Simple Harmonic Motion


 Barrie Sherman
 1 years ago
 Views:
Transcription
1 Simple Harmonic Motion Objective: In this exercise you will investigate the simple harmonic motion of mass suspended from a helical (coiled) spring. Apparatus: Spring 1 Table Post 1 Short Rod 1 Rightangled Clamp 1 Ring with Hook 1 Mass Set 1 Stopwatch 1 Introduction: A mass attached to an ideal coiled spring is said to experience a restoring force proportional to the amount of extension or compression of the spring from its rest, or equilibrium, position. This restoring force can be written as: F = k(x x 0 ) (1) where x is the stretched or compressed length and x 0 is the equilibrium length (in metres), and k is the spring constant (in N/m). The negative sign indicates that the force exerted by the spring is opposite to the direction of the stretch or compression so that the force always acts to restore the spring to its equilibrium position. In the circular motion lab, you studied the static properties of a spring by attaching a mass and measuring how far the spring stretched. Here you will study the dynamic properties by attaching a mass and setting it in motion. A useful way to analyse this, and many other systems where the force can be described, is to recognize that Newton s second law (F = ma) can be applied. Since F is known (i.e., the spring force), the unknown is the acceleration. But the acceleration is the rate of change of the rate of change of position, so the unknown is really the position of the mass on the end of the spring. And that can, in principle, be measured. If the mass is allowed to oscillate vertically, it will do so about the position x 0, with some frequency, ω (omega), and will take a time T (the period) to make one full oscillation. Without loss of generality, we can set x 0 = 0, since it is simply a reference point. Newton s second law (usually written F=ma) states that a force applied to a mass, m, will cause the mass to experience an acceleration, a, proportional to the applied force, inversely proportional to the mass, and acting in the same direction as the applied force.
2 2 Newton s second law applied to this system gives: ma = kx (2) d 2 x dt 2 or = " k % $ 'x (3) # m & The term on the left is the acceleration of the mass written as the second derivative (with respect to time) of the displacement of the mass from equilibrium. This is an example of a second order differential equation. It is important to recognise that: the acceleration is directed opposite the extension of the spring (i.e., in the same direction as the restoring force), and the acceleration is proportional to the displacement These are characteristics of simple harmonic motion. Equation 3 does not help directly with the experiment, because the acceleration is not easy to measure, and depending on how fast the mass oscillates, the position may not be easy to measure either. What you need is to find something measurable. Instead of focusing on the position of the mass, perhaps it is better to think about time instead, as in the period, or how long it takes for the mass to complete one oscillation. Equation 3 does not contain time explicitly, so you need to find a relationship between position and period. One solution (but not the most general one) to equation 3 is where A is the amplitude of the oscillation (in metres), ω is the frequency of oscillation (in radians/second), and t is time (in seconds). x = Acos( ωt) (4) Recall that the derivative of cosine is sine, and the derivative of sine is cosine. If you substitute equation 4 into 3, and solve the derivatives, you get: ω 2 = k m (5)
3 3 The relationship between period and frequency is defined to be: T = 2π ω (6) This is the time required for one complete oscillation (since the cosine function repeats after 2π radians). Combining equations 5 and 6 gives the result:! T = 2π m $ # & " k % 1 2 (7) So the above analysis predicts that the period of oscillation of an applied mass suspended from an ideal, massless spring is proportional to the square root of the applied mass. The purpose of the lab is to investigate the behaviour of a real spring, however. Generally, the mass of a real spring causes the spring to oscillate, even when no additional mass is added to the spring. Equation 7 must be modified to account for this:! T = 2π m + m $ 0 # & " k % 1 2 (8) m is the mass applied to the spring, m 0 is the contribution the mass of the spring itself makes to the oscillation of the spring. It is not the entire mass of the spring, but rather a fraction of the spring mass (sometimes quoted as 1 3 m spring ). The term m+m 0 is called the effective mass. You will test equation 8. Procedure: To test equation 8, you will systematically add masses to a vertically hanging spring and measure the time required for multiple oscillations of the masses to occur. Use seven masses in the range from 200 to 500 grams in steps of 50 grams. Select a mass and hang it from the spring. Allow the mass to come to rest. Pull the mass down a short (5 mm or less) distance and release it. Time how long it takes for 10 oscillations to occur. Do three time trials for each mass. Performing the runs in succession from smaller to larger mass is not necessarily the best method. The main drawback is the risk of inadvertently recording the expected outcome instead of the actual outcome (as in That doesn t look right. It should be this. ) Try randomising the order of the runs.
