Simple Harmonic Motion

Transcription

1 Simple Harmonic Motion James H Dann, Ph.D. Say Thanks to the Authors Click (No sign in required)

2 To access a customizable version of this book, as well as other interactive content, visit CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. AUTHOR James H Dann, Ph.D. CONTRIBUTORS Chris Addiego Antonio De Jesus López Catherine Pavlov Copyright 2014 CK-12 Foundation, The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at Printed: April 26, 2014

3 Chapter 1. Simple Harmonic Motion CHAPTER 1 Simple Harmonic Motion CHAPTER OUTLINE 1.1 Harmonic Period and Frequency 1.2 Springs 1.3 Pendulum Introduction The Big Idea The development of devices to measure time, like the pendulum, led to the analysis of periodic motion. Such motion repeats itself in equal intervals of time (called periods) and is also referred to as harmonic motion. When an object moves back and forth over the same path in harmonic motion it is said to be oscillating. If the distance such an object travels in one oscillation remains constant, it is called simple harmonic motion (SHM). A grandfather clock s pendulum and the quartz crystal in a modern watch are examples of SHM. 1

4 1.1. Harmonic Period and Frequency Harmonic Period and Frequency Describe the concepts of period and frequency. Read the period and frequency off a graph and calculate them for systems in harmonic motion. Students will learn the concepts of period and frequency and how to read them off a graph and how to calculate them for systems in harmonic motion. In addition, students will learn how to graph simple harmonic motion. Key Equations T = 1 f m T spring = 2π Period Equations k L T pendulum = 2π g Period is the inverse of frequency Period of mass m on a spring with constant k Period of a pendulum of length L Kinematics of SHM { x(t) = x 0 + Acos2π f (t t 0 ) v(t) = 2π f Acos2π f (t t 0 ) Position of an object in SHM of Amplitude A Velocity of an object in SHM of Amplitude A Guidance Example 1 A bee flaps its wings at a rate of approximately 190 Hz. How long does it take for a bee to flap its wings once (down and up)? Question: T =? [sec] Given: f = 190 Hz Equation: T = 1 f Plug n Chug: T = 1 f = Hz = s = 5.26 ms Answer: 5.26 ms Example 2 Question: The effective k of a diving board is 800N/m (we say effective because it bends in the direction of motion instead of stretching like a spring, but otherwise behaves the same). A pudgy diver is bouncing up and down at the end of the diving board. The y vs. t graph is shown below. a) What is the distance between the lowest and the highest point of oscillation? b) What is the Period and frequency of the diver? c) What is the diver s mass? 2

5 Chapter 1. Simple Harmonic Motion d) Write the sinusoidal equation of motion for the diver. Solution: a) As we can see from the graph the highest point is 2m and the lowest point is 2m. Therefore the distance is 2m ( 2m) = 4m b)we know that f = 1 T From the graph we know that the period is 2 seconds, so the frequency is 1 2 hz. c) To find the diver s mass we will use the equation T = 2π m k and solve for m. Then it is a simple matter to plug in the known values to get the mass. m T = 2π k T m 2π = k ( T 2π )2 = m k k( T 2π )2 = m Now we plug in what we know. m = k( T 2π )2 = 800 N m ( πs 2π )2 = 200kg d) To get the sinusoidal equation we must first choose whether to go with a cosine graph or a sine graph. Then we must find the amplitude (A), vertical shift (D), horizontal shift (C), and period (B). Cosine is easier in this case so we will work with it instead of sine. As we can see from the graph, the amplitude is 2, the vertical shift is 0, and the horizontal shift is.4. We solved for the period already. Therefore, we can write the sinusoidal equation of this graph. AcosB(x C) + D = 2cosπ(x +.4) 3

