Chapter 16 Lectures. Oscillatory Motion and Waves

Transcription

1 Chapter 16 Lectures January :48 PM Oscillatory Motion and Waves Oscillations are back-and-forth motions. Sometimes the word vibration is used in place of oscillation; for our purposes, we can consider the two words to represent the same class of motions. Oscillations often cause waves, so the two concepts are connected. In the ideal case of no friction, free oscillations are a sub-class of periodic motions; that is, in the absence of friction, all free oscillations are periodic motions, but not all periodic motions are oscillations. For example, uniform circular motion (which we studied in PHYS 1P21/1P91) is periodic, but not considered an oscillation. Uniform circular motion can be modelled by sine or cosine functions of time; think back to the unit circle in high-school math when you were learning trigonometry. (Sine and cosine functions are collectively known as sinusoidal functions, or sinusoids for short.) If the restoring force that causes oscillation is a linear function of displacement, then the resulting oscillatory motion can also be modelled by a sinusoidal function of time. There is a close relationship between uniform circular motion and oscillatory motion caused by a linear restoring force; we'll discuss this later. Ch16Lectures2018Jan Page 1

2 Q: What is a restoring force? Q: What is a linear restoring force? The graph above illustrates Hooke's law. Ch16Lectures2018Jan Page 2

3 Q: What is Hooke's law? Notice that the formula for Hooke's law is represented by the graph; the magnitude of the restoring force is proportional to the magnitude of the displacement from equilibrium, and the force is in the direction opposite to the displacement. The constant of proportionality is called the stiffness constant of the spring. Our textbook uses "force constant" instead of stiffness constant, and other books use "spring constant" instead of stiffness constant. Q: What are the units of k? Q: What are some typical values for the stiffness constant for coil springs in your experience (ones in your car's shock absorbers, in your ball-point pen, attached to your aluminum door, etc.)? Example: A spring has equilibrium length 18 cm and stiffness constant 100 N/m. Determine the magnitude of the force needed to stretch the spring so that its length is a fixed 20 cm. Solution: Ch16Lectures2018Jan Page 3

4 Example: A spring has equilibrium length 30 cm and stiffness constant 500 N/m. A force of magnitude 10 N stretches the spring. Determine the new length of the spring. Solution: Calculate the displacement of the end of the spring first: Therefore, the new length of the spring is Frequency and Period of a Periodic Motion Q: What is the period of a periodic motion? Q: What is the frequency of a periodic motion? Q: How are the frequency and period of a periodic motion related? Ch16Lectures2018Jan Page 4

5 Example: The rotating tires of a car make 27.3 revolutions in 8.6 s. (a) Determine the period of the rotation of the tires. (b) Determine the frequency of the rotation of the tires. Solution: Animation of a simple harmonic motion (SHM) Here is an example position-time diagram for an SHM: Ch16Lectures2018Jan Page 5

6 Notice that the position-time diagram for the simple harmonic motion (SHM) resembles the graph of a sinusoidal function. You'll get a chance to see that this must be so for an oscillator that is subject to a linear restoring force (Hooke's law) and no friction later in the chapter. Now is a good time to review sinusoidal functions, so let's do it: Review of sine and cosine functions Ch16Lectures2018Jan Page 6

7 Play with the values of A and f in the cosine graph, and see what happens: Now repeat the same play, only set up a sine function instead of a cosine function. Q: Determine the amplitude of the cosine function x(t) = 4.3 cos (2.1t) where x is measured in cm and t is measured in seconds. Solution: Q: Determine the angular frequency of the cosine function x(t) = 4.3 cos (2.1t) where x is measured in cm and t is measured in seconds. Solution: Q: Determine the frequency of the cosine function Ch16Lectures2018Jan Page 7

8 x(t) = 4.3 cos (2.1t) where x is measured in cm and t is measured in seconds. Solution: Q: Determine the period of the cosine function x(t) = 4.3 cos (2.1t) where x is measured in cm and t is measured in seconds. Solution: Summary of sine and cosine functions: Ch16Lectures2018Jan Page 8