4 4 Weigh the masses in the combinations you use. Do not weigh the masses singly only, then add the numbers in combination, because this will increase the measurement uncertainty. Also weigh the spring. You should have 21 trials altogether: 7 different masses with three time trials each. Remember to estimate uncertainties in timing. These are primarily reaction times (try the drop test method to estimate reaction time, or look it up from the Projectile lab experiment where you tested your reaction time). If you can time some oscillations better than others, then make a note of this. Measurement uncertainties are not necessarily equal for all measurements. Note the behaviour of the massspring system: Are the oscillations vertical only? Does the spring oscillate when no mass is attached? Try to measure the period. Analysis: From your data of three trials of 10 oscillations, determine the average period of oscillation of each mass (Don t forget: a period of oscillation means for one oscillation, not for ten!). Also calculate the standard deviations (σ T ) in the period data for three runs: the smallest, largest, and one middle value. Plot a graph of T 2 (yaxis: average period squared) vs. applied mass. You can plot this by hand or use a plotting program. Calculate the slope of the line, and the yintercept, and include the correct units and error estimates (if you use a plotting program, use it to find a linear least squares fit, or trendline ). Error bars: Calculate error bars for the T 2 data (just the three runs for which you have calculated the standard deviations): δ ( T 2 ) = 2TδT (9) where δt is the error in the average period. This error is the bigger of: σ T 3 or 2 ( δt ) 3 (10) where δt is the reaction time error for each period measurement. The 2 is there because there are two reaction times for each trial, one for starting, and one for stopping. Error bars for the xaxis masses are just reading errors from the digital balance scale. Note that these are too small to draw on your graph.
5 5 Generate a maximum and minimum line using the error bars for the lowest and highest points. Find the slopes and intercepts. Relating the graph to equation 8: First, note that you plotted period squared, not period, so a good first step would be to square both sides of equation 8: which can be rewritten as:! T 2 = 4π 2 m + m 0 $ # & (11) " k % T 2 = 4π 2 k m + 4π 2 k m 0 (Recall that m is the mass you hung on the spring, and m 0 is a correction factor (which, at the moment, is an unknown)). Next, note that the equation of a line is: y = ax + b (12) Make a direct correspondence between equations 11 and 12 so that you know how the slope of the line and the yintercept relate to k and m 0. (Note: k is not the slope!) Once you have correctly determined how these terms relate to the graph and the best fit line, calculate the spring constant, k, and correction term m 0, and error estimates. Compare your value of m 0 with the mass of the spring. Testing your result: According to your graph, what is the expected period for a mass of 150 grams? Hang the required mass on the spring, set it oscillating, and measure the period. Compare the measured period against the expected period. Use your error estimates from above to determine if the predicted and measured masses agree within error. PHYS120 students: write out the full error term for the period T based on equation 8. Additional activity Resonance Frequency is the inverse of the period. Predict what mass you should hang from the spring so that the natural frequency is 1.0 Hertz. Hang this mass (or as close as you can manage to this mass). Try driving the spring (with very low amplitude!) at this frequency, and at frequencies different from this. What do you notice about the amplitude of the oscillations?
6 6 Some sources of error to think about: The analysis above assumes that the motion of the mass is in one dimension only. Was the mass moving only in one dimension? Was there any evidence of damping behavior (i.e., did the oscillations get smaller as the ten periods were measured)? Equation 1 (and hence equation 8) is valid if the spring compressions and extensions are small compared with the length of the spring. Was this condition observed?
