Chapter 14 Oscillations В© 2010 Pearson Education, Inc. PowerPointВ® Lectures for College Physics: A Strategic Approach, Second Edition 14 Oscillations В© 2010 Pearson Education, Inc. Slide 14-2 В© 2010 Pearson Education, Inc. Slide 14-3 В© 2010 Pearson Education, Inc. Slide 14-4 В© 2010 Pearson Education, Inc. Slide 14-5 Equilibrium and Oscillation В© 2010 Pearson Education, Inc. Slide 14-12 Linear Restoring Forces and Simple Harmonic Motion В© 2010 Pearson Education, Inc. Slide 14-13 Frequency and Period The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation. В© 2010 Pearson Education, Inc. Slide 14-14 Checking Understanding A set of springs all have initial length 10 cm. Each spring now has a mass suspended from its end, and the different springs stretch as shown below. Now, each mass is pulled down by an additional 1 cm and released, so that it oscillates up and down. Rank the frequencies of the oscillating systems A, B, C and D, from highest to lowest. A. B. C. D. BпЂЅDпЂѕCпЂЅA BпЂѕAпЂЅDпЂѕC CпЂѕAпЂЅDпЂѕB AпЂЅCпЂѕBпЂЅD В© 2010 Pearson Education, Inc. Slide 14-15 Answer A set of springs all have initial length 10 cm. Each spring now has a mass suspended from its end, and the different springs stretch as shown below. Now, each mass is pulled down by an additional 1 cm and released, so that it oscillates up and down. Rank the frequencies of the oscillating systems A, B, C and D, from highest to lowest. A. B. C. D. BпЂЅDпЂѕCпЂЅA BпЂѕAпЂЅDпЂѕC CпЂѕAпЂЅDпЂѕB AпЂЅCпЂѕBпЂЅD В© 2010 Pearson Education, Inc. Slide 14-16 Checking Understanding A series of pendulums with different length strings and different masses is shown below. Each pendulum is pulled to the side by the same (small) angle, the pendulums are released, and they begin to swing from side to side. Rank the frequencies of the five pendulums, from highest to lowest. A. B. C. D. AпЂЅEпЂѕBпЂЅDпЂѕC DпЂѕAпЂЅCпЂѕBпЂЅE AпЂЅBпЂЅCпЂЅDпЂЅE BпЂѕEпЂѕCпЂѕAпЂѕD В© 2010 Pearson Education, Inc. Slide 14-17 Answer A series of pendulums with different length strings and different masses is shown below. Each pendulum is pulled to the side by the same (small) angle, the pendulums are released, and they begin to swing from side to side. Rank the frequencies of the five pendulums, from highest to lowest. A. B. C. D. AпЂЅEпЂѕBпЂЅDпЂѕC DпЂѕAпЂЅCпЂѕBпЂЅE AпЂЅBпЂЅCпЂЅDпЂЅE BпЂѕEпЂѕCпЂѕAпЂѕD В© 2010 Pearson Education, Inc. Slide 14-18 Example Problems The first astronauts to visit Mars are each allowed to take along some personal items to remind them of home. One astronaut takes along a grandfather clock, which, on earth, has a pendulum that takes 1 second per swing, each swing corresponding to one tick of the clock. When the clock is set up on Mars, will it run fast or slow? A 5.0 kg mass is suspended from a spring. Pulling the mass down by an additional 10 cm takes a force of 20 N. If the mass is then released, it will rise up and then come back down. How long will it take for the mass to return to its starting point 10 cm below its equilibrium position? В© 2010 Pearson Education, Inc. Slide 14-19 Energy in Simple Harmonic Motion As a mass on a spring goes through its cycle of oscillation, energy is transformed from potential to kinetic and back to potential. В© 2010 Pearson Education, Inc. Slide 14-20 Sinusoidal Relationships В© 2010 Pearson Education, Inc. Slide 14-21 Mathematical Description of Simple Harmonic Motion В© 2010 Pearson Education, Inc. Slide 14-22 Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: 1. 2. 3. 4. 5. 6. At which of the above times is the displacement zero? At which of the above times is the velocity zero? At which of the above times is the acceleration zero? At which of the above times is the kinetic energy a maximum? At which of the above times is the potential energy a maximum? At which of the above times is kinetic energy being transformed to potential energy? 7. At which of the above times is potential energy being transformed to kinetic energy? В© 2010 Pearson Education, Inc. Slide 14-23 Example Problem A pendulum is pulled to the side and released. Its subsequent motion appears as follows: 1. 2. 3. 4. 5. 6. At which of the above times is the displacement zero? At which of the above times is the velocity zero? At which of the above times is the acceleration zero? At which of the above times is the kinetic energy a maximum? At which of the above times is the potential energy a maximum? At which of the above times is kinetic energy being transformed to potential energy? 