6 1.1. Harmonic Period and Frequency Watch this Explanation Time for Practice 1. While treading water, you notice a buoy way out towards the horizon. The buoy is bobbing up and down in simple harmonic motion. You only see the buoy at the most upward part of its cycle. You see the buoy appear 10 times over the course of one minute. a. What kind of force that is leading to simple harmonic motion? b. What is the period (T ) and frequency ( f ) of its cycle? Use the proper units. 2. The pitch of a Middle C note on a piano is 263 Hz. This means when you hear this note, the hairs in your ears wiggle back and forth at this frequency. a. What is the period of oscillation for your ear hairs? b. What is the period of oscillation of the struck wire within the piano? 3. The Sun tends to have dark, Earth-sized spots on its surface due to kinks in its magnetic field. The number of visible spots varies over the course of years. Use the graph of the sunspot cycle above to answer the following questions. (Note that this is real data from our sun, so it doesn t look like a perfect sine wave. What you need to do is estimate the best sine wave that fits this data.) a. Estimate the period T in years. b. When do we expect the next solar maximum? Answers to Selected Problems 4 1. a. Buoyant force and gravity b. T = 6 s, f = 1/6 Hz 2. a s b s

7 Chapter 1. Simple Harmonic Motion 3. a. About 11 years b. About

8 1.2. Springs Springs Calculate periods, frequencies, etc. of spring systems in harmonic motion. Students will learn to calculate periods, frequencies, etc. of spring systems in harmonic motion. Key Equations T = 1 f ; Period is the inverse of frequency m T spring = 2π ; Period of mass m on a spring with constant k k F sp = kx ; the force of a spring equals the spring constant multiplied by the amount the spring is stretched or compressed from its equilibrium point. The negative sign indicates it is a restoring force (i.e. direction of the force is opposite its displacement from equilibrium position. U sp = 1 2 kx2 ; the potential energy of a spring is equal to one half times the spring constant times the distance squared that it is stretched or compressed from equilibrium Guidance The oscillating object does not lose any energy in SHM. Friction is assumed to be zero. In harmonic motion there is always a restorative force, which attempts to restore the oscillating object to its equilibrium position. The restorative force changes during an oscillation and depends on the position of the object. In a spring the force is given by Hooke s Law: F = kx The period, T, is the amount of time needed for the harmonic motion to repeat itself, or for the object to go one full cycle. In SHM, T is the time it takes the object to return to its exact starting point and starting direction. The frequency, f, is the number of cycles an object goes through in 1 second. Frequency is measured in Hertz (Hz). 1 Hz = 1 cycle per sec. The amplitude, A, is the distance from the equilibrium (or center) point of motion to either its lowest or highest point (end points). The amplitude, therefore, is half of the total distance covered by the oscillating object. The amplitude can vary in harmonic motion, but is constant in SHM. Example 1 6

9 Chapter 1. Simple Harmonic Motion Watch this Explanation Simulation Mass & Springs (PhET Simulation) Time for Practice 1. A rope can be considered as a spring with a very high spring constant k, so high, in fact, that you don t notice the rope stretch at all before it pulls back. a. What is the k of a rope that stretches by 1 mm when a 100 kg weight hangs from it? b. If a boy of 50 kg hangs from the rope, how far will it stretch? c. If the boy kicks himself up a bit, and then is bouncing up and down ever so slightly, what is his frequency of oscillation? Would he notice this oscillation? If so, how? If not, why not? 2. If a 5.0 kg mass attached to a spring oscillates 4.0 times every second, what is the spring constant k of the spring? 7

10 1.2. Springs 3. A horizontal spring attached to the wall is attached to a block of wood on the other end. All this is sitting on a frictionless surface. The spring is compressed 0.3 m. Due to the compression there is 5.0 J of energy stored in the spring. The spring is then released. The block of wood experiences a maximum speed of 25 m/s. a. Find the value of the spring constant. b. Find the mass of the block of wood. c. What is the equation that describes the position of the mass? d. What is the equation that describes the speed of the mass? e. Draw three complete cycles of the block s oscillatory motion on an x vs. t graph. 4. A spider of 0.5 g walks to the middle of her web. The web sinks by 1.0 mm due to her weight. You may assume the mass of the web is negligible. a. If a small burst of wind sets her in motion, with what frequency will she oscillate? b. How many times will she go up and down in one s? In 20 s? c. How long is each cycle? d. Draw the x vs t graph of three cycles, assuming the spider is at its highest point in the cycle at t = 0 s. Answers to Selected Problems 1. a N/m b. 0.5 mm c. 22 Hz N/m 3. a. 110 N/m d. v(t) = (25)cos(83t) 4. a. 16 Hz b. 16 complete cycles but 32 times up and down, 315 complete cycles but 630 times up and down c s Investigation 1. Your task: Match the period of the circular motion system with that of the spring system. You are only allowed to change the velocity involved in the circular motion system. Consider the effective distance between the block and the pivot to be to be fixed at 1m. The spring constant(13.5n/m) is also fixed. You should view the charts to check whether you have succeeded. Instructions: To alter the velocity, simply click on the Select Tool, and select the pivot. The Position tab below will allow you to numerically adjust the rotational speed using the Motor field. To view the graphs of their respective motion in order to determine if they are in sync, click on Chart tab below Now the mass on the spring has been replaced by a mass that is twice the rotating mass. Also, the distance between the rotating mass and the pivot has been changed to 1.5 m. What velocity will keep the period the same now? 4. 8

