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Work done on the system is force multiplied by distance which equals the area under the curve or A in the | [
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] | 8,894 |
Another way to determine the work is to note that the force increases linearly from 0 to kx so that the average force is the distance moved isx and thus B in the Chapter 16 Oscillatory Motion and Waves 685Figure 16 6 A graph of applied force versus distance for the deformation of a system that can be described by s law is displayed | [
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The work done on the system equals the area under the graph or the area of the triangle which is half its base multiplied by its height or Example 16 2 Calculating Stored A Tranquilizer Gun Spring We can use a toy spring mechanism to ask and answer two simple How much energy is stored in the spring of a tranquilizer gun that has a force constant of 50 0 and is compressed 0 150 If you neglect friction and the mass of the spring at what speed will a 2 projectile be ejected from the Figure 16 7 In this image of the gun the spring is uncompressed before being cocked The spring has been compressed a distance x and the projectile is in place When released the spring converts elastic potential energy PEelinto kinetic energy Strategy for a The energy stored in the spring can be found directly from elastic potential energy equation because kandxare given Solution for a Entering the given values for kandxyields 0 150 0 563 N m 0 563 J Strategy for b Because there is no friction the potential energy is converted entirely into kinetic energy | [
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] | 8,896 |
The expression for kinetic energy can be solved for the speed Solution for b686 Chapter 16 Oscillatory Motion and Waves This OpenStax book is available for free at 141 | [
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] | 8,900 |
The numbers in this problem seem reasonable | [
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] | 8,901 |
The force needed to compress the spring is small enough for an adult to manage and the energy imparted to the dart is small enough to limit the damage it might do | [
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Y et the speed of the dart is great enough for it to travel an acceptable distance Check your Understanding Envision holding the end of a ruler with one hand and deforming it with the other | [
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] | 8,903 |
When you let go you can see the oscillations of the ruler | [
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] | 8,904 |
In what way could you modify this simple experiment to increase the rigidity of the Solution You could hold the ruler at its midpoint so that the part of the ruler that oscillates is half as long as in the original experiment Check your Understanding If you apply a deforming force on an object and let it come to equilibrium what happened to the work you did on the Solution It was stored in the object as potential energy 16 2Period and Frequency in Oscillations Learning Objectives By the end of this section you will be able Relate recurring mechanical vibrations to the frequency and period of harmonic motion such as the motion of a guitar string Compute the frequency and period of an oscillation The information presented in this section supports the following learning objectives and science B 3 3 The student can analyze data to identify qualitative or quantitative relationships between given values and variables e force displacement acceleration velocity period of motion frequency spring constant string length associated with objects in oscillatory motion to use that data to determine the value of an unknown P 2 2 5 Figure 16 8 The strings on this guitar vibrate at regular time intervals 16 Oscillatory Motion and Waves 687When you pluck a guitar string the resulting sound has a steady tone and lasts a long time | [
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] | 8,905 |
Each successive vibration of the string takes the same time as the previous one | [
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] | 8,906 |
We define periodic motion to be a motion that repeats itself at regular time intervals such as exhibited by the guitar string or by an object on a spring moving up and down | [
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] | 8,908 |
Its units are usually seconds but may be any convenient unit of time The word period refers to the time for some event whether repetitive or but we shall be primarily interested in periodic motion which is by definition repetitive | [
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] | 8,909 |
A concept closely related to period is the frequency of an event | [
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] | 8,911 |
Frequency fis defined to be the number of events per unit time | [
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] | 8,912 |
For periodic motion frequency is the number of oscillations per unit time The relationship between frequency and period is T The SI unit for frequency is the cycle per second which is defined to be a hertz Hz 1cycle secor 1 Hz A cycle is one complete oscillation | [
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] | 8,913 |
Note that a vibration can be a single or multiple event whereas oscillations are usually repetitive for a significant number of cycles Example 16 3 Determine the Frequency of Two Medical Ultrasound and the Period of Middle C We can use the formulas presented in this module to determine both the frequency based on known oscillations and the oscillation based on a known frequency | [
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] | 8,914 |
try one example of each A medical imaging device produces ultrasound by oscillating with a period of 0 400 | [
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] | 8,915 |
What is the frequency of this The frequency of middle C on a typical musical instrument is 264 Hz | [
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] | 8,916 |
What is the time for one complete Strategy Both questions and can be answered using the relationship between period and frequency | [