LABORATORY 9. Simple Harmonic Motion
LABORATORY 9 Simple Harmonic Motion Purpose In this experiment we will investigate two examples of simple harmonic motion: the massspring system and the simple pendulum. For the massspring system we
More informationSimple Harmonic Motion
Simple Harmonic Motion Simple harmonic motion is one of the most common motions found in nature and can be observed from the microscopic vibration of atoms in a solid to rocking of a supertanker on the
More informationHOOKE'S LAW AND A SIMPLE SPRING DONALD C. PECKHAM PHYSICS 307 FALL 1983 ABSTRACT
HOOKE'S LAW AND A SIMPLE SPRING DONALD C. PECKHAM PHYSICS 307 FALL 983 (Digitized and Revised, Fall 005) ABSTRACT The spring constant of a screendoor spring was determined both statically, by measuring
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationSimple Harmonic Motion Concepts
Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called
More informationSIMPLE HARMONIC MOTION
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 informationp = F net t (2) But, what is the net force acting on the object? Here s a little help in identifying the net force on an object:
Harmonic Oscillator Objective: Describe the position as a function of time of a harmonic oscillator. Apply the momentum principle to a harmonic oscillator. Sketch (and interpret) a graph of position as
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationAdvanced Higher Physics: MECHANICS. Simple Harmonic Motion
Advanced Higher Physics: MECHANICS Simple Harmonic Motion At the end of this section, you should be able to: Describe examples of simple harmonic motion (SHM). State that in SHM the unbalanced force is
More informationExperiment 4: Harmonic Motion Analysis
Experiment 4: Harmonic Motion Analysis Background In this experiment you will investigate the influence of damping on a driven harmonic oscillator and study resonant conditions. The following theoretical
More informationLab M1: The Simple Pendulum
Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of
More informationPrelab Exercises: Hooke's Law and the Behavior of Springs
59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically
More informationHOOKE 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 informationChapter 1. Oscillations. Oscillations
Oscillations 1. A mass m hanging on a spring with a spring constant k has simple harmonic motion with a period T. If the mass is doubled to 2m, the period of oscillation A) increases by a factor of 2.
More informationTHE SPRING CONSTANT. Apparatus: A spiral spring, a set of weights, a weight hanger, a balance, a stop watch, and a twometer
THE SPRING CONSTANT Objective: To determine the spring constant of a spiral spring by Hooe s law and by its period of oscillatory motion in response to a weight. Apparatus: A spiral spring, a set of weights,
More information11/27/2014 Partner: Diem Tran. Bungee Lab I: Exploring the Relationship Between Bungee Cord Length and Spring Force Constant
Bungee Lab I: Exploring the Relationship Between Bungee Cord Length and Spring Force Constant Introduction: This lab relies on an understanding of the motion of a spring and spring constant to facilitate
More informationHOOKE 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 springmass oscillator. INTRODUCTION The force which restores a spring to its equilibrium
More informationPHYS 130 Laboratory Experiment 11 Hooke s Law & Simple Harmonic Motion
PHYS 130 Laboratory Experiment 11 Hooke s Law & Simple Harmonic Motion NAME: DATE: SECTION: PARTNERS: OBJECTIVES 1. Verify Hooke s Law and use it to measure the force constant of a spring. 2. Investigate
More informationHOOKE'S LAW AND SIMPLE HARMONIC MOTION OBJECT
5 M19 M19.1 HOOKE'S LAW AND SIMPLE HARMONIC MOTION OBJECT The object of this experiment is to determine whether a vertical massspring system obeys Hooke's Law and to study simple harmonic motion. THEORY
More informationUpdated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum
Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are
More informationSimple 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 informationDetermination of g using a spring
INTRODUCTION UNIVERSITY OF SURREY DEPARTMENT OF PHYSICS Level 1 Laboratory: Introduction Experiment Determination of g using a spring This experiment is designed to get you confident in using the quantitative
More informationPeriodic Motion or Oscillations. Physics 232 Lecture 01 1
Periodic Motion or Oscillations Physics 3 Lecture 01 1 Periodic Motion Periodic Motion is motion that repeats about a point of stable equilibrium Stable Equilibrium Unstable Equilibrium A necessary requirement
More informationHooke s Law and Simple Harmonic Motion
Hooke s Law and Simple Harmonic Motion OBJECTIVE to measure the spring constant of the springs using Hooke s Law to explore the static properties of springy objects and springs, connected in series and
More informationName: Lab Partner: Section:
Chapter 10 Simple Harmonic Motion Name: Lab Partner: Section: 10.1 Purpose Simple harmonic motion will be examined in this experiment. 10.2 Introduction A periodic motion is one that repeats itself in
More informationSimple Harmonic Motion
Simple Harmonic Motion 9M Object: Apparatus: To determine the force constant of a spring and then study the harmonic motion of that spring when it is loaded with a mass m. Force sensor, motion sensor,
More informationPhysics 53. Oscillations. You've got to be very careful if you don't know where you're going, because you might not get there.
Physics 53 Oscillations You've got to be very careful if you don't know where you're going, because you might not get there. Yogi Berra Overview Many natural phenomena exhibit motion in which particles
More informationAP Physics C. Oscillations/SHM Review Packet
AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete
More informationComputer 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 informationSimple Harmonic Motion
Simple Harmonic Motion Theory Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is eecuted by any quantity obeying the Differential
More informationSTANDING WAVES. Objective: To verify the relationship between wave velocity, wavelength, and frequency of a transverse wave.
