with a speed of 4.0. What is the agnitude of the average

More information

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

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

More information

Lab M1: The Simple Pendulum

Lab 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 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

All you wanted to know about Formulae and Graphs (but were afraid to ask!)

All you wanted to know about Formulae and Graphs (but were afraid to ask!) All you wanted to know about Forulae and Graphs (but were afraid to ask!) o be able to carry out practical investigations in Physics you ust understand the following: 1. What variables are you investigating

More information

SIMPLE HARMONIC MOTION

SIMPLE 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 information

Simple Harmonic Motion

Simple 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 Right-angled

More information

A Gas Law And Absolute Zero Lab 11

A Gas Law And Absolute Zero Lab 11 HB 04-06-05 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution

More information

Simple Harmonic Motion MC Review KEY

Simple Harmonic Motion MC Review KEY Siple Haronic Motion MC Review EY. A block attache to an ieal sprin uneroes siple haronic otion. The acceleration of the block has its axiu anitue at the point where: a. the spee is the axiu. b. the potential

More information

p = 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:

p = 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 information

MECHANICS IV - SIMPLE HARMONIC MOTION

MECHANICS IV - SIMPLE HARMONIC MOTION M-IV-p.1 A. OSCILLATIONS B. SIMPLE PENDULUM C. KINEMATICS OF SIMPLE HARMONIC MOTION D. SPRING-AND-MASS SYSTEM E. ENERGY OF SHM F. DAMPED HARMONIC MOTION G. FORCED VIBRATION A. OSCILLATIONS A to-and-fro

More information

Version 001 Summer Review #04 tubman (IBII ) 1

Version 001 Summer Review #04 tubman (IBII ) 1 Version 001 Suer Review #04 tuban (IBII20142015) 1 This print-out should have 20 questions. Multiple-choice questions ay continue on the next colun or page find all choices before answering. Conceptual

More information

LAWS OF MOTION PROBLEM AND THEIR SOLUTION

LAWS OF MOTION PROBLEM AND THEIR SOLUTION http://www.rpauryascienceblog.co/ LWS OF OIO PROBLE D HEIR SOLUIO. What is the axiu value of the force F such that the F block shown in the arrangeent, does not ove? 60 = =3kg 3. particle of ass 3 kg oves

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

8 SIMPLE HARMONIC MOTION

8 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 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

Simple Harmonic Motion

Simple 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 information

The Mathematics of Pumping Water

The Mathematics of Pumping Water The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role

More information

Physics 101 Exam 1 NAME 2/7

Physics 101 Exam 1 NAME 2/7 Physics 101 Exam 1 NAME 2/7 1 In the situation below, a person pulls a string attached to block A, which is in turn attached to another, heavier block B via a second string (a) Which block has the larger

More information

Answer: Same magnitude total momentum in both situations.

Answer: Same magnitude total momentum in both situations. Page 1 of 9 CTP-1. In which situation is the agnitude of the total oentu the largest? A) Situation I has larger total oentu B) Situation II C) Sae agnitude total oentu in both situations. I: v 2 (rest)

More information

Chapter 6: Energy and Oscillations. 1. Which of the following is not an energy unit? A. N m B. Joule C. calorie D. watt E.

Chapter 6: Energy and Oscillations. 1. Which of the following is not an energy unit? A. N m B. Joule C. calorie D. watt E. Chapter 6: Energy and Oscillations 1. Which of the following is not an energy unit? A. N m B. Joule C. calorie D. watt E. kwh 2. Work is not being done on an object unless the A. net force on the object

More information

7. Kinetic Energy and Work

7. 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 information

turn-table 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

turn-table 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 turn-table in terms of SHM and UCM: If you look at the ball from the side, its motion could be plotted

More information

Advanced Higher Physics: MECHANICS. Simple Harmonic Motion

Advanced 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 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

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

Experiment P19: Simple Harmonic Motion - Mass on a Spring (Force Sensor, Motion Sensor)