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] | 8,917 |
In question the period Tis given and we are asked to find frequency f | [
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] | 8,918 |
In question the frequency fis given and we are asked to find the periodT Solution a 1 | [
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] | 8,919 |
Substitute 0 400 0 Solve to find 2 Discussion a The frequency of sound found in is much higher than the highest frequency that humans can hear and therefore is called ultrasound | [
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Appropriate oscillations at this frequency generate ultrasound used for noninvasive medical diagnoses such as observations of a fetus in the womb Solution b 1 | [
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] | 8,921 |
Identify the known The time for one complete oscillation is the period T 2 | [
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] | 8,922 |
Solve for f 3 | [
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] | 8,923 |
Substitute the given value for the frequency into the resulting 264 264 3 3 79 ms Discussion The period found in is the time per cycle but this value is often quoted as simply the time in convenient units or milliseconds in this 688 Chapter 16 Oscillatory Motion and Waves This OpenStax book is available for free at 14Check your Understanding Identify an event in your life as receiving a that occurs regularly | [
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] | 8,924 |
Identify both the period and frequency of this event Solution I visit my parents for dinner every other Sunday | [
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The period is two weeks 16 3Simple Harmonic A Special Periodic Motion Learning Objectives By the end of this section you will be able Describe a simple harmonic oscillator Relate physical characteristics of a vibrating system to aspects of simple harmonic motion and any resulting waves The information presented in this section supports the following learning objectives and science B 3 1 The student is able to predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the motion is on those properties P 6 4 7 B 3 4 The student is able to construct a qualitative a quantitative explanation of oscillatory behavior given evidence of a restoring force P 2 2 6 A 3 1 The student is able to use graphical representation of a periodic mechanical wave to determine the amplitude of the wave P 1 B 1 1 The student is able to use a graphical representation of a periodic mechanical wave versus to determine the period and frequency of the wave and describe how a change in the frequency would modify features of the representation P 1 4 2 The oscillations of a system in which the net force can be described by s law are of special importance because they are very common | [
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] | 8,927 |
They are also the simplest oscillatory systems | [
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Simple Harmonic Motion is the name given to oscillatory motion for a system where the net force can be described by s law and such a system is called a simple harmonic oscillator | [
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] | 8,929 |
If the net force can be described by law and there is no damping friction or other then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position as shown for an object on a spring in Figure 16 9 | [
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] | 8,930 |
The maximum displacement from equilibrium is called the amplitude X | [
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] | 8,931 |
The units for amplitude and displacement are the same but depend on the type of oscillation | [
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] | 8,932 |
For the object on the spring the units of amplitude and displacement are whereas for sound oscillations they have units of pressure other types of oscillations have yet other | [
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Get a feel for the force required to maintain this periodic motion | [
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What is the restoring force and what role does the force you apply play in the simple harmonic motion of the 16 Oscillatory Motion and Waves 689Figure 16 9 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator | [
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] | 8,937 |
When displaced from equilibrium the object performs simple harmonic motion that has an amplitude Xand a period T | [
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] | 8,939 |
The stiffer the spring is the smaller the period T | [
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] | 8,940 |
The greater the mass of the object is the greater the period T What is so significant about simple harmonic One special thing is that the period Tand frequency fof a simple harmonic oscillator are independent of amplitude | [
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] | 8,941 |
The string of a guitar for example will oscillate with the same frequency whether plucked gently or hard | [
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] | 8,942 |
Because the period is constant a simple harmonic oscillator can be used as a clock Two important factors do affect the period of a simple harmonic oscillator | [
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] | 8,944 |
A very stiff object has a large force constant k which causes the system to have a smaller period | [
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] | 8,945 |
For example you can adjust a diving stiffer it is the faster it vibrates and the shorter its period | [
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] | 8,947 |
The more massive the system is the longer the period | [
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For example a heavy person on a diving board bounces up and down more slowly than a light one In fact the mass mand the force constant kare the only factors that affect the period and frequency of simple harmonic motion Period of Simple Harmonic Oscillator Theperiod of a simple harmonic oscillator is given by k and because the frequency of a simple harmonic oscillator is Note that neither Tnorfhas any dependence on amplitude Example 16 4 Mechanical Waves What do sound waves water waves and seismic waves have in They are all governed by s laws and they can exist only when traveling in a medium such as air water or rocks | [
