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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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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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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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Identify known PEelor 0 563 J 2 | [
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Solve for 563 0 002 23 3 | [
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Convert 23 7 Discussion and This projectile speed is impressive for a tranquilizer gun than 80 | [
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The numbers in this problem seem reasonable | [
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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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When you let go you can see the oscillations of the ruler | [
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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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Each successive vibration of the string takes the same time as the previous one | [
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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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The time to complete one oscillation remains constant and is called the periodT | [
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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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A concept closely related to period is the frequency of an event | [
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For example if you get a paycheck twice a month the frequency of payment is two per month and the period between checks is half a month | [
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Frequency fis defined to be the number of events per unit time | [
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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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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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try one example of each A medical imaging device produces ultrasound by oscillating with a period of 0 400 | [
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What is the frequency of this The frequency of middle C on a typical musical instrument is 264 Hz | [
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