7. At which of the above times is potential energy being transformed to kinetic energy? В© 2010 Pearson Education, Inc. Slide 14-24 Solving Problems В© 2010 Pearson Education, Inc. Slide 14-25 Example Problem We think of butterflies and moths as gently fluttering their wings, but this is not always the case. Tomato hornworms turn into remarkable moths called hawkmoths whose flight resembles that of a hummingbird. To a good approximation, the wings move with simple harmonic motion with a very high frequencyвЂ”about 26 Hz, a high enough frequency to generate an audible tone. The tips of the wings move up and down by about 5.0 cm from their central position during one cycle. Given these numbers, A. What is the maximum velocity of the tip of a hawkmoth wing? B. What is the maximum acceleration of the tip of a hawkmoth wing? В© 2010 Pearson Education, Inc. Slide 14-26 Example Problem A car rides on four wheels that are connected to the body of the car by springs that allow the car to move up and down as the wheels go over bumps and dips in the road. Each spring supports approximately 1/4 the mass of the vehicle. A lightweight car has a mass of 2400 lbs. When a 160-lb person sits on the left front fender, this corner of the car dips by about 1/2 in. A. B. What is the spring constant of this spring? When four people of this mass are in the car, what is the oscillation frequency of the vehicle on the springs? В© 2010 Pearson Education, Inc. Slide 14-27 Example Problem Manufacturers are now making shoes with springs in the soles. A spring in the heel will decrease the impact force when your heel strikes, but, equally important, it can store energy that is returned when the foot rolls forward to push off. Ideally, a shoe would be designed so that, when your heel strikes, the spring compresses and then rebounds at a natural point in your stride. We can model this rebound as a segment of an oscillation to get some idea of how such a shoe should be designed. A. In a moderate running gait, your heel is in contact with the ground for 0.1 s or so. For optimal timing, what should the period of the motion for the mass (the runner) and spring (the spring in the shoe) system be? B. What should be the spring constant of the spring for a 70-kg man? C. When the man puts all his weight on the heel, how much should the spring compress? В© 2010 Pearson Education, Inc. Slide 14-28 Example Problem A 204 g block is suspended from a vertical spring, causing the spring to stretch by 20 cm. The block is then pulled down an additional 10 cm and released. What is the speed of the block when it is 5.0 cm above the equilibrium position? В© 2010 Pearson Education, Inc. Slide 14-29 Damping В© 2010 Pearson Education, Inc. Slide 14-30 Resonance В© 2010 Pearson Education, Inc. Slide 14-31 Summary В© 2010 Pearson Education, Inc. Slide 14-32 Summary В© 2010 Pearson Education, Inc. Slide 14-33 Additional Questions Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a пЂѕ b пЂѕ c пЂѕ d B. d пЂѕ c пЂѕ b пЂѕ a C. a пЂЅ b пЂЅ c пЂЅ d В© 2010 Pearson Education, Inc. Slide 14-34 Answer Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a пЂѕ b пЂѕ c пЂѕ d B. d пЂѕ c пЂѕ b пЂѕ a C. a пЂЅ b пЂЅ c пЂЅ d В© 2010 Pearson Education, Inc. Slide 14-35 Additional Questions Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a пЂѕ b пЂѕ c пЂѕ d B. d пЂѕ c пЂѕ b пЂѕ a C. a пЂЅ b пЂЅ c пЂЅ d В© 2010 Pearson Education, Inc. Slide 14-36 Answer Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a пЂѕ b пЂѕ c пЂѕ d B. d пЂѕ c пЂѕ b пЂѕ a C. a пЂЅ b пЂЅ c пЂЅ d В© 2010 Pearson Education, Inc. Slide 14-37 Additional Questions A pendulum is pulled to the side and released. Rank the following positions in terms of the speed, from highest to lowest. There may be ties. В© 2010 Pearson Education, Inc. Slide 14-38 Additional Questions A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? A. 1 s B. 3.3 s C. 6.7 s D. 13 s В© 2010 Pearson Education, Inc. Slide 14-39 Answer A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? A. 1 s B. 3.3 s C. 6.7 s D. 13 s В© 2010 Pearson Education, Inc. Slide 14-40 Additional Example Problem Walter has a summer job babysitting an 18 kg youngster. He takes his young charge to the playground, where the boy immediately runs to the swings. The seat of the swing the boy chooses hangs down 2.5 m below the top bar. вЂњPush me,вЂќ the boy shouts, and Walter obliges. He gives the boy one small shove for each period of the swing, in order keep him going. Walter earns $6 per hour. While pushing, he has time for his mind to wander, so he decides to compute how much he is paid per push. How much does Walter earn for each push of the swing? В© 2010 Pearson Education, Inc. Slide 14-41 Additional Example Problems A 500 g block is attached to a spring on a frictionless horizontal surface. The block is pulled to stretch the spring by 10 cm, then gently released. A short time later, as the block passes through the equilibrium position, its speed is 1.0 m/s. What is the blockвЂ™s period of oscillation? What is the blockвЂ™s speed at the point where the spring is compressed by 5.0 cm? A mass bounces up and down on a spring. The oscillation decays with a time constant of 50 s. If the oscillation begins with an amplitude of 20 cm, how long will it take until the amplitude has decreased by half to 10 cm? If the oscillation begins with an amplitude of 20 cm, how long will it take until the energy of the oscillation has decreased by half? В© 2010 Pearson Education, Inc. Slide 14-42

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