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] | 8,949 |
Waves that require a medium to travel are collectively known as waves Mass and Ruler Oscillations Find two identical wooden or plastic rulers | [
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] | 8,950 |
Tape one end of each ruler firmly to the edge of a table so that the length of each690 Chapter 16 Oscillatory Motion and Waves This OpenStax book is available for free at 14ruler that protrudes from the table is the same | [
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] | 8,951 |
On the free end of one ruler tape a heavy object such as a few large coins Pluck the ends of the rulers at the same time and observe which one undergoes more cycles in a time period and measure the period of oscillation of each of the rulers Example 16 5 Calculate the Frequency and Period of Bad Shock Absorbers in a Car If the shock absorbers in a car go bad then the car will oscillate at the least provocation such as when going over bumps in the road and after stopping Figure 16 | [
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0.04074176400899887,
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0.014917700551450253,
-0.03488381579518318,
-0.02825220674276352
] | 8,952 |
Calculate the frequency and period of these oscillations for such a car if the mass its is 900 kg and the force constant of the suspension system is 6 Strategy The frequency of the oscillations will be that of a simple harmonic oscillator as given in the equation | [
-0.01972915232181549,
0.10373778641223907,
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0.03375045955181122,
0.00046848805504851043,
0.041329026222229004,
0.06341879069805145,
-1.6558292605591062e-33,
0.07281334698200226,
0.07601511478424072,
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0.033559650182724,
0.0453176312148571,
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0.05432293936610222,
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0.0881614089012146,
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0.007867023348808289,
0.02786758542060852,
0.005434542894363403,
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0.09974732249975204,
0.001252189977094531,
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-2.676592458783489e-8,
0.0632508248090744,
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-0.045626282691955566
] | 8,953 |
The mass and the force constant are both given Solution 1 | [
0.0072377268224954605,
0.07248209416866302,
0.01230521872639656,
0.07003021985292435,
0.019737612456083298,
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] | 8,954 |
Enter the known values of 900 kg 2 | [
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] | 8,955 |
Calculate the 1 1 1 1 36 Hz 3 | [
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] | 8,956 |
You could use kto calculate the period but it is simpler to use the relationship substitute the value just found for 1 356 0 738 s Discussion The values of Tandfboth seem about right for a bouncing car | [
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] | 8,957 |
You can observe these oscillations if you push down hard on the end of a car and let go The Link between Simple Harmonic Motion and Waves If a photograph of the bouncing car were taken as it drove by the headlight would make a wavelike streak as shown in Figure 16 10 | [
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] | 8,958 |
Similarly Figure 16 11 shows an object bouncing on a spring as it leaves a wavelike of its position on a moving strip of paper | [
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] | 8,960 |
All simple harmonic motion is intimately related to sine and cosine waves Figure 16 10 The bouncing car makes a wavelike motion | [
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If the restoring force in the suspension system can be described only by s law then the wave is a sine function wave is the trace produced by the headlight as the car moves to the right 16 Oscillatory Motion and Waves 691Figure 16 11 The vertical position of an object bouncing on a spring is recorded on a strip of moving paper leaving a sine wave The displacement as a function of time tin any simple harmonic is one in which the net restoring force can be described by law is given by T whereXis amplitude | [
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] | 8,962 |
At 0 the initial position is and the displacement oscillates back and forth with a period T we because 1 | [
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] | 8,963 |
Furthermore from this expression for x the velocity vas a function of time is given | [
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] | 8,965 |
The minus sign in the first equation for the correct direction for the velocity | [
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] | 8,966 |
Just after the start of the motion for instance the velocity is negative because the system is moving back toward the equilibrium point | [
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Finally we can get an expression for acceleration using second law we have t the quantities needed for kinematics and a description of simple harmonic motion According to s second law the acceleration is So also a cosine T Hence directly proportional to and in the opposite direction to Figure 16 12 shows the simple harmonic motion of an object on a spring and presents graphs of time 692 Chapter 16 Oscillatory Motion and Waves This OpenStax book is available for free at 14Figure 16 12 Graphs the motion of an object on a spring | [
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] | 8,968 |
The net force on the object can be described by law and so the object undergoes simple harmonic motion | [
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] | 8,969 |
Note that the initial position has the vertical displacement at its maximum value vis initially zero and then negative as the object moves and the initial acceleration is negative back toward the equilibrium position and becomes zero at that point The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion | [
