1. Oscillations and Simple Harmonic Motion : Mechanics C

2. Oscillatory Motion Oscillatory Motion is repetitive back and forth motion about an equilibrium position Oscillatory Motion is periodic. Swinging motion and vibrations are forms of Oscillatory Motion. Objects that undergo Oscillatory Motion are called Oscillators.

3. Simple Harmonic Motion The time to complete one full cycle of oscillation is a Period. The amount of oscillations per second is called frequency and is measured in Hertz.

4. What is the oscillation period for the broadcast of a 100MHz FM radio station? Heinrich Hertz produced the first artificial radio waves back in 1887!

5. What is the oscillation period for the broadcast of a 100MHz FM radio station? Heinrich Hertz produced the first artificial radio waves back in 1887!

6. Simple Harmonic Motion The most basic of all types of oscillation is depicted on the bottom sinusoidal graph. Motion that follows this pattern is called simple harmonic motion or SHM.

7. Simple Harmonic Motion An objects maximum displacement from its equilibrium position is called the Amplitude (A) of the motion.

8. What shape will a velocity-time graph have for SHM? Everywhere that the slope (first derivative) of the position graph is zero, the velocity graph crosses through zero. Either sine or cosine. The choice is up to where you call time = 0

9. We need a position function to describe the motion above.

10. Mathematical Models of SHM x(t) to symbolize position as a function of time A=x max =x min When t=T, cos(2 π )=cos(0) x(t)=A

11. Mathematical Models of SHM In this context we will call omega Angular Frequency What is the physical meaning of the product (Aω)? The maximum speed of an oscillation!

12. Recall: Hooke’s Law Here is what we want to do: DERIVE AN EXPRESSION THAT DEFINES THE DISPLACEMENT FROM EQUILIBRIUM OF THE SPRING IN TERMS OF TIME. WHAT DOES THIS MEAN? THE SECOND DERIVATIVE OF A FUNCTION THAT IS ADDED TO A CONSTANT TIMES ITSELF IS EQUAL TO ZERO. What kind of function will ALWAYS do this?

14. Example: An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds. What is the period of oscillation?

15. Example: An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds. What is the object’s maximum speed?

16. Example: An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds. What are the position and velocity at t=0.8s?

17. Example: A mass oscillating in SHM starts at x=A and has period T. At what time, as a fraction of T, does the object first pass through 0.5A?

18. Model of SHM When collecting lab data of SHM your mathematical model was: What if your clock didn’t start at x=A or x=-A? This value represents our initial conditions. We call it the phase angle: This value represents our initial conditions. We call it the phase angle:

19. SHM and Circular Motion Uniform circular motion projected onto one dimension is simple harmonic motion.

20. SHM and Circular Motion Start with the x-component of position of the particle in UCM End with the same result as the spring in SHM! Notice it started at angle zero

21. Initial conditions: We will not always start our clocks at one amplitude.

22. The Phase Constant: Phi is called the phase of the oscillation Phi naught is called the phase constant or phase shift. This value specifies the initial conditions. Different values of the phase constant correspond to different starting points on the circle and thus to different initial conditions

23. Phase Shifts:

24. An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2s? Initial conditions: From the period we get:

25. An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2s?

26. We have modeled SHM mathematically. Now comes the physics. Total mechanical energy is conserved for our SHM example of a spring with constant k, mass m, and on a frictionless surface. The particle has all potential energy at x=A and x=–A, and the particle has purely kinetic energy at x=0.

27. At turning points: At x=0: From conservation: Maximum speed as related to amplitude:

28. From energy considerations: From kinematics: Combine these:

29. a 500g block on a spring is pulled a distance of 20cm and released. The subsequent oscillations are measured to have a period of 0.8s. at what position or positions is the block’s speed 1.0m/s? The motion is SHM and energy is conserved.

30. Dynamics of SHM Acceleration is at a maximum when the particle is at maximum and minimum displacement from x=0.

31. Dynamics of SHM Acceleration is proportional to the negative of the displacement.

32. Dynamics of SHM As we found with energy considerations: According to Newton’s 2 nd Law: Acceleration is not constant: This is the equation of motion for a mass on a spring. It is of a general form called a second order differential equation.

33. 2 nd -Order Differential Equations: Unlike algebraic equations, their solutions are not numbers, but functions. In SHM we are only interested in one form so we can use our solution for many objects undergoing SHM. Solutions to these diff. eqns. are unique (there is only one). One common method of solving is guessing the solution that the equation should have… From evidence, we expect the solution:

34. 2 nd -Order Differential Equations: Let’s put this possible solution into our equation and see if we guessed right! IT WORKS. Sinusoidal oscillation of SHM is a result of Newton’s laws!

35. Vertical springs oscillate differently than horizontal springs because there are 2 forces acting. The equilibrium position gets shifted downward

36. What about vertical oscillations of a spring-mass system?? Hanging at rest: this is the equilibrium position of the system.

37. Now we let the system oscillate. At maximum: But: So: Everything that we have learned about horizontal oscillations is equally valid for vertical oscillations!

38. You need to show how to derive the Period of a Pendulum equation T = 2∏√l/g

39. The Pendulum Equation of motion for a pendulum

40. Small Angle Approximation: When θ is about 0.1rad or less, h and s are about the same.

41. The Pendulum Equation of motion for a pendulum

42. A Pendulum Clock What length pendulum will have a period of exactly 1s?

43. Conditions for SHM Notice that all objects that we look at are described the same mathematically. Any system with a linear restoring force will undergo simple harmonic motion around the equilibrium position.