Experiment P19: Simple Harmonic Motion - Mass on a Spring (Force Sensor, Motion Sensor) PASCO scientific Physics Lab Manual: P19-1 Science Workshop S. H. M. Mass on a Spring Experiment P19: Simple Harmonic Motion - Mass on a Spring (Force Sensor, Motion Sensor) Concept Time SW Interface Macintosh

More information

Rotational Mechanics - 1

Rotational Mechanics - 1 Rotational Mechanics - 1 The Radian The radian is a unit of angular measure. The radian can be defined as the arc length s along a circle divided by the radius r. s r Comparing degrees and radians 360

More information

F=ma From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.edu

F=ma From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.edu Chapter 4 F=a Fro Probles and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, orin@physics.harvard.edu 4.1 Introduction Newton s laws In the preceding two chapters, we dealt

More information

Version 001 test 1 review tubman (IBII201516) 1

Version 001 test 1 review tubman (IBII201516) 1 Version 001 test 1 review tuban (IBII01516) 1 This print-out should have 44 questions. Multiple-choice questions ay continue on the next colun or page find all choices before answering. Crossbow Experient

More information

Phys101 Lectures 14, 15, 16 Momentum and Collisions

Phys101 Lectures 14, 15, 16 Momentum and Collisions Phs0 Lectures 4, 5, 6 Moentu and ollisions Ke points: Moentu and ipulse ondition for conservation of oentu and wh How to solve collision probles entre of ass Ref: 9-,,3,4,5,6,7,8,9. Page Moentu is a vector:

More information

Lab 5: Conservation of Energy

Lab 5: Conservation of Energy Lab 5: Conservation of Energy Equipment SWS, 1-meter stick, 2-meter stick, heavy duty bench clamp, 90-cm rod, 40-cm rod, 2 double clamps, brass spring, 100-g mass, 500-g mass with 5-cm cardboard square

More information

Chapter 14 Oscillations

Chapter 14 Oscillations Chapter 4 Oscillations Conceptual Probles 3 n object attached to a spring exhibits siple haronic otion with an aplitude o 4. c. When the object is. c ro the equilibriu position, what percentage o its total

More information

Homework 8. problems: 10.40, 10.73, 11.55, 12.43

Homework 8. problems: 10.40, 10.73, 11.55, 12.43 Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks

More information

Application of Second Order Differential Equations in Mechanical Engineering Analysis

Application of Second Order Differential Equations in Mechanical Engineering Analysis ME 13 Applied Engineering Analysis Chapter 4 Application of Second Order Differential Equations in Mechanical Engineering Analysis Tai-Ran Hsu, Professor Departent of Mechanical and Aerospace Engineering

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

MAJOR FIELD TEST IN PHYSICS SAMPLE QUESTIONS

MAJOR FIELD TEST IN PHYSICS SAMPLE QUESTIONS MAJOR FIELD TEST IN PHYSICS SAMPLE QUESTIONS The following questions illustrate the range of the test in ters of the abilities easured, the disciplines covered, and the difficulty of the questions posed.

More information

Chapter 13, example problems: x (cm) 10.0

Chapter 13, example problems: x (cm) 10.0 Chapter 13, example problems: (13.04) Reading Fig. 13-30 (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 information

(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 3)

(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 information

Physics 201 Homework 5

Physics 201 Homework 5 Physics 201 Homework 5 Feb 6, 2013 1. The (non-conservative) force propelling a 1500-kilogram car up a mountain -1.21 10 6 joules road does 4.70 10 6 joules of work on the car. The car starts from rest

More information

People s Physics book 3e Ch 25-1

People s Physics book 3e Ch 25-1 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 information

Sample Questions for the AP Physics 1 Exam

Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each

More information

Centripetal Force. 1. Introduction

Centripetal Force. 1. Introduction 1. Introduction Centripetal Force When an object travels in a circle, even at constant speed, it is undergoing acceleration. In this case the acceleration acts not to increase or decrease the magnitude

More information

The Virtual Spring Mass System

The Virtual Spring Mass System The Virtual Spring Mass Syste J. S. Freudenberg EECS 6 Ebedded Control Systes Huan Coputer Interaction A force feedbac syste, such as the haptic heel used in the EECS 6 lab, is capable of exhibiting a