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They are very useful in visualizing waves associated with simple harmonic motion including visualizing how waves add with one another Check Your Understanding Suppose you pluck a banjo string | [
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You hear a single note that starts out loud and slowly quiets over time | [
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] | 8,972 |
Describe what happens to the sound waves in terms of period frequency and amplitude as the sound decreases in volume Solution Frequency and period remain essentially unchanged | [
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] | 8,973 |
Only amplitude decreases as volume decreases Check Your Understanding A babysitter is pushing a child on a swing | [
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] | 8,974 |
At the point where the swing reaches x where would the corresponding point on a wave of this motion be SolutionChapter 16 Oscillatory Motion and Waves 693xis the maximum deformation which corresponds to the amplitude of the wave | [
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The point on the wave would either be at the very top or the very bottom of the curve PhET Masses and Springs A realistic mass and spring laboratory | [
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] | 8,976 |
Hang masses from springs and adjust the spring stiffness and damping | [
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You can even slow time | [
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Transport the lab to different planets | [
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A chart shows the kinetic potential and thermal energy for each spring Figure 16 13 Masses and Springs 16 4The Simple Pendulum Learning Objectives By the end of this section you will be able Determine the period of oscillation of a hanging pendulum The information presented in this section supports the following learning objectives and science B 3 1 The student is able to predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the motion is on those properties P 6 4 7 B 3 2 The student is able to design a plan and collect data in order to ascertain the characteristics of the motion of a system undergoing oscillatory motion caused by a restoring force P 4 B 3 3 The student can analyze data to identify qualitative or quantitative relationships between given values and variables e force displacement acceleration velocity period of motion frequency spring constant string length associated with objects in oscillatory motion to use that data to determine the value of an unknown P 2 2 5 B 3 4 The student is able to construct a qualitative a quantitative explanation of oscillatory behavior given evidence of a restoring force P 2 2 6 Figure 16 14 A simple pendulum has a bob and a string that has a very small mass but is strong enough not to stretch appreciably | [
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] | 8,980 |
The linear displacement from equilibrium is s the length of the arc | [
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] | 8,981 |
Also shown are the forces on the bob which result in a net force of toward the equilibrium is a restoring force Pendulums are in common usage | [
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] | 8,983 |
For small displacements a pendulum is a simple harmonic oscillator | [
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] | 8,984 |
A simple pendulum is defined to have an object that has a small mass also known as the pendulum bob which is suspended from a light wire or string such as shown in Figure 16 14 | [
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] | 8,985 |
Exploring the simple pendulum a bit further we can discover the conditions under which it performs simple harmonic motion and we can derive an interesting expression for its period We begin by defining the displacement to be the arc length s | [
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] | 8,986 |
We see from Figure 16 14 that the net force on the bob is tangent to the arc and equals weight mghas components the string and to the arc Tension in the string exactly cancels the component to the string | [
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This leaves a netrestoring force back694 Chapter 16 Oscillatory Motion and Waves This OpenStax book is available for free at 14toward the equilibrium position at 0 Now if we can show that the restoring force is directly proportional to the displacement then we have a simple harmonic oscillator | [
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-0.020156435668468475,
0.037930868566036224,
0.07819726318120956,
-4.198594538706857e-8,
0.007073752582073212,
-0.041126467287540436,
-0.015826884657144547,
0.08072859048843384,
0.058925826102495193,
0.0276546161621809,
0.057844169437885284,
-0.08541794121265411,
-0.037517186254262924,
-0.07587648928165436,
0.014393962919712067,
0.03996366634964943,
0.047078248113393784,
-0.04099287465214729,
-0.04461695998907089,
0.012680506333708763,
-0.02699957974255085,
0.011835424229502678,
-0.04925961792469025,
-0.0387529656291008,
-0.022883165627717972,
-0.004722975194454193,
0.060155853629112244,
0.043243568390607834,
0.07542496919631958,
0.02682368829846382,
-0.07926390320062637,
-0.02954685501754284,
-0.047431960701942444,
-0.021929506212472916,
-0.030477499589323997,
-0.0324888713657856,
-0.0015816409140825272,
-0.09031416475772858,
-0.028704093769192696,
0.07929803431034088,
-0.0731884092092514,
0.009644659236073494,
-0.0004024870286229998,
0.00002958032382593956,
-0.05526398867368698,
0.11031293123960495,
-0.04068421199917793,
0.004215239081531763,
0.016208717599511147,
0.03275907784700394,
-0.06669749319553375,
-0.022452877834439278,
0.05315477028489113,
-0.007340582087635994,
-0.008298857137560844,
-0.0020948369055986404,
0.048250988125801086,
-0.03892209753394127,
0.0003241001395508647,
0.07200606167316437,
-0.05516184866428375,
0.0028654239140450954,
-0.09957029670476913,
0.001213942072354257,
-0.005116040352731943,
0.04606234282255173,
0.02133423276245594,
-0.03163636848330498
] | 8,988 |