9 Mathematical models for SHM Simple harmonic motion can be modelled by a sinusoidal position-time function. If we simplify the situation, and allow only two types of starts to the motion (in the language of differential equations, we allow only two possible initial conditions), then we can model the motion either by a simple sine function or a simple cosine function. (In general, one needs either a phase shifted sinusoid, or a linear combination of sine and cosine functions to model all possible initial conditions.) Sine model for an SHM (relevant for an oscillator that begins at the equilibrium position and is "kicked" to begin the motion): Cosine model for an SHM (relevant for an oscillator that is displaced away from the equilibrium position and then released to begin the motion): Ch16Lectures2018Jan Page 9

10 Here's a graph illustrating the cosine model; a graph illustrating the sine model is similar. Ch16Lectures2018Jan Page 10

11 Summary of mathematical models of SHM: Example: Consider an oscillation with position function Ch16Lectures2018Jan Page 11

12 x = 8 cos (2t) where x is measured in cm and t is measured in s. Determine the positions on the x-axis that are turning points. Solution: Example: Consider an oscillation with position function x = 8 cos (2t) where x is measured in cm and t is measured in s. Determine the first two times at which the object is at a turning point. Solution: Ch16Lectures2018Jan Page 12

13 Example: Consider an oscillation with position function x = 8 cos (2t) where x is measured in cm and t is measured in s. Determine the first two times at which the object is at its equilibrium position. Solution: Ch16Lectures2018Jan Page 13

14 Example: Consider an oscillation with position function x = 8 cos (2t) where x is measured in cm and t is measured in s. Determine the first two times at which the object has maximum speed. Solution: The maximum speed occurs when the oscillating object is at its equilibrium position. Thus, the solution to this problem is exactly the same as the solution to a previous problem, where we determined the first two times at which the object was at the equilibrium position. Ch16Lectures2018Jan Page 14

15 Example: Consider an oscillation with position function x = 8 cos (2t) where x is measured in cm and t is measured in s. Determine the first two times at which the object has minimum speed. Solution: The minimum speed (which is zero) occurs when the object is at either of its turning points. Thus, the solution to this problem is exactly the same as a previous problem, where we determined the first two times that the object is at a turning point. Note that for both the sine and cosine models of SHM (and indeed, for any model of SHM, even the more complex ones), the acceleration-time function is a constant multiple of the position-time function. This is a consequence of the linearity of the restoring force (i.e., Hooke's law), as can be verified using Newton's second law of motion: Ch16Lectures2018Jan Page 15

16 Comparing this relation with the relation obtained earlier from the mathematical models, we can obtain useful relations for the frequency and period of an SHM: The period of an SHM is the reciprocal of its frequency, so the period of an SHM is Q: Do the formulas for angular frequency and frequency of SHM make sense physically? Discuss. Ch16Lectures2018Jan Page 16

17 Q: Note that the period (and therefore also both the frequency and angular frequency) does not depend on the amplitude of the oscillation. This is interesting. Does this match with your experience? Although we won't be discussing pendulums, it's interesting to at least write down the analogous formula for the period of a pendulum of length L (valid as long as the amplitude is small): where g is the acceleration due to gravity. It's interesting to note that the period decreases (and therefore the frequency increases) when the length of the pendulum decreases. Q: Do you have examples of this pendulum behaviour in your experience? Q: Doesn't this make sense when you think about what naturally happens to your arms when you are walking, you speed up, and then you begin to run? Ch16Lectures2018Jan Page 17

18 Example: A block of mass 0.45 kg is attached to a spring with stiffness constant 8.3 N/m. The block oscillates on a horizontal frictionless surface, and the spring stays in line as it stretches and compresses. Determine the frequency of the block's oscillation. Solution: Ch16Lectures2018Jan Page 18

