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Reading Quiz

Reading Quiz. The type of function that describes simple harmonic motion is linear exponential quadratic sinusoidal inverse. Answer. The type of function that describes simple harmonic motion is linear exponential quadratic sinusoidal inverse. Reading Quiz.

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Reading Quiz

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  1. Reading Quiz • The type of function that describes simple harmonic motion is • linear • exponential • quadratic • sinusoidal • inverse

  2. Answer • The type of function that describes simple harmonic motion is • linear • exponential • quadratic • sinusoidal • inverse

  3. Reading Quiz • A mass is bobbing up and down on a spring. If you increase the amplitude of the motion, how does this affect the time for one oscillation? • The time increases. • The time decreases. • The time does not change.

  4. Answer • A mass is bobbing up and down on a spring. If you increase the amplitude of the motion, how does this affect the time for one oscillation? • The time increases. • The time decreases. • The time does not change.

  5. Reading Quiz • If you drive an oscillator, it will have the largest amplitude if you drive it at its _______ frequency. • special • positive • natural • damped • pendulum

  6. Answer • If you drive an oscillator, it will have the largest amplitude if you drive it at its _______ frequency. • special • positive • natural • damped • pendulum

  7. Equilibrium and Oscillation

  8. Linear Restoring Forces and Simple Harmonic Motion • Simple harmonic motion (SHM) is the motion of an object subject to a force that is proportional to the object's displacement. One example of SHM is the motion of a mass attached to a spring. In this case, the relationship between the spring force and the displacement is given by Hooke's Law, F = -kx, where k is the spring constant, x is the displacement from the equilibrium length of the spring, and the minus sign indicates that the force opposes the displacement.

  9. Linear Restoring Forces and Simple Harmonic Motion

  10. Frequency and Period The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.

  11. Sinusoidal Relationships

  12. Each dimension of circular motion is a sinusoidal funtion. • Animation of sine and cosine

  13. Mathematical Description of Simple Harmonic Motion

  14. What is the largest value cos(2pft) can ever have? • 0 • 1 • 2pft •  • ½

  15. What is the largest value cos(2pft) can ever have? • 0 • 1 • 2pft •  • ½

  16. What is the largest value sin(2pft) can ever have? • 0 • 1 • 2pft •  • ½

  17. What is the largest value sin(2pft) can ever have? • 0 • 1 • 2pft •  • ½

  18. Maximum displacement, velocity and acceleration • What is the largest displacement possible? • 0 • A • 2pft •  • ½ A

  19. Maximum displacement, velocity and acceleration • What is the largest displacement possible? • 0 • A • 2pft •  • ½ A

  20. Maximum displacement, velocity and acceleration • What is the largest velocity possible? • 0 • 2pf • 2Apf • Apf • 

  21. Maximum displacement, velocity and acceleration • What is the largest velocity possible? • 0 • 2pf • 2Apf • Apf • 

  22. Maximum displacement, velocity and acceleration • What is the largest acceleration possible? • 0 • 2A(pf)2 • 4p2Af2 •  • 2Apf

  23. Maximum displacement, velocity and acceleration • What is the largest acceleration possible? • 0 • 2A(pf)2 • 4p2Af2 •  • 2Apf

  24. 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.

  25. Solving Problems

  26. Equations

  27. 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. “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?

  28. 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. “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? T = 2p √(l/g) = 3.17 s 3600 s/hr (1 push/3.17s) = 1136 pushes/ hr $6/hr (1hr/1136 pushes) = $0.0053 / push Or half a penny per push!

  29. 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. 25 • 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

  30. 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. 25 • 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 f = 1/2p √(g/l) Frequency is proportional to the inverse of the length. So longer has a lower frequency

  31. 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 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?

  32. 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 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? • vmax = 2pfA = 2 p 26Hz 0.05m = 8.17 m/s • amax= A (2pf)2 = (2 p 26Hz) 2 0.05m = 1334 m/s2

  33. 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?

  34. 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? C, G A, E, I B, F D, H

  35. 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? C, G A, E, I B, F D, H

  36. 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 velocity zero? C, G A, E, I B, F D, H

  37. 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 velocity zero? C, G A, E, I B, F D, H

  38. 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 acceleration zero? C, G A, E, I B, F D, H

  39. Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: Without ball At which of the above times is the acceleration zero? C, G A, E, I B, F D, H

  40. 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 kinetic energy a maximum? C, G A, E, I B, F D, H

  41. 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 kinetic energy a maximum? C, G A, E, I B, F D, H

  42. 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 potential energy a maximum? C, G A, E, I B, F D, H

  43. 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 potential energy a maximum? C, G A, E, I B, F D, H

  44. 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 kinetic energy being transformed to potential energy? C, G A, E, I B, F D, H

  45. 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 kinetic energy being transformed to potential energy? C, G A, E, I B, F D, H

  46. 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 potential energy being transformed to kinetic energy? C, G A, E, I B, F D, H

  47. 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 potential energy being transformed to kinetic energy? C, G A, E, I B, F D, H

  48. Example Problem A pendulum is pulled to the side and 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?

  49. Example Problem A pendulum is pulled to the side and released. Its subsequent motion appears as follows: At which of the above times is the displacement zero? C, G At which of the above times is the velocity zero? A, E, I At which of the above times is the acceleration zero? C, G At which of the above times is the kinetic energy a maximum? C, G At which of the above times is the potential energy a maximum? A, E, I At which of the above times is kinetic energy being transformed to potential energy? D, H At which of the above times is potential energy being transformed to kinetic energy? B, F

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