44. A Physical Pendulum when there is mass in the entire pendulum, not just the bob. Small Angle Approx.

45. Damped Oscillations All real oscillators are damped oscillators. these are any that slow down and eventually stop. a model of drag force for slow objects: b is the damping constant (sort of like a coefficient of friction).

46. Damped Oscillations Another 2 nd -order diff eq. Solution to 2 nd - order diff eq:

47. Damped Oscillations A slowly changing line that provides a border to a rapid oscillation is called the envelope of the oscillations.

49. Driven Oscillations Not all oscillating objects are disturbed from rest then allowed to move undisturbed. Some objects may be subjected to a periodic external force.

50. Driven Oscillations All objects have a natural frequency at which they tend to vibrate when disturbed. Objects may be exposed to a periodic force with a particular driving frequency. If the driven frequency matches the natural frequency of an object, RESONANCE occurs

51. Spring Best example of simple harmonic oscillator. T = 2  m/k m

52. Simple Pendulum Acts as simple harmonic oscillator only when angle of swing is small. T = 2  L/g 

53. Conical Pendulum Not really a simple harmonic oscillator, but equation is similar to simple pendulum. T = 2  L(cos  )/g 

54. The small angle approximation for a simple pendulum mg  mgcos  mgsin  A simple pendulum is one where a mass is located at the end of string. The string’s length represents the radius of a circle and has negligible mass. Once again, using our sine function model we can derive using circular motion equations the formula for the period of a pendulum. If the angle is small, the “radian” value for theta and the sine of the theta in degrees will be equal.

55. Torsional Pendulum Twists back and forth through equilibrium position. T = 2  I/ 

56. Physical Pendulum Anything that doesn’t fall into any of the other categories of pendulums. T = 2  I/   = Mgd

57. K 1 + U 1 = K 2 + U 2 ◦K = 1 / 2 mv 2 ◦U = mgh 1 / 2 mv 1 2 +mgh 1 = 1 / 2 mv 2 2 + mgh 2 v 1 2 + 2gh 1 = v 2 2 + 2gh 2 Energy Conservation in Pendulums h

58. The Physical Pendulum A physical pendulum is an oscillating body that rotates according to the location of its center of mass rather than a simple pendulum where all the mass is located at the end of a light string. It is important to understand that “d” is the lever arm distance or the distance from the COM position to the point of rotation. It is also the same “d” in the Parallel Axes theorem.

59. Example A spring is hanging from the ceiling. You know that if you elongate the spring by 3.0 meters, it will take 330 N of force to hold it at that position: The spring is then hung and a 5.0-kg mass is attached. The system is allowed to reach equilibrium; then displaced an additional 1.5 meters and released. Calculate the: Spring Constant Angular frequency 110 N/m 4.7 rad/s

60. Example A spring is hanging from the ceiling. You know that if you elongate the spring by 3.0 meters, it will take 330 N of force to hold it at that position: The spring is then hung and a 5.0-kg mass is attached. The system is allowed to reach equilibrium; then displaced an additional 1.5 meters and released. Calculate the: Amplitude Frequency and Period Stated in the question as 1.5 m 0.75 Hz 1.34 s

61. Example A spring is hanging from the ceiling. You know that if you elongate the spring by 3.0 meters, it will take 330 N of force to hold it at that position: The spring is then hung and a 5.0-kg mass is attached. The system is allowed to reach equilibrium; then displaced an additional 1.5 meters and released. Calculate the: Total Energy Maximum velocity 123.75 J 7.05 m/s

62. Position of mass at maximum velocity Maximum acceleration of the mass Position of mass at maximum acceleration At the equilibrium position 33.135 m/s/s At maximum amplitude, 1.5 m A spring is hanging from the ceiling. You know that if you elongate the spring by 3.0 meters, it will take 330 N of force to hold it at that position: The spring is then hung and a 5.0-kg mass is attached. The system is allowed to reach equilibrium; then displaced an additional 1.5 meters and released. Calculate the:

63. a)Find the length of a simple seconds pendulum. b)What assumption have you made in this calculation? A “seconds pendulum” beats seconds; that is, it takes 1 s for half a cycle.

64. Problem #2 A thin uniform rod of mass 0.112 kg and length 0.096 m is suspended by a wire through its center and perpendicular to its length. The wire is twisted and the rod set to oscillating. The period is found to be 2.14 s. When a flat body in the shape of an equilateral triangle is suspended similarly through its center of mass, the period is 5.83. Find the rotational inertia of the triangle.

65. Problem #3 A uniform disk is pivoted at its rim. Find the period for small oscillations and the length of the equivalent simple pendulum.

67. Student slides

68. An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds. What is the period of oscillation?

69. What is the object’s maximum speed?

70. What are the position and velocity at t=0.8s?

71. A mass oscillating in SHM starts at x=A and has period T. At what time, as a fraction of T, does the object first pass through 0.5A?

72. An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2s?

73. a 500g block on a spring is pulled a distance of 20cm and released. The subsequent oscillations are measured to have a period of 0.8s. at what position or positions is the block’s speed 1.0m/s? The motion is SHM and energy is conserved.

74. Vertical springs oscillate differently than horizontal springs because there are 2 forces acting. The equilibrium position gets shifted downward

76. The Pendulum Equation of motion for a pendulum