19 Example: A block of mass 0.45 kg is attached to a spring with stiffness constant 8.3 N/m. The block oscillates on a horizontal frictionless surface, and the spring stays in line as it stretches and compresses. Determine the period of the block's oscillation. Solution: Energy in Simple Harmonic Motion When a spring is stretched or compressed it stores energy that we call elastic potential energy. Using Hooke's law, the general relation between force and energy for a conservative force, and calculus, it's possible to derive the following expression for the elastic potential energy U stored in a spring that has been displaced from its equilibrium position by x: Ch16Lectures2018Jan Page 19

20 Recall the expression for kinetic energy and therefore the total mechanical energy of a simple harmonic oscillator is In the ideal situation of no friction, the total mechanical energy is conserved, and this is a very useful problem-solving tool in such cases. As a simple harmonic oscillator moves, its total mechanical energy is continually transferred back and forth between kinetic energy and elastic potential energy. An energy diagram is useful for illustrating and understanding these energy transfers. Ch16Lectures2018Jan Page 20

21 Note that the horizontal axis in the energy diagram represents displacement, NOT time. Therefore, an energy diagram is interpreted differently from a position-time graph. For example, the energy diagram does not explicitly tell us how the motion evolves in time. However, because we know how the motion evolves in time, an energy diagram allows us to understand the motion more deeply. Ch16Lectures2018Jan Page 21

22 Example: Consider a block of mass 3.7 kg attached to a spring. The position-time function of this oscillator is x = 5.3 cos(1.9t), where x is measured in cm and t is measured in Ch16Lectures2018Jan Page 22

23 seconds. Determine the spring's stiffness constant. Solution: We can read off the angular frequency of the oscillation from the position-time function. Then we can use that information, together with the mass of the block, to determine the stiffness constant of the spring. Example: Consider a block of mass 3.7 kg attached to a spring. The position-time function of this oscillator is x = 5.3 cos(1.9t), where x is measured in cm and t is measured in seconds. Determine the total mechanical energy of the oscillator. Solution: The total mechanical energy of the oscillator is equal to the maximum potential energy of the oscillator. Thus, the total mechanical energy of the oscillator is Ch16Lectures2018Jan Page 23

24 Example: Consider a block of mass 3.7 kg attached to a spring. The position-time function of this oscillator is x = 5.3 cos(1.9t), where x is measured in cm and t is measured in seconds. Determine the maximum potential energy. Solution: This is exactly the same as the previous problem, just phrased in a different way. Do you understand that these two problems are the same? Example: Consider a block of mass 3.7 kg attached to a spring. The position-time function of this oscillator is x = 5.3 cos(1.9t), where x is measured in cm and t is measured in seconds. Determine the maximum kinetic energy. Solution: This is exactly the same as the previous two problems, just phrased in a different way. Do you understand that these three problems are the same? Example: Consider a block of mass 3.7 kg attached to a spring. The position-time function of this oscillator is x = 5.3 cos(1.9t), where x is measured in cm and t is measured in seconds. Determine the positions at which the kinetic energy and the Ch16Lectures2018Jan Page 24

25 potential energy of the oscillator are equal. Solution: In problems such as this one, the potential energy formula is easier to work with than the kinetic energy formula, so rephrase the problem to eliminate the kinetic energy. The reason for this is that we are asked to calculate positions, and the potential energy formula is expressed in terms of position. The kinetic energy formula is expressed in terms of speed, and using it will make life more difficult for us. Let's think through this "rephrasing" of the problem. When the kinetic energy is equal to the potential energy, the potential energy is half of the total mechanical energy. Do you understand this? Thus: Ch16Lectures2018Jan Page 25

26 Example: When the position of a simple harmonic oscillator is half of the amplitude, what fraction of the total energy is potential energy? Solution: Let U 1 represent the total mechanical energy and let U 2 represent the potential energy when the position is half of the amplitude. Then: Ch16Lectures2018Jan Page 26