11 Chapter 1. Simple Harmonic Motion 9

12 1.3. Pendulum Pendulum Describe the harmonic motion of a pendulum and calculate its period. Students will learn about the harmonic motion of a pendulum and how to calculate its period. Key Equations T pendulum = 2π L ; Period of a pendulum of length L g Guidance In harmonic motion there is always a restorative force, which acts in the opposite direction of the velocity. The restorative force changes during oscillation and depends on the position of the object. In a pendulum it is the component of gravity along the path of motion. The force on the oscillating object is directly opposite that of the direction of velocity. For pendulums, the period gets larger as the length of the pendulum increases. Example 1 You have a mass swinging on the end of 1 m pendulum. If the maximum linear velocity of the mass is 2 m/s, (a)calculate the period of the pendulum and (b) calculate the amplitude of the pendulum. Solution To calculate the period of the pendulum, we can just plug in the given length into the equation above. T = 2π T = 2π T = 2 s l g 1 m 9.8 m/s 2 To find the amplitude, we ll use the equation given in the Period and Frequency lesson that gives us the velocity as a function of time. Since the problem says that the given velocity is the maximum velocity, we know that the pendulum is at the bottom of it s arc and 1/4th (or 3/4th s) of it s way through one period. Based on this knowledge, we can plug in 1/4 of the period for the change in time. We also know the frequency because we just found the period, so all we have to do is solve for the amplitude. 10

13 Chapter 1. Simple Harmonic Motion v(t) = 2π f A cos(2π f t) v max = 2π f Acos(2π f 1 4 T ) v max A = 2π f cos(2π f 1 4 T ) 2 m/s A = 2π 1 2 Hz cos(2π 1 2 Hz s) A =.63 m start with the equation for velocity put in the terms we know solve for the amplitude plug in the known values Watch this Explanation Simulation 11

14 1.3. Pendulum Pendulum Lab (PhET Simulation) Time for Practice 1. Why doesn t the period of a pendulum depend on the mass of the pendulum weight? Shouldn t a heavier weight feel a stronger force of gravity? 2. The pendulum of a small clock is cm long. How many times does it go back and forth before the second hand goes forward one second? 3. On the moon, how long must a pendulum be if the period of one cycle is one second? The acceleration of gravity on the moon is one sixth that of Earth. Answers to Selected Problems times m Investigation We have explored two examples of simple harmonic motion: the pendulum and the mass-spring system in the previous lesson. The purpose of this investigation is to get you to notice the connections between the two systems. Your task: Match the period of the pendulum system with that of the spring system. You are only allowed to change the mass involved in the spring system. Consider the effective length of the pendulum to be fixed at 2m because that is the distance between the center of mass and the pivot. The spring constant(13.5n/m) is also fixed. You may use any relationships you have learned about to help you. You should view the charts to check whether you have succeeded. Instructions: To alter the mass, simply click on the select tool in the menu, and select the mass. Then use the tab at the bottom of your screen to change the density or dimensions of the block to get the mass that you want. To view chart legend, click on Settings and you can plot velocity or position of the pendulum or mass on the spring. 12

15 Chapter 1. Simple Harmonic Motion The mass and the spring constant have now been changed. What is the new period of the mass-spring system? Can you change the length of the pendulum to match the periods now? Summary In these lessons students will learn the concepts of frequency and period. In addition students will learn how to graph harmonic motion and how to calculate the periods of a spring mass system and a pendulum system. 13