STANDING WAVES Objective: To verify the relationship between wave velocity, wavelength, and frequency of a transverse wave. Apparatus: Magnetic oscillator, string, mass hanger and assorted masses, pulley,
More informationSHM Simple Harmonic Motion revised June 16, 2012
SHM Simple Harmonic Motion revised June 16, 01 Learning Objectives: During this lab, you will 1. communicate scientific results in writing.. estimate the uncertainty in a quantity that is calculated from
More informationCentripetal 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 informationSimple 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 informationExperiment P19: Simple Harmonic Motion  Mass on a Spring (Force Sensor, Motion Sensor)
PASCO scientific Physics Lab Manual: P191 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 informationHooke s Law. Spring. Simple Harmonic Motion. Energy. 12/9/09 Physics 201, UWMadison 1
Hooke s Law Spring Simple Harmonic Motion Energy 12/9/09 Physics 201, UWMadison 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, UWMadison 2 We know
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationPhysics 1022: Chapter 14 Waves
Phys 10: Introduction, Pg 1 Physics 10: Chapter 14 Waves Anatomy of a wave Simple harmonic motion Energy and simple harmonic motion Phys 10: Introduction, Pg Page 1 1 Waves New Topic Phys 10: Introduction,
More informationLab 5: Conservation of Energy
Lab 5: Conservation of Energy Equipment SWS, 1meter stick, 2meter stick, heavy duty bench clamp, 90cm rod, 40cm rod, 2 double clamps, brass spring, 100g mass, 500g mass with 5cm cardboard square
More information8 SIMPLE HARMONIC MOTION
8 SIMPLE HARMONIC MOTION Chapter 8 Simple Harmonic Motion Objectives After studying this chapter you should be able to model oscillations; be able to derive laws to describe oscillations; be able to use
More informationMeasurement of the Acceleration Due to Gravity
Measurement of the Acceleration Due to Gravity Phys 303 Lab Experiment 0 Justin M. Sanders January 12, 2004 Abstract Near the surface of the earth, all objects freely fall downward with the same acceleration
More informationChapter 13, example problems: x (cm) 10.0
Chapter 13, example problems: (13.04) Reading Fig. 1330 (reproduced on the right): (a) Frequency f = 1/ T = 1/ (16s) = 0.0625 Hz. (since the figure shows that T/2 is 8 s.) (b) The amplitude is 10 cm.
More informationPHY 157 Standing Waves on a String (Experiment 5)
PHY 157 Standing Waves on a String (Experiment 5) Name: 1 Introduction In this lab you will observe standing waves on a string. You will also investigate the relationship between wave speed and tension
More informationboth 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 informationStanding Waves on a String
1 of 6 Standing Waves on a String Summer 2004 Standing Waves on a String If a string is tied between two fixed supports, pulled tightly and sharply plucked at one end, a pulse will travel from one end
More informationMeasurement of Gravity with a Projectile Experiment Daniel Brown Aberystwyth University
Measurement of Gravity th a Projectile Experiment Daniel Brown Aberystwyth University Abstract The aim of this experiment is to measure the value of g, the acceleration of gravity at the surface of the
More informationSimple Harmonic Motion
5 Simple Harmonic Motion Note: this section is not part of the syllabus for PHYS26. You should already be familiar with simple harmonic motion from your first year course PH115 Oscillations and Waves.
More informationPhysics 2305 Lab 11: Torsion Pendulum
Name ID number Date Lab CRN Lab partner Lab instructor Physics 2305 Lab 11: Torsion Pendulum Objective 1. To demonstrate that the motion of the torsion pendulum satisfies the simple harmonic form in equation
More informationSimple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines
Simple Harmonic Motion(SHM) Vibration (oscillation) Equilibrium position position of the natural length of a spring Amplitude maximum displacement Period and Frequency Period (T) Time for one complete
More informationSIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE
MISN026 SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE by Kirby Morgan 1. Dynamics of Harmonic Motion a. Force Varies in Magnitude and Direction................
