Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Equilibrium, restoring forces, and oscillation Mathematical description of oscillatory motion Energy in oscillatory motion Damped oscillations Resonance Chapter 14 Oscillations Topics: Slide 14-1

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Resonance Slide Show movie

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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. Slide 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.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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. Slide Rank the frequencies of the five pendulums, from highest to lowest.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: Slide )At which of the above times is the displacement zero? 2)At which of the above times is the velocity zero? 3)At which of the above times is the acceleration zero? 4)At which of the above times is the kinetic energy a maximum? 5)At which of the above times is the potential energy a maximum? 6)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?

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. A pendulum is pulled to the side and released. Its subsequent motion appears as follows: Slide )At which of the above times is the displacement zero? 2)At which of the above times is the velocity zero? 3)At which of the above times is the acceleration zero? 4)At which of the above times is the kinetic energy a maximum? 5)At which of the above times is the potential energy a maximum? 6)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?

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Examples 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? Slide 14-19

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Damping A 500 g mass on a string oscillates as a pendulum. The pendulum’s energy decays to 50% of its initial value in 30 s. What is the value of the damping constant? Slide 14-24

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Demo Wave table Demo - the wave Slide Wave Example

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Slide 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 frequency—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? Example

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Slide The deflection of the end of a diving board produces a linear restoring force, as we saw in Chapter 8. A diving board dips by 15 cm when a 65 kg person stands on its end. Now, this person jumps and lands on the end of the board, depressing the end by another 10 cm, after which they move up and down with the oscillations of the end of the board. A.Treating the person on the end of the diving board as a mass on a spring, what is the spring constant? B.For a 65 kg diver, what will be the oscillation period? C.For the noted oscillation, what will be the maximum speed? D.What amplitude would lead to an acceleration greater than that of gravity—meaning the person would leave the board at some point during the cycle? Example

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. In Chapter 10, we saw that the Achilles tendon will stretch and then rebound, storing and returning energy during a step. We can model this motion as that of a mass on a spring. It’s far from a perfect model, but it does give some insight. Suppose a 60 kg person stands on a low wall with her full weight on the balls of one foot and the heel free to move. The stretch of the Achilles tendon will cause her center of mass to lower by about 2.5 mm. A.What is the value of k for this system? B.Given the mass and the spring constant, what would you expect for the period of this system were it to undergo an oscillation? C.When the balls of the feet take the weight of a stride, the tendon spring begins to stretch as the body moves down; kinetic energy is being converted into elastic potential energy. Ideally, when the foot is leaving the ground, the cycle of the motion will have advanced so that potential energy is being converted to kinetic energy. What fraction of an oscillation period should the time between landing and lift off correspond to? Given the period you calculated above, what is this time? D.Sprinters running a short race keep their foot in contact with the ground for about 0.10 s, some of which corresponds to the heel strike and subsequent rolling forward of the foot. Given this, does the final number you have calculated above make sense? Slide Example

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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? Slide Example

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram: Slide Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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: Slide Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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. Slide 14-28

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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 Slide 14-29

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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? B.3.3 s Slide Answer

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Additional Example Walter has a summer job babysitting a 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. “Push me,” 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? Slide 14-31

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 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 block’s period of oscillation? What is the block’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? Slide Additional Examples