More information

Exercise 4 INVESTIGATION OF THE ONE-DEGREE-OF-FREEDOM SYSTEM

Exercise 4 INVESTIGATION OF THE ONE-DEGREE-OF-FREEDOM SYSTEM Eercise 4 IVESTIGATIO OF THE OE-DEGREE-OF-FREEDOM SYSTEM 1. Ai of the eercise Identification of paraeters of the euation describing a one-degree-of- freedo (1 DOF) atheatical odel of the real vibrating

More information

Chapter 5: Work, Power & Energy

Chapter 5: Work, Power & Energy Chapter 5: Work, Power & Energy Abition is like a vector; it needs agnitude and direction. Otherwise, it s just energy. Grace Lindsay Objectives 1. Define work and calculate the work done by a force..

More information

Chapter 6 Work and Energy

Chapter 6 Work and Energy Chapter 6 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

More information

Lecture L26-3D Rigid Body Dynamics: The Inertia Tensor

Lecture L26-3D Rigid Body Dynamics: The Inertia Tensor J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L6-3D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall

More information

( C) CLASS 10. TEMPERATURE AND ATOMS

( C) CLASS 10. TEMPERATURE AND ATOMS CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressure-volue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between

More information

Type: Double Date: Simple Harmonic Motion III. Homework: Read 10.3, Do CONCEPT QUEST #(7) Do PROBLEMS #(5, 19, 28) Ch. 10

Type: 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 information

Math Review: Circular Motion 8.01

Math Review: Circular Motion 8.01 Math Review: Circular Motion 8.01 Position and Displacement r ( t) : position vector of an object moving in a circular orbit of radius R Δr ( t) : change in position between time t and time t+δt Position

More information

16 OSCILLATORY MOTION AND WAVES

16 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 information

1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!

1.10 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 information

Lesson 04: Newton s laws of motion

Lesson 04: Newton s laws of motion www.scimsacademy.com Lesson 04: Newton s laws of motion If you are not familiar with the basics of calculus and vectors, please read our freely available lessons on these topics, before reading this lesson.

More information

C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N

C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a

More information

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc. Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

More information

physics 111N rotational motion

physics 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 information

Chapter 24 Physical Pendulum

Chapter 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 information

E k = ½ m v 2. (J) (kg) (m s -1 ) FXA KINETIC ENERGY (E k ) 1. Candidates should be able to : This is the energy possessed by a moving object.

E k = ½ m v 2. (J) (kg) (m s -1 ) FXA KINETIC ENERGY (E k ) 1. Candidates should be able to : This is the energy possessed by a moving object. KINETIC ENERGY (E k ) 1 Candidates should be able to : This is the energy possessed by a oing object. Select and apply the equation for kinetic energy : E k = ½ 2 KINETIC ENERGY = ½ x MASS x SPEED 2 E

More information

Doppler effect, moving sources/receivers

Doppler effect, moving sources/receivers Goals: Lecture 29 Chapter 20 Work with a ew iportant characteristics o sound waves. (e.g., Doppler eect) Chapter 21 Recognize standing waves are the superposition o two traveling waves o sae requency Study

More information

Potential Energy and Conservation of Energy. 8.01t Oct 18, 2004

Potential Energy and Conservation of Energy. 8.01t Oct 18, 2004 Potential Energy and Conservation of Energy 8.01t Oct 18, 2004 Conservative Forces Definition: Conservative Work Whenever the work done by a force in moving an object from an initial point to a final point

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

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

Computer Experiment. Simple Harmonic Motion. Kinematics and Dynamics of Simple Harmonic Motion. Evaluation copy

Computer Experiment. Simple Harmonic Motion. Kinematics and Dynamics of Simple Harmonic Motion. Evaluation copy INTRODUCTION Simple Harmonic Motion Kinematics and Dynamics of Simple Harmonic Motion Computer Experiment 16 When you suspend an object from a spring, the spring will stretch. If you pull on the object,

More information