More informationDetermination 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 informationExperiment 5: Newton s Second Law
Name Section Date Introduction Experiment : Newton s Second Law In this laboratory experiment you will consider Newton s second law of motion, which states that an object will accelerate if an unbalanced
More information1 of 10 11/23/2009 6:37 PM
hapter 14 Homework Due: 9:00am on Thursday November 19 2009 Note: To understand how points are awarded read your instructor's Grading Policy. [Return to Standard Assignment View] Good Vibes: Introduction
More informationExperiment Type: OpenEnded
Simple Harmonic Oscillation Overview Experiment Type: OpenEnded In this experiment, students will look at three kinds of oscillators and determine whether or not they can be approximated as simple harmonic
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationUNIT 14: HARMONIC MOTION
Name St.No.  Date(YY/MM/DD) / / Section UNIT 14: HARMONIC MOTION Approximate Time three 100minute sessions Back and Forth and Back and Forth... Cameo OBJECTIVES 1. To learn directly about some of the
More informationChapter 14. Oscillations. PowerPoint Lectures for College Physics: A Strategic Approach, Second Edition Pearson Education, Inc.
Chapter 14 Oscillations PowerPoint Lectures for College Physics: A Strategic Approach, Second Edition 14 Oscillations Reading Quiz 1. The type of function that describes simple harmonic motion is A.
More information1: (ta initials) 2: first name (print) last name (print) brock id (ab13cd) (lab date)
1: (ta initials) 2: first name (print) last name (print) brock id (ab13cd) (lab date) Experiment 5 Harmonic motion In this Experiment you will learn that Hooke s Law F = kx can be used to model the interaction
More informationPENDULUM 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 informationChapter 24 Physical Pendulum
Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...
More informationPHYS 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 information1.2 ERRORS AND UNCERTAINTIES Notes
1.2 ERRORS AND UNCERTAINTIES Notes I. UNCERTAINTY AND ERROR IN MEASUREMENT A. PRECISION AND ACCURACY B. RANDOM AND SYSTEMATIC ERRORS C. REPORTING A SINGLE MEASUREMENT D. REPORTING YOUR BEST ESTIMATE OF
More informationturntable in terms of SHM and UCM: be plotted as a sine wave. n Think about spinning a ball on a string or a ball on a
RECALL: Angular Displacement & Angular Velocity Think about spinning a ball on a string or a ball on a turntable in terms of SHM and UCM: If you look at the ball from the side, its motion could be plotted
More informationTwo massthree spring system. Math 216 Differential Equations. Forces on mass m 1. Forces on mass m 2. Kenneth Harris
Two massthree spring system Math 6 Differential Equations Kenneth Harris kaharri@umich.edu m, m > 0, two masses k, k, k 3 > 0, spring elasticity t), t), displacement of m, m from equilibrium. Positive
More informationy = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)
5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If
More informationLab M3: The Physical Pendulum
M3.1 Lab M3: The Physical Pendulum Another pendulum lab? Not really. This lab introduces anular motion. For the ordinary pendulum, we use Newton's second law, F = ma to describe the motion. For the physical
More informationPHYS 202 Laboratory #4. Activity 1: Thinking about Oscillating Systems
SHM Lab 1 Introduction PHYS 202 Laboratory #4 Oscillations and Simple Harmonic Motion In this laboratory, we examine three simple oscillatory systems: a mass on a spring, a pendulum, and a mass on a rubber
More informationBungee Constant per Unit Length & Bungees in Parallel. Skipping school to bungee jump will get you suspended.
Name: Johanna Goergen Section: 05 Date: 10/28/14 Partner: Lydia Barit Introduction: Bungee Constant per Unit Length & Bungees in Parallel Skipping school to bungee jump will get you suspended. The purpose
More information1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!
1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating
More informationSIMPLE HARMONIC MOTION Ken Cheney
SIMPLE HARMONIC MOTION Ken Cheney INTRODUCTION GENERAL Probably no tools that you will learn in Physics are more widely used than those that deal with simple harmonic motion. Here we will be investigating
More informationOscillations: Mass on a Spring and Pendulums
Chapter 3 Oscillations: Mass on a Spring and Pendulums 3.1 Purpose 3.2 Introduction Galileo is said to have been sitting in church watching the large chandelier swinging to and fro when he decided that
More informationPHYS2020: General Physics II Course Lecture Notes Section VII
PHYS2020: General Physics II Course Lecture Notes Section VII Dr. Donald G. Luttermoser East Tennessee State University Edition 4.0 Abstract These class notes are designed for use of the instructor and
More informationSolving the Harmonic Oscillator Equation. Morgan Root NCSU Department of Math
Solving the Harmonic Oscillator Equation Morgan Root NCSU Department of Math SpringMass System Consider a mass attached to a wall by means of a spring. Define y to be the equilibrium position of the block.
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More information2. The graph shows how the displacement varies with time for an object undergoing simple harmonic motion.