27 Damped Simple Harmonic Motion Cars have shock absorbers to make the ride smoother. A shock absorber consists of a stiff spring together with a damping tube. (The damping tube consists of a piston in an enclosed cylinder that is filled with a thick (i.e., viscous) oil.) Without the damping tube, a car would oscillate for a long time after going over a bump in a road; the damping tube helps to limit both the amplitude and duration of the oscillations. When the damping tube doesn't work anymore, the car tends to oscillate for a long time after going over a bump, which is annoying. The same thing happens with a screen door when its damping tube malfunctions. Ch16Lectures2018Jan Page 27

28 (There is a spring attached to the door, but it is not shown in the photograph.) The position function for a damped oscillator is modified as follows: where is called the "time constant" for the oscillation. You can think of this position function as representing a sort of sinusoid, but one with a steadily decreasing amplitude; the amplitude is the constant A times the exponential factor, which steadily decreases as time passes. Ch16Lectures2018Jan Page 28

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30 The damping tube in a screen door is adjustable. If the resistance is too great, then the door will take a long time to shut after it is opened. If the resistance is too little, then the door will swing back and forth many times before shutting. There is an ideal medium amount of resistance ("critical damping") which works best; you'll learn more about this, and see how these three cases (overdamping, critical damping, and underdamping) follow naturally from different classes of solutions to the appropriate differential equation, in PHYS 2P20 and MATH 2P08. Ch16Lectures2018Jan Page 30

31 Forced Oscillations and Resonance Examples of forced oscillations (also known as driven oscillations): child sloshing around in a bathtub parent pushing a child on a playground swing Some systems have a natural oscillating frequency; if you drive the system at its natural oscillating frequency, its amplitude can increase dramatically. This phenomenon is called resonance. In many situations, one tries to avoid resonant oscillations. For example, that annoying vibration in your dashboard when you are driving on the highway at a certain speed is very annoying. More seriously, soldiers are trained to break ranks when they Ch16Lectures2018Jan Page 31

32 march across a bridge, because if their collective march is at the same frequency as the natural frequency of the bridge, then there is danger that they could collapse the bridge. (This is an ancient custom, from an age when many bridges were made of wood.) Engineers must be careful to design bridges and tall buildings so that they don't have natural vibration frequencies; otherwise an unlucky wind could cause dangerous large-amplitude vibrations. The next two diagrams illustrate the fact that when an oscillator is driven at its resonant frequency, its amplitude increases. If the damping is small, the increase in amplitude can be enormous. Ch16Lectures2018Jan Page 32

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34 Another good example of resonance is tuned electrical circuits, such as the ones used in radio or television reception. Radio waves of many different frequencies are incident on a radio receiver in your home; each tries to "drive" electrical oscillations in an electrical circuit. The natural frequency of the electrical circuit can be adjusted so that it will resonate with only a certain frequency of radio wave; this is how you "tune in" to a certain radio station. The oscillations due to the resonant frequency persist, while all the other frequencies are rapidly damped. Waves Q: What is a wave? Examples of waves: sound waves, seismic waves (also called earthquakes), other mechanical waves (waves on a guitar string or piano string), light waves, radio waves, etc. In PHYS 1P21/1P91 (Mechanics) we mainly used a particle model of nature. In this course we widen our perspective to include phenomena that are more "extended" and which therefore can be better described as waves and/or fields. As an initial example, consider a mechanical wave on a string, such as a plucked guitar string. Ch16Lectures2018Jan Page 34

35 There are two basic types of waves that we'll study in this course, transverse waves and longitudinal waves: Longitudinal and Transverse Waves Ch16Lectures2018Jan Page 35

36 Q: Write a sentence or two to describe what a transverse wave is and what a longitudinal wave is. Picturing a wave as it moves is a little challenging, because it's not a point object moving (as we would model in PHYS 1P21/1P91 using a position-time graph), but rather an extended object moving. One way to do this is to plot "snapshots" of the wave at a sequence of times, as follows for example. None of the snapshots in the previous diagrams are positiontime diagrams; it's only the series of snapshots that gives you a sense for the motion of the wave. Each snapshot by itself tells you nothing about how the wave moves. Ch16Lectures2018Jan Page 36

