Physics 151 Week 13 Day 1

1. Physics 151 Week 13 Day 1 Topics: Harmonic Oscillations (Ch. 14) Oscillations Period & Frequency Angular Frequency Forces Acceleration Energy Damping Resonance Mathematical Models Mass and Spring Pendulum

2. Announcements Slide 14-23

3. Recent Physics Models Energy Model Momentum Model Oscillator Model Slide 14-23

4. Solving Problems Slide 14-14

5. Linear Restoring Forces and Simple Harmonic Motion Slide 14-13

6. Frequency and Period The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation. Slide 14-13

7. 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. Slide 14-15

8. 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. • B  D  C  A • B  A  D  C • C  A  D  B • A  C  B  D Slide 14-15

9. 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. • B  D  C  A • B  A  D  C • C  A  D  B • A  C  B  D Slide 14-16

10. 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  E  B  D  C • D  A  C  B  E • A  B  C  D  E • B  E  C  A  D Slide 14-17

11. 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  E  B  D  C • D  A  C  B  E • A  B  C  D  E • B  E  C  A  D Slide 14-18

12. Mathematical Description of Simple Harmonic Motion Slide 14-22

13. Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: 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? At which of the above times is potential energy being transformed to kinetic energy? Slide 14-23

14. Analyzing Simple Harmonic Motion Looking at motion, forces, and Energy over a harmonic oscillation cycle.

15. 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. Slide 14-12

16. Damping Slide 14-30

17. Damping Slide 14-30

18. 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

19. Example • 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, • What is the maximum velocity of the tip of a hawkmoth wing? • What is the maximum acceleration of the tip of a hawkmoth wing? Slide 14-20

20. Example • 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. • Treating the person on the end of the diving board as a mass on a spring, what is the spring constant? • For a 65 kg diver, what will be the oscillation period? • For the noted oscillation, what will be the maximum speed? • 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? Slide 14-21

21. Example • 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. • What is the value of k for this system? • Given the mass and the spring constant, what would you expect for the period of this system were it to undergo an oscillation? • 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? • 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 14-22

22. Example 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 14-23