In trying to determine if we have a simple harmonic oscillator we should note that for small angles than about by about or less at smaller | [
-0.014840278774499893,
0.03932441398501396,
-0.0159600842744112,
-0.050396375358104706,
-0.03728540986776352,
-0.027071114629507065,
-0.08866991102695465,
0.03417976200580597,
0.033430155366659164,
0.022398579865694046,
0.08653250336647034,
-0.05273663252592087,
0.00018013814406003803,
-0.00010719589045038447,
0.10049764811992645,
-0.0749073475599289,
0.002993157133460045,
0.012781311757862568,
-0.01795286126434803,
0.05417615920305252,
0.010024853982031345,
-0.04293236881494522,
-0.06231383979320526,
-0.027464186772704124,
-0.03967803344130516,
-0.0012615140294656157,
0.0130807776004076,
-0.004896034020930529,
0.05632313713431358,
-0.06739123165607452,
-0.029541147872805595,
0.02406342327594757,
-0.017543887719511986,
-0.03201892226934433,
0.005551778711378574,
0.004718149080872536,
0.04779087379574776,
0.015013320371508598,
0.0060425493866205215,
-0.014276894740760326,
0.03224315121769905,
0.1251179575920105,
0.08456059545278549,
0.06573852896690369,
-0.05766186863183975,
-0.029684431850910187,
-0.04568406939506531,
-0.06812857836484909,
-0.07764027267694473,
-0.060099899768829346,
-0.013552543707191944,
-0.019486911594867706,
-0.03328444063663483,
-0.013884423300623894,
-0.06635475158691406,
-0.002073565497994423,
-0.02369662933051586,
0.02095830999314785,
0.04403591528534889,
0.02729111723601818,
0.037331193685531616,
-0.0791337788105011,
0.023890074342489243,
-0.041409969329833984,
0.00001356087886961177,
-0.05938630551099777,
0.03041446954011917,
-0.10569748282432556,
-0.04534938931465149,
-0.009528431110084057,
-0.052679553627967834,
-0.03667494282126427,
-0.06100128963589668,
0.08784711360931396,
0.010990709066390991,
-0.08159073442220688,
-0.08832202106714249,
0.06717774271965027,
-0.018806835636496544,
0.04996686428785324,
-0.1150042936205864,
-0.07651736587285995,
-0.09054697304964066,
-0.01659487560391426,
-0.016614174470305443,
-0.009941482916474342,
-0.012808935716748238,
-0.03766351565718651,
0.017686661332845688,
-0.022516243159770966,
-0.023251906037330627,
0.019343532621860504,
-0.024345597252249718,
-0.06561965495347977,
0.058017320930957794,
0.036754507571458817,
0.03342713415622711,
0.05347789451479912,
-0.043384112417697906,
0.04192093759775162,
0.040137652307748795,
-0.035680077970027924,
0.058081407099962234,
0.026099497452378273,
-0.025638431310653687,
0.03964880108833313,
0.04899098724126816,
-0.0200031865388155,
0.03256971761584282,
0.022899389266967773,
-0.03179638087749481,
0.054872334003448486,
0.008486524224281311,
0.07403076440095901,
0.09809058904647827,
-0.04678425192832947,
-0.038886599242687225,
-0.0028414088301360607,
0.009400341659784317,
-0.011118510738015175,
0.05620169639587402,
0.00555380480363965,
-0.06073591113090515,
0.038302600383758545,
-0.0137045131996274,
0.03887903690338135,
-0.05026252567768097,
1.5606377280295174e-33,
-0.002794395899400115,
0.14802488684654236,
0.09973765909671783,
-0.03466692939400673,
-0.006070950534194708,
-0.005449036601930857,
-0.043718546628952026,
0.08091773837804794,
0.08839614689350128,
0.047062911093235016,
-0.02462597005069256,
0.03805315122008324,
0.02094147726893425,
-0.0162795577198267,
0.03052065707743168,
-0.024250945076346397,
0.01622875966131687,
0.02034682035446167,
-0.05970965698361397,
-0.033401552587747574,
0.020232558250427246,
-0.03703870624303818,
0.03747496008872986,
-0.007458873558789492,
-0.06197639927268028,
-0.050590723752975464,
0.04862320050597191,
-0.00004730588989332318,
-0.12136185169219971,
-0.010771731846034527,
0.049024030566215515,
-0.02084827795624733,
-0.03140084072947502,
-0.02019764482975006,
0.08340679854154587,
-0.008469468913972378,
0.0011773717124015093,
0.050641223788261414,
0.013521740213036537,
-0.012462087906897068,
-0.07965926080942154,
0.0011119113769382238,
-0.0466877743601799,
0.06699415296316147,
0.08684006333351135,
-0.03344954550266266,
0.03417300432920456,
0.008690537884831429,
-0.0384192131459713,
0.03846777603030205,
-0.0371478870511055,
-0.009292203933000565,
-0.006186755374073982,
0.03623226657509804,
0.13506311178207397,
0.0991247370839119,
0.04922611638903618,
0.010085009969770908,
-0.11133426427841187,
0.01617451198399067,
-0.006917121820151806,
-0.009959901683032513,
0.06576953083276749,
-0.07110344618558884,
-0.06600472331047058,
0.034354422241449356,
-0.010489429347217083,
-0.020802637562155724,
0.00025508206454105675,
-0.012640643864870071,
0.0332413986325264,
0.036874573677778244,
0.04076773673295975,
0.025152597576379776,
0.06770393997430801,
-0.01889154314994812,
0.038120970129966736,
0.07722150534391403,
0.017084304243326187,
-0.016959069296717644,
0.03884335234761238,
-0.04754028841853142,
0.03605300933122635,
0.011719420552253723,
-0.09653498232364655,
-0.08846435695886612,
0.07791269570589066,
0.032170020043849945,
-0.09459017962217331,
0.04831283912062645,
-0.0514628067612648,
0.02983366698026657,
-0.043132390826940536,
0.03666505590081215,
-0.020432325080037117,
-4.4264587925296835e-33,
-0.03320004418492317,
-0.03543008491396904,
0.05813737213611603,
-0.02450620010495186,
-0.07907973229885101,
0.08386453986167908,
0.03828446939587593,
0.043071404099464417,
-0.08438913524150848,
-0.08063338696956635,
0.08989230543375015,
0.018114503473043442,
0.029118096455931664,
-0.058942314237356186,
-0.03727426752448082,
0.10937084257602692,
0.0012875483371317387,
-0.05176557973027229,
0.08900780975818634,
0.07604849338531494,
0.011651131324470043,
0.04475415498018265,
-0.009997394867241383,
0.02988009713590145,
0.07526146620512009,
0.0143721429631114,
0.015184758231043816,
-0.04564662650227547,
-0.07193301618099213,
-0.009611577726900578,
-0.029720783233642578,
-0.0034038755111396313,
0.03892553970217705,
-0.12835942208766937,
-0.02178843319416046,
0.03519400581717491,
0.00013478921027854085,
-0.017906751483678818,
-0.008938017301261425,
0.038560569286346436,
-0.05394231900572777,
0.0017668254440650344,
0.11213180422782898,
-0.038869716227054596,
-0.06085120514035225,
-0.03766094520688057,
0.10547938942909241,
0.02177431248128414,
-0.049253880977630615,
0.08293511718511581,
-0.006815754342824221,
-0.10212715715169907,
0.022737890481948853,
0.014573285356163979,
-0.028926203027367592,
0.06575440615415573,
-0.029773935675621033,
0.03564921021461487,
0.009689797647297382,