37 Note that in a transverse mechanical wave, each individual particle in the medium simply executes an oscillation in a line perpendicular to the direction of the wave's motion. A transverse mechanical wave is a complex combination of these oscillations. You can see this for yourself by focusing your attention on one of the green spots in this PHeT simulation: Thus, in the sequence of hand-drawn diagrams above, each individual particle on the string executes oscillation in the y- direction only, but the wave itself moves in the x-direction only. This is interesting and worth reflecting on. Q: How can this possibly happen? What's going on at a microscopic level that makes this possible? There is a relationship connecting the speed of a wave, its wavelength, frequency, and period, which we'll now explore. Ch16Lectures2018Jan Page 37

38 Although we reasoned our way to the above equations based on transverse waves, they are also valid for longitudinal waves; it's just that it might be a little harder to visualize the relationships for longitudinal waves. Q: Isn't what we labelled the wavelength in the diagram above what we called the period in high-school math class? Why are we labelling things differently now? Where do we see the period in the diagrams above? What's going on?? Example: A sound wave has a speed of about 300 m/s and a frequency of about 10 khz. Determine the wavelength of the sound wave. Solution: Ch16Lectures2018Jan Page 38

39 Superposition and Interference of Waves In mechanics we learned that if two forces act on one object, the net force acting on the object is the sum of the two forces, added together in the way vectors are added. If three forces act on one object, then the net force acting on the object is the vector sum of all three forces, and so on. This is an example of a general phenomenon called the principle of superposition. If more than one cause is acting, the net effect can be calculated by "adding" together the causes. Experiments show that this principle is valid for a wide range of phenomena, but not all phenomena. The validity of the principle of superposition is connected to the fact that the differential equations that describe many phenomena in nature are linear differential equations. As we'll see later in the course, Maxwell's equations that describe electric and magnetic phenomena (at least in vacuum; within some materials, called nonlinear materials, this doesn't work) are linear differential equations, and so electric and magnetic fields in vacuum also satisfy the principle of superposition. Waves in some media satisfy the principle of superposition, but waves in other media (called nonlinear media) do not satisfy the principle of superposition. We'll focus in this course on the Ch16Lectures2018Jan Page 39

40 simplest examples of wave motion for which the principle of superposition is valid. Historically, the experimental verification of the principle of superposition for light waves was powerful evidence for the wave theory of light. Newton's theory of light envisioned light as a swarm of particles (he called them "corpuscles"), and he was able to explain many of the observed light phenomena to his satisfaction. However, it was observed that two beams of light could pass through each other with apparently no disturbance to either. This was quite surprising; how could two swarms of particles pass through each other without scattering in a million different directions?? This didn't make any sense to Newton's contemporaries and successors, and it doesn't make any sense to us nowadays either. However, if light is a wave phenomenon, then there is no problem with two waves passing through each other without disturbance, as the following simulation illustrates. (The principle of superposition!) THE PRINCIPLE OF SUPERPOSITION OF WAVES_PART 01 Ch16Lectures2018Jan Page 40

41 Another term for superposition of waves that means the same thing, and is often used in physics textbooks, is interference of waves. Physics textbooks speak about constructive interference and destructive interference. Q: Describe constructive interference. Q: Describe destructive interference. The Original Double Slit Experiment Ch16Lectures2018Jan Page 41

42 PhET interference simulation: Standing waves When two waves travel along a string in opposite directions, they interfere as they pass each other. If a pulse is created, such as by plucking a guitar string, the pulse reflects from both of the fixed ends of the string, creating waves of the same wavelength moving in opposite directions. The result will be some complex combination of constructive and destructive interference that typically looks like a mess. However, if the wavelength of the waves is just right, one gets a very simple kind of vibration, called a standing wave. Wave Reflection and Standing Waves 2.mp4 Ch16Lectures2018Jan Page 42