Practice Test: 29 marks (37 minutes) Additional Problem: 31 marks (45 minutes) 1. A transverse wave travels from left to right. The diagram on the right shows how, at a particular instant of time, the
More informationMechanical 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 informationELASTIC FORCES and HOOKE S LAW
PHYS101 LAB03 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 informationEXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY
EXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY Objective: To learn how to use the Excel spreadsheet to record your data, calculate values and make graphs. To analyze the data from the Acceleration Due
More information= mg [down] =!mg [up]; F! x
Section 4.6: Elastic Potential Energy and Simple Harmonic Motion Mini Investigation: Spring Force, page 193 Answers may vary. Sample answers: A. The relationship between F g and x is linear. B. The slope
More informationThe Pendulum. Experiment #1 NOTE:
The Pendulum Experiment #1 NOTE: For submitting the report on this laboratory session you will need a report booklet of the type that can be purchased at the McGill Bookstore. The material of the course
More informationVELOCITY, 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 informationHomework #7 Solutions
MAT 0 Spring 201 Problems Homework #7 Solutions Section.: 4, 18, 22, 24, 4, 40 Section.4: 4, abc, 16, 18, 22. Omit the graphing part on problems 16 and 18...4. Find the general solution to the differential
More informationGRAPHING LINEAR EQUATIONS IN TWO VARIABLES
GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: SlopeIntercept Form: y = mx+ b In an equation
More informationExperiment 08: RLC Circuits and Resonance Dr. Pezzaglia
Mar9 RLC Circuit Page Experiment 8: RLC Circuits and Resonance Dr. Pezzaglia Theory When a system at a stable equilibrium is displaced, it will tend to oscillate. An Inductor combined with Capacitor will
More informationMECHANICS IV  SIMPLE HARMONIC MOTION
MIVp.1 A. OSCILLATIONS B. SIMPLE PENDULUM C. KINEMATICS OF SIMPLE HARMONIC MOTION D. SPRINGANDMASS SYSTEM E. ENERGY OF SHM F. DAMPED HARMONIC MOTION G. FORCED VIBRATION A. OSCILLATIONS A toandfro
More informationPhysics 231 Lecture 15
Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)
More informationEquilibrium. To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium
Equilibrium Object To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium situation. 2 Apparatus orce table, masses, mass pans, metal loop, pulleys, strings,
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
More informationSIMPLE HARMONIC MOTION
PERIODIC MOTION SIMPLE HARMONIC MOTION If a particle moves such that it repeats its path regularly after equal intervals of time, its motion is said to be periodic. The interval of time required to complete
More informationANALYTICAL 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 informationSample lab procedure and report. The Simple Pendulum
Sample lab procedure and report The Simple Pendulum In this laboratory, you will investigate the effects of a few different physical variables on the period of a simple pendulum. The variables we consider
More informationRotational Motion & Moment of Inertia
Rotational Motion & Moment of nertia Physics 161 ntroduction n this experiment we will study motion of objects is a circular path as well as the effect of a constant torque on a symmetrical body. n Part
More informationSimple Harmonic Motion
Periodic motion Earth around the sun Elastic ball bouncing up an down Quartz in your watch, computer clock, ipod clock, etc. Heart beat, and many more In taking your pulse, you count 70.0 heartbeats in
More informationLecture Presentation Chapter 14 Oscillations
Lecture Presentation Chapter 14 Oscillations Suggested Videos for Chapter 14 Prelecture Videos Describing Simple Harmonic Motion Details of SHM Damping and Resonance Class Videos Oscillations Basic Oscillation
More informationPhysics 53. Wave Motion 1
Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can
More informationThe moment of inertia of a rod rotating about its centre is given by:
Pendulum Physics 161 Introduction This experiment is designed to study the motion of a pendulum consisting of a rod and a mass attached to it. The period of the pendulum will be measured using three different
More informationForce. Net Force Mass. Acceleration = Section 1: Weight. Equipment Needed Qty Equipment Needed Qty Force Sensor 1 Mass and Hanger Set 1 Balance 1
Department of Physics and Geology Background orce Physical Science 1421 A force is a vector quantity capable of producing motion or a change in motion. In the SI unit system, the unit of force is the Newton
More informationGraphing Quadratic Functions
Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x
More informationExperiment 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 informationMAKE SURE TA & TI STAMPS EVERY PAGE BEFORE YOU START
Laboratory Section: Last Revised on September 21, 2016 Partners Names: Grade: EXPERIMENT 11 Velocity of Waves 0. PreLaboratory Work [2 pts] 1.) What is the longest wavelength at which a sound wave will
More information