-0.07422391325235367,
-0.10030615329742432,
0.013200800865888596,
-0.025382965803146362,
-0.01650431379675865,
-0.04708948731422424,
0.0031347551848739386,
-0.006757070776075125,
-0.03644467145204544,
0.02722296491265297,
-0.01696251891553402,
0.009647072292864323,
-0.014119015075266361,
0.04408003389835358,
0.005724503193050623,
-0.05382847413420677,
-0.026263274252414703,
0.06705843657255173,
-0.12543529272079468,
-0.10220684111118317,
-0.015845563262701035,
0.008839176036417484,
0.07408595830202103,
-0.06722769141197205,
-0.08899666368961334,
-0.032991666346788406,
-0.06361529976129532,
-0.02538197860121727,
-0.06908009201288223,
-0.014434890821576118,
0.05915374681353569,
-0.04842411354184151,
0.147643581032753,
-0.03641975298523903,
-0.10103626549243927,
0.060782574117183685,
-3.2175243802612385e-8,
0.03436760604381561,
-0.06048404425382614,
0.007869786582887173,
0.010706765577197075,
0.07990556955337524,
0.028016388416290283,
-0.012243399396538734,
-0.08613225072622299,
-0.09496115148067474,
-0.022491103038191795,
0.04297195374965668,
0.030347591266036034,
-0.0011898702941834927,
-0.0074701677076518536,
0.025081053376197815,
-0.02841424196958542,
0.004242825787514448,
0.04076153784990311,
0.03275666758418083,
0.028945449739694595,
-0.01757156103849411,
0.011254587210714817,
0.08224514871835709,
-0.02732527069747448,
0.0453057698905468,
0.022779405117034912,
-0.0703846737742424,
0.010970205068588257,
-0.06892646849155426,
-0.00800038781017065,
0.05641305446624756,
0.02669716440141201,
0.03670723736286163,
-0.033428825438022614,
0.027402428910136223,
0.047027986496686935,
-0.04496001824736595,
0.01736951619386673,
0.00722174858674407,
-0.08957788348197937,
-0.07721492648124695,
0.05123318359255791,
-0.04989727959036827,
0.05682440102100372,
0.04039648920297623,
0.012695337645709515,
0.006388102192431688,
-0.014363477937877178,
0.02246389165520668,
-0.03180653229355812,
-0.06787443161010742,
0.009732346050441265,
0.03201302886009216,
-0.05536498874425888,
-0.07130388915538788,
0.05773238465189934,
0.04830678924918175,
-0.01572166383266449,
-0.11711225658655167,
-0.023089714348316193,
0.06412019580602646,
0.09602972120046616,
-0.009208602830767632,
-0.04853081703186035
] | 8,989 |
Thus for angles less than about the restoring force Fis The displacement sis directly proportional to | [
-0.05555053427815437,
0.03410898894071579,
0.026756010949611664,
0.03623275086283684,
-0.009920302778482437,
0.050631921738386154,
-0.08932121843099594,
0.04931696504354477,
0.07070343941450119,
0.05030432716012001,
0.08632424473762512,
0.03971736133098602,
0.02814646251499653,
0.03118021786212921,
-0.0669117346405983,
-0.07990523427724838,
0.010452238842844963,
0.13912779092788696,
-0.07513711601495743,
0.10001380741596222,
-0.0016187397995963693,
-0.013555346988141537,
0.04464791715145111,
0.027160828933119774,
-0.0012726126005873084,
0.004489311017096043,
-0.008523409254848957,
0.033922113478183746,
0.11297550052404404,
-0.07889357954263687,
-0.023188982158899307,
0.1041518896818161,
-0.04533123970031738,
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-0.07045347988605499,
0.029893847182393074,
-0.038470905274152756,
-0.02161676064133644,
-0.05341729521751404,
-0.01604524627327919,
-0.038111601024866104,
0.08345185220241547,
0.03306536749005318,
0.028505785390734673,
0.04819415509700775,
0.04938311502337456,
0.006075022742152214,
0.012114076875150204,
0.0049783834256231785,
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0.025652389973402023,
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0.010166323743760586,
0.006594504229724407,
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0.05040300637483597,
-0.04940298944711685,
0.07241427153348923,
-0.01056059543043375,
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0.040435854345560074,
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-0.08311265707015991,
0.03777153044939041,
-0.055279817432165146,
0.007128057535737753,
-0.03643828630447388,
0.003056498244404793,
0.06221093237400055,
-0.07400237768888474,
-0.029289253056049347,
0.14920809864997864,
0.061849117279052734,
0.013190539553761482,
0.014479272998869419,
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-0.1059255301952362,
0.0421355739235878,
-0.07848241180181503,
0.007189668715000153,
-0.0642567053437233,
-0.07960368692874908,
0.06764084100723267,
-0.010552107356488705,
0.06676403433084488,
0.03431287035346031,
-0.04135938733816147,
-0.03660013526678085,
0.0008431152673438191,
-0.05899258702993393,
-0.010277099907398224,
0.050457436591386795,
-0.012476122938096523,
0.025517091155052185,
0.029567258432507515,
0.025399496778845787,
-0.08863429725170135,
0.09702879935503006,
-0.054843343794345856,
-0.01654151827096939,
0.04061521589756012,
0.02003423124551773,
0.00613057054579258,
0.11204837262630463,
-0.003518184181302786,
-0.017547670751810074,
-0.048573266714811325,
0.0013571473537012935,
0.006393113639205694,
-0.03541567549109459,
0.013174782507121563,
-0.036717116832733154,
-0.0057409643195569515,
-0.04240506887435913,
0.04936859384179115,
-0.016150524839758873,
-0.03182907775044441,
0.0754576250910759,
-0.002603689907118678,
-0.016272855922579765,
-0.017017928883433342,
-3.4380532756838904e-34,
0.006416631862521172,
0.11468909680843353,
0.052455876022577286,
-0.06594044715166092,
-0.011478499509394169,
-0.07847504317760468,
-0.032527271658182144,
0.054973896592855453,
0.1050175279378891,
-0.005083062220364809,
-0.1341140866279602,
0.08687712252140045,
-0.010265294462442398,
-0.038652289658784866,
-0.023313011974096298,
-0.006590163800865412,
0.006299685221165419,
0.09099039435386658,
-0.06338459998369217,
0.0017799708293750882,
0.004399820230901241,
-0.005413152743130922,
-0.02374357357621193,
-0.10195993632078171,
-0.08133487403392792,
0.0027217420283704996,
-0.012280236929655075,
0.01930643990635872,
-0.12488095462322235,
-0.016291150823235512,
0.017423896118998528,
0.0040497491136193275,
0.00821743719279766,
-0.02747419849038124,
-0.013617652468383312,
-0.003805303480476141,
0.050624821335077286,
-0.040564194321632385,
-0.04446100443601608,
0.0352272093296051,
-0.015527194365859032,
0.02644466795027256,
0.016864674165844917,
-0.016125846654176712,
0.08253750950098038,
-0.032315220683813095,
0.06072167679667473,
-0.035162292420864105,
-0.05432809889316559,