43 Standing Waves Part I: Demonstration We can calculate the wavelengths that result in standing waves Ch16Lectures2018Jan Page 43

44 by noting that standing waves occur when a whole number of half-wavelengths fit inside the length of the string. That is, the wavelengths that result in standing waves satisfy That is, The speed of a wave along a string depends on the properties of the string (specifically the tension in the string and linear density of the string). The frequencies that result in standing waves can then be calculated by knowing the wave speed and using the relationship between speed, frequency, and wavelength of a wave, as the following example shows. Example: Waves on a guitar string of length 78 cm travel at a rate of 390 m/s. Determine the wavelengths and frequencies of the first few standing waves (of longest wavelength). Solution: Ch16Lectures2018Jan Page 44

45 Solution: Beats When two waves of similar frequency, but not exactly the same frequency, are combined, the result is a wave that has a complex structure. If the two waves are sound waves, the result is that we hear a sound with a frequency that is the average of the two frequencies, but whose loudness varies with a frequency that is the difference of the two frequencies. This Ch16Lectures2018Jan Page 45

46 slow variation of the loudness of two sounds with similar frequencies is called "beats" and the difference of the two frequencies is called the beat frequency. Easy Beats- Physics - TRY THIS EXPERIMENT - AAPT Films Ch16Lectures2018Jan Page 46

47 Beats are typically annoying, which is why musical bands and orchestras are so careful to tune their instruments every time before they perform. However, there are some practical uses of beats; for example, piano tuners used tuning forks and beats in the old days to tune pianos. (Nowadays electronic devices are often used to detect frequencies of sounds, and the tuning is adjusted until the frequency is correct.) Energy and Intensity of Waves Energy in a mechanical wave is proportional to the square of its amplitude. The textbook makes a good attempt at providing an intuitive understanding of this. Q: Do you have a good sense for why the energy of a mechanical wave is proportional to the square of its amplitude? Ch16Lectures2018Jan Page 47

48 A wave is an extended object, and the energy of a wave is spread out throughout the region of space in which the wave exists. As the wave moves, it's possible that the energy of the wave becomes more or less concentrated. The intensity of a wave is, roughly speaking, a measure of how concentrated is the energy in the wave. Q: What is the precise definition of the intensity of a wave? Example: The intensity of sunlight on a warm summer day in St. Catharines is about 750 W/m 2. Determine how much sunlight energy is incident on your skin in 15 minutes if you are lying perpendicular to the sun's rays and the area of your exposed skin is 0.82 m 2. Solution: Ch16Lectures2018Jan Page 48

49 Example: The energy of a wave is proportional to the square of its amplitude. If the amplitude of the wave decreases by 5%, by what percentage does the energy of the wave decrease? Interesting WRONG way to approach the problem: 5 2 = 25, so the energy decreases by 25%. This is clearly wrong, because 10 2 = 100, and it makes no sense that all of the energy should disappear if the amplitude decreases by only 10%. Solution: Thus, a 5% decrease in amplitude leads to about a 10% decrease in energy. For all you calculus lovers out there, think about the tangent line approximation for a quadratic function: Ch16Lectures2018Jan Page 49

50 Now divide both sides of the previous equation by y to obtain Notice the factor of 2 in the previous relationship. Also notice that 2 times 5% is 10%, so this calculus derivation gives us a very good approximation to the actual percentage decrease in the energy of the wave. (It's not quite exact, because it is based on the tangent line approximation, not on the graph of the quadratic function, but it's quite close.) The error propagation formula for a power function (the squaring function in this case) works in exactly the same way: If x is a measured quantity, m is a numerical factor that has no error, and y is a calculated quantity, where Then, by the above calculus reasoning, Ch16Lectures2018Jan Page 50

51 Q: What are your summarizing statements for this chapter? Q: What did you find difficult? Q: What did you find interesting? Q: What was your most instructive mistake in this chapter? What did you learn from this mistake? Ch16Lectures2018Jan Page 51