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PHY 113 A General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 15:

PHY 113 A General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 15: Chapter 15 – Simple harmonic motion Object attached to a spring; pendulum Displacement as a function of time Kinetic and potential energy Driven harmonic motion; resonance. From Webassign Assignment #13

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PHY 113 A General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 15:

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  1. PHY 113 A General Physics I • 11 AM – 12:15 PM TR Olin 101 • Plan for Lecture 15: • Chapter 15 – Simple harmonic motion • Object attached to a spring; pendulum • Displacement as a function of time • Kinetic and potential energy • Driven harmonic motion; resonance PHY 113 C Fall 2013 -- Lecture 15

  2. PHY 113 C Fall 2013 -- Lecture 15

  3. From Webassign Assignment #13 A uniform beam of length L = 7.45 m and weight 4.95 102 N is carried by two workers, Sam and Joe, as shown in the figure below. Determine the force that each person exerts on the beam. FSam FJoe d1 X d2 mg PHY 113 C Fall 2013 -- Lecture 15

  4. Illustration of “simple harmonic motion”. PHY 113 C Fall 2013 -- Lecture 15

  5. Simple harmonic motion; mass and spring Behavior of materials: Hooke’s law Fs = -k(x-x0) Young’s modulus x0 =0 PHY 113 C Fall 2013 -- Lecture 15

  6. Microscopic picture of material with springs representing bonds between atoms Measurementof elastic response: k PHY 113 C Fall 2013 -- Lecture 15

  7. Potential energy associated with Hooke’s law: Form of Hooke’s law for ideal system: Us = ½ k x2 Fs = -kx Fs (N) Us (J) x x Us = ½ k (x-1)2 General potential energy curve: U (J) U(x) x PHY 113 C Fall 2013 -- Lecture 15

  8. Motion associated with Hooke’s law forces Newton’s second law: F = -k x = m a  “second-order” linear differential equation PHY 113 C Fall 2013 -- Lecture 15

  9. How to solve a second order linear differential equation: Earlier example – constant force F0 acceleration a0 x(t) = x0 +v0t + ½ a0 t2 2 constants (initial values) PHY 113 C Fall 2013 -- Lecture 15

  10. Hooke’s law motion: Forms of solution: where: 2 constants (initial values) PHY 113 C Fall 2013 -- Lecture 15

  11. Verification: Differential relations: Therefore: PHY 113 C Fall 2013 -- Lecture 15

  12. PHY 113 C Fall 2013 -- Lecture 15

  13. iclicker exercise: • Which of the following other possible guesses would provide a solution to Newton’s law for the mass-spring system: PHY 113 C Fall 2013 -- Lecture 15

  14. iclicker question How do you decide whether to use Either can be a solution, depending on initial position and/or velocity Pure guess PHY 113 C Fall 2013 -- Lecture 15

  15. PHY 113 C Fall 2013 -- Lecture 15

  16. PHY 113 C Fall 2013 -- Lecture 15

  17. PHY 113 C Fall 2013 -- Lecture 15

  18. iclicker exercise: A certain mass m on a spring oscillates with a characteristic frequency of 2 cycles per second. Which of the following changes to the mass would increase the frequency to 4 cycles per second? (a) 2m (b) 4m (c) m/2 (d) m/4 PHY 113 C Fall 2013 -- Lecture 15

  19. Energy associated with simple harmonic motion PHY 113 C Fall 2013 -- Lecture 15

  20. Energy diagram: U(x)=1/2 k x2 E=1/2 k A2 K x A PHY 113 C Fall 2013 -- Lecture 15

  21. Simple harmonic motion for a pendulum: L Q Approximation for small Q: PHY 113 C Fall 2013 -- Lecture 15

  22. Pendulum example: Suppose L=2m, what is the period of the pendulum? L Q PHY 113 C Fall 2013 -- Lecture 15

  23. The notion of resonance: Suppose F=-kx+F0 sin(Wt) According to Newton: PHY 113 C Fall 2013 -- Lecture 15

  24. Physics of a “driven” harmonic oscillator: “driving” frequency “natural” frequency (mag) (W=2rad/s) w PHY 113 C Fall 2013 -- Lecture 15

  25. F(t)=1 N sin (3t) Examples: Suppose a mass m=0.2 kg is attached to a spring with k=1.81N/m and an oscillating driving force as shown above. Find the steady-state displacement x(t). Note: If k=1.90 N/m then: PHY 113 C Fall 2013 -- Lecture 15

  26. Tacoma Narrows bridge resonance Images from Wikipedia Rebuilt bridge in Tacoma WA PHY 113 C Fall 2013 -- Lecture 15

  27. Bridge in July, 1940 PHY 113 C Fall 2013 -- Lecture 15

  28. Collapse of bridge on Nov. 7, 1940 http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_%281940%29 PHY 113 C Fall 2013 -- Lecture 15

  29. Effects of gravity on spring motion x=0 PHY 113 C Fall 2013 -- Lecture 15

  30. Example Suppose a mass of 2 kg is attached to a spring with spring constant k=20 N/m. At t=0 the mass is displaced by 0.4 m and released from rest. What is the subsequent displacement as a function of time? What is the total energy?: PHY 113 C Fall 2013 -- Lecture 15

  31. Example Suppose a mass of 2 kg is attached to a spring with spring constant k=20 N/m. The mass is attached to a device which supplies an oscillatory force of the form What is the subsequent displacement as a function of time? PHY 113 C Fall 2013 -- Lecture 15

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