0.03208368271589279,
-0.000902744650375098,
0.0756930410861969,
0.013616370037198067,
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0.06564506143331528,
0.022532420232892036,
-0.046806592494249344,
0.005498305894434452,
-0.1268751323223114,
0.03424043953418732,
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0.05552472546696663,
0.01907264068722725,
-0.0004660797421820462,
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-0.04408014565706253,
0.06062539294362068,
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0.030751429498195648,
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0.03389054909348488,
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0.08551225811243057,
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0.02271558716893196,
-0.014759378507733345,
-0.04075651988387108,
-0.032596405595541,
-0.03993413224816322,
0.01266889926046133,
0.09366638958454132,
-1.872669948390409e-33,
-0.019334236159920692,
0.009122258983552456,
-0.12159158289432526,
-0.02361363358795643,
-0.0767139196395874,
0.015339633449912071,
0.03215731680393219,
0.029666120186448097,
-0.06882735341787338,
-0.025217950344085693,
-0.012637709267437458,
-0.05105791985988617,
0.010542363859713078,
-0.010988699272274971,
-0.047666437923908234,
0.051272325217723846,
0.00102325773332268,
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0.0514802411198616,
0.07652445882558823,
0.12208908051252365,
0.04024546965956688,
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0.05820835754275322,
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0.050864461809396744,
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0.049471382051706314,
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0.07010292261838913,
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0.019103439524769783,
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0.03401056304574013,
0.050081439316272736,
0.0345943421125412,
0.013878568075597286,
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0.054671552032232285,
0.08815254271030426,
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0.034686218947172165,
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0.05770929530262947,
0.03095964342355728,
0.00528901070356369,
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0.009150654077529907,
0.06859536468982697,
-0.041822317987680435,
-0.12093372642993927,
-0.01213517040014267,
-0.04419347643852234,
0.01288928184658289,
-0.022073063999414444,
-0.02262091636657715,
-0.08639629185199738,
-0.04124252125620842,
-0.08677444607019424,
-0.008742040954530239,
0.02904263697564602,
0.08868476003408432,
-0.031152551993727684,
0.02974018082022667,
0.09795216470956802,
-0.058110181242227554,
0.017054764553904533,
-2.0528942101805114e-8,
-0.01900039240717888,
0.02546539157629013,
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0.04111534357070923,
-0.03745974600315094,
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-0.05818900465965271,
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0.062797911465168,
0.11217331886291504,
0.07896188646554947,
0.0292804092168808,
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0.009985019452869892,
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-0.04712850600481033,
0.06467890739440918,
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0.13872234523296356,
0.012116879224777222,
0.02076505683362484,
0.025952192023396492,
-0.045953549444675446,
-0.008920845575630665,
0.03446168079972267,
0.04927074909210205,
0.04098004847764969,
-0.06284633278846741,
0.03833219036459923,
-0.04999060183763504,
0.04484288766980171,
0.060723911970853806,
-0.012057078070938587,
-0.0009732609032653272,
0.031763624399900436,
0.029240621253848076,
-0.056119997054338455,
0.06354644894599915,
0.01812305860221386,
0.05549457669258118,
0.05851132050156593,
0.053508348762989044,
-0.04475550726056099,
0.008653897792100906,
0.003793613752350211,
-0.036411307752132416,
-0.040006380528211594,
0.044970959424972534,
0.019134027883410454,
0.016040345653891563,
0.09500565379858017,
-0.007823911495506763,
0.04362523928284645,
-0.023956570774316788,
-0.008876892738044262,
-0.018346549943089485,
0.062040720134973526,
0.05419318005442619,
0.0606369748711586,
-0.06234339624643326
] | 8,990 |
expressed in radians the arc length in a circle is related to its radius this so that L For small angles then the expression for the restoring force Ls This expression is of the where the force constant is given by the displacement is given by | [
-0.011459625326097012,
0.037751100957393646,
0.018451234325766563,
0.11048577725887299,
0.07603023946285248,
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0.07223302870988846,
0.03562534227967262,
0.051845911890268326,
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0.02322569116950035,
0.05573057755827904,
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0.02201598696410656,
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0.025850698351860046,
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0.08612018078565598,
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-0.09016580879688263,
-0.0576758086681366,
-0.01840178109705448,
-0.04978662729263306,
0.0070496816188097,
0.008503466844558716,
-0.07590053230524063,
-0.004420687910169363,
-0.03155919164419174,
0.006798959337174892,
-0.06403671205043793,
0.0664072260260582,
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0.025841133669018745,
-0.0352964885532856,
0.02977302297949791,
0.10034490376710892,
0.011497960425913334,
0.01400833111256361,
-0.02365989424288273,
0.04098895937204361,
0.0775572657585144,
0.02400214411318302,
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0.0069839246571063995,
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0.05070789158344269,
0.013138090260326862,
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0.038339655846357346,
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0.03541240096092224,
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0.007861047051846981,
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0.07191972434520721,
0.09826549142599106,
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0.08378874510526657,
0.03040323220193386,
-0.01683657243847847,
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0.038123052567243576,
0.10932467877864838,
0.031121162697672844,
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0.05231347307562828,
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0.05923375114798546,
0.012946823611855507,
0.015006051398813725,
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0.05199730023741722,
0.02890046499669552,
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0.04627080634236336,
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0.03519994392991066,
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0.05315611883997917,
0.006514774169772863,
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0.040650736540555954,
0.012809486128389835,
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0.031080860644578934,
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0.03996209055185318,
0.05497671663761139,
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0.03278788924217224,
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-0.03392576798796654,
0.054627425968647,
0.10657910257577896,
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0.016154784709215164,
0.04607376083731651,
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0.0581267885863781,
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0.01350028719753027,
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0.08285358548164368,
-0.041354142129421234,
0.0196673683822155,
-0.037192452698946,
-0.025062281638383865,
-0.006400636397302151,
-0.010003444738686085,
0.042800936847925186,
0.11224352568387985,
-1.9874351149001196e-33,
-0.01925048604607582,
0.025229882448911667,
-0.07455851882696152,
-0.045281536877155304,
-0.01548408716917038,
-0.004892004653811455,
-0.0025042807683348656,
0.0481613352894783,
-0.0432591587305069,
-0.06963558495044708,
-0.062088385224342346,
-0.06187968701124191,
0.0013593400362879038,
-0.03142891824245453,
-0.00580018013715744,
0.0006169438711367548,
-0.0022722026333212852,
-0.07213735580444336,
-0.0557381846010685,
0.09605869650840759,
0.004165282938629389,
0.06931182742118835,
0.036166995763778687,
-0.04432441294193268,
0.06272426247596741,
0.008268017321825027,
0.026284383609890938,
-0.038803182542324066,
-0.06684983521699905,
0.013389536179602146,
-0.05189599469304085,
-0.052220895886421204,
0.03970327228307724,
0.0036165958736091852,
-0.014333492144942284,
-0.0338331013917923,
0.053488973528146744,
0.008041108958423138,
-0.01308221835643053,
-0.03151382878422737,
0.0632093995809555,
-0.029845280572772026,
0.12428416311740875,
-0.05198810249567032,
0.019060106948018074,
-0.08322826772928238,
0.05787188559770584,
-0.01009236928075552,
-0.027998454868793488,
-0.02604982815682888,
0.02314920164644718,
-0.09085972607135773,
0.05876833572983742,
-0.04720234125852585,
0.029908349737524986,
-0.024032413959503174,
-0.04111505672335625,
0.0685604065656662,
0.03968995809555054,
-0.0442495234310627,
-0.0756884515285492,
-0.052406370639801025,
0.03845107555389404,
0.07317359000444412,
-0.00417330814525485,
0.0425281897187233,
0.06358754634857178,
0.013553799130022526,
-0.025457214564085007,
-0.06147395819425583,
-0.002787710865959525,
0.08485421538352966,
0.010754589922726154,
0.03594454005360603,
-0.054687127470970154,
-0.05454588681459427,
0.012348361313343048,
-0.09140615165233612,
-0.10501405596733093,
-0.006021135486662388,
-0.03190680220723152,
0.0006745764403603971,
-0.051975030452013016,
-0.04591268301010132,
-0.07702691853046417,
-0.09135010093450546,
-0.04682762175798416,
0.031016552820801735,
-0.041961196810007095,
0.01807432807981968,
-0.06868389248847961,
0.04542131721973419,
0.005083514377474785,
-0.04868276044726372,
0.1149756982922554,
-3.038411833244936e-8,
-0.03626490384340286,
0.04337810352444649,
-0.05319545790553093,
0.038572993129491806,
0.07483717799186707,
0.05920760706067085,
-0.006922772154211998,
-0.003598553128540516,
-0.07840175181627274,
-0.018284698948264122,
0.02634064480662346,
0.09574704617261887,
0.08938068896532059,
0.022808466106653214,
0.04134545102715492,
0.02995268628001213,
0.040458474308252335,
-0.01681637391448021,
0.01082124188542366,
0.08483004570007324,
0.042017750442028046,
-0.011344236321747303,
0.12758690118789673,
-0.05318654701113701,
0.05044930800795555,
-0.020753104239702225,
-0.0039817895740270615,
0.016057226806879044,
-0.01687113381922245,
0.00939246267080307,
0.045216936618089676,
-0.011097181588411331,
0.009752397425472736,
-0.06414972990751266,
0.021056661382317543,
-0.003183302702382207,
0.02594727650284767,
-0.0398944653570652,
0.03625469282269478,
0.04352043569087982,
-0.07428329437971115,
0.11214184761047363,
-0.015837810933589935,
-0.01967572048306465,
0.03688748925924301,
0.0491645410656929,
0.029563751071691513,
-0.03973810374736786,
-0.05692183971405029,
-0.06280701607465744,
-0.044724348932504654,
-0.011533882468938828,
0.001189494039863348,
0.03890298679471016,
0.07333077490329742,
-0.045844051986932755,
0.0947028398513794,
0.022415485233068466,
-0.04435867443680763,
-0.011352930217981339,
0.009129106998443604,
0.10696826130151749,
0.11989487707614899,
-0.04061929136514664
] | 8,991 |
For angles less than about the restoring force is directly proportional to the displacement and the simple pendulum is a simple harmonic oscillator Using this equation we can find the period of a pendulum for amplitudes less than about | [
-0.06366512179374695,
0.01069615501910448,
0.04619159922003746,
0.031079309061169624,
-0.04383285343647003,
0.05693209543824196,
-0.10793082416057587,
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0.021242184564471245,
0.02594083361327648,
0.07024826854467392,
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0.07293301075696945,
-0.051335155963897705,
0.03346751630306244,
-0.038401439785957336,
-0.02064882591366768,
0.08138401806354523,
0.022127924486994743,
0.03506788983941078,
0.0555751733481884,
-0.06728817522525787,
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-0.029190868139266968,
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0.030711757019162178,
-0.04054192453622818,
0.027764391154050827,
0.06617488712072372,
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0.04264070466160774,
-0.030317068099975586,
-0.08520472794771194,
-0.045680612325668335,
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] | 8,992 |
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Subsets and Splits