2. Chapter 23 Oscillatory Motion

3. 23-1 The Kinematics of Simple Harmonic Motion We know that if we stretch a spring with a mass on the end and let it go, the mass will oscillate back and forth (if there is no friction). k m k m k m This oscillation is called Simple Harmonic Motion Any motion that repeats itself at regular intervals is called periodic motion Examples: circular motion, oscillatory motion

4. The position of the object is angular frequency ω: determined by the inertia of the moving objects and the restoring force acting on it. amplitude A: The maximum distance of displacement to the equilibrium point phase  t+  phase angle (constant)  The value of A and  depend on the displacement and velocity of the particle at time t = 0 (the initial conditions) SI: rad/s

5. Period T: the time for one complete oscillation (or cycle); Frequency f: number of oscillations that are completed each second.

6. (a) only in that its amplitude is greater (b) only in that its period is T´ = T/2 (c) only in that  = -  /4 rad rather than zero The red curve differs from the blue curve a phase difference

7.    T = 2  /  A A       

8. o A1A1 -A 1 A2A2 - A 2 x1x1 x2x2 T t xx o A1A1 -A 1 A2A2 - A 2 x1x1 x2x2 T t They are in phraseThey have a phrase difference of 

9. We can take derivatives to find velocity and acceleration: Relations Among Position, Velocity, and Acceleration in Simple Harmonic Motion v(t) is phrase –shifted to the left from x(t) by  /2 x(t) is phrase –shifted to the right from v(t) by -  /2 v(t) leads x(t) by  /2 x(t) lags behind v(t) by  /2

10. In SHM, the acceleration is proportional to the displacement but opposite in sign, and the two quantities are related by the square of the angular frequency. a(t) is phrase –shifted to the left from x(t) by 

11. 23-2 A Connection to Circular Motion Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs. A reference particle P´ moving in a reference circle of radius A with steady angular velocity . Its projection P on the x axis executes simple harmonic motion. demo

12. ACT A mass oscillates up & down on a spring. Its position as a function of time is shown below. Write down the displacement as the function of time t (s) y(t)(cm) 4 2 1 y

13. 23-3 Springs and Simple Harmonic Motion k x m F F = -kxa a differential equation for x(t) Hooke’s law Simple harmonic motion is the motion executed by a particle of mass m subject to a force that is proportional to the displacement of the particle but opposite in sign ( a restoring force). The block–spring system forms a linear simple harmonic oscillator Combining with Newton’s second law

14. Solution The period of the motion is independent of the amplitude t=0 ； x=x 0 ， v=v 0 The initial conditions

15. X=0 X=A X=-A X=A; v=0; a=-a max X=0; v=-v max ; a=0X=-A; v=0; a=a max X=0; v=v max ; a=0 X=A; v=0; a=-a max

16. ok Another solution is is equivalent to where

17. Example A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m. The block is pulled a distance x = 11 cm from its equilibrium position at x = 0 on a frictionless surface and released from rest at t = 0. (a) What are the angular frequency, the frequency, and the period of the resulting motion? (b) What is the amplitude of the oscillation? (c) What is the maximum speed v m of the oscillating block, and where is the block when it occurs? (d) What is the magnitude a m of the maximum acceleration of the block? (e) What is the phase constant  for the motion? (f) What is the displacement function x(t) for the spring–block system?

18. Solution At equilibrium point

19. Example At t = 0, the displacement x(0) of the block in a linear oscillator is -8.50 cm. The block's velocity v(0) then is -0.920 m/s, and its acceleration a(0) is +47.0 m/s 2. (a) What is the angular frequency  of this system? (b) What are the phase constant  and amplitude A? Solution Correct phase constant is155 0 155 -25

20. Additional Constant Forces Solution Simple harmonic motion with the same frequency, but equilibrium point is shifted from x=0 to x=x 1

21. Vertical Springs Choose the origin at equilibrium position Solution Simple harmonic motion with equilibrium point at y=0 FsFs mg

22. ACTA mass hanging from a vertical spring is lifted a distance d above equilibrium and released at t = 0. Which of the following describes its velocity and acceleration as a function of time? k m y 0 d (a) v(t) = -v max sin(  t) a(t) = -a max cos(  t) (b) v(t) = v max sin(  t) a(t) = a max cos(  t) (c) v(t) = v max cos(  t) a(t) = -a max cos(  t) (both v max and a max are positive numbers) t = 0

23. 23-4 Energy and Simple Harmonic Motion This is not surprising since there are only conservative forces present, hence the total energy is conserved.

24. (a)Potential energy U(t), kinetic energy K(t), and mechanical energy E as functions of time t for a linear harmonic oscillator. (b)Potential energy U(x), kinetic energy K(x), and mechanical energy E as functions of position x for a linear harmonic oscillator with amplitude x m. For x = 0 the energy is all kinetic, and for x = ±x m it is all potential. The mechanical energy is conserved

25. Note  The mechanical energy is conserved for a linear harmonic oscillator  The potential energy and the kinetic energy peak twice during every period  The dependence of energy on the square of the amplitude is typical of Simple Harmonic Motion

26. ACT In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation is the same as in Case 1. In which case is the maximum potential energy of the mass and spring the biggest? A. Case 1 B. Case 2 C. Same Look at time of maximum displacement x = A Energy = ½ k A 2 + 0 Same for both!

27. Besides springs, there are many other systems that exhibit simple harmonic motion. Here are some examples: It’s Not Just About Springs

28. Almost all systems that are in stable equilibrium exhibit simple harmonic motion when they depart slightly from their equilibrium position For example, the potential between H atoms in an H 2 molecule looks something like this: U x

29. Restoring force If we do a Taylor expansion of this function about the minimum, we find that for small displacements, the potential is quadratic: since x 0 is minimum of potential then U x x0x0 U x

30. Identifying SHM c, c’ positive constant

31. o x Example A mass m on a frictionless table is attached to two springs with spring constants k 1 and k 2 respectively. The mass can move along the straight line. Find the period. Solution

32. Transport Tunnel A straight tunnel is dug through the Earth. A student jumps into the hole at noon. What time does he get back? He gets back 84 minutes later, at 1:24 p.m. x g = 9.81 m/s 2 R E = 6.38 x 10 6 m

33. Strange but true:Strange but true: The period of oscillation does not depend on the length of the tunnel. Any straight tunnel gives the same answer, as long as it is frictionless and the density of the Earth is constant. g =  2 R 9.81 =  2 6.38(10) 6 m   =.00124 s -1 so T = = 5067 s 84 min  Another strange but true fact:Another strange but true fact: An object orbiting the earth near the surface will have a period of the same length as that of the transport tunnel.

34. 23-5 The Simple Pendulum a simple pendulum consists of a pointlike mass m (called the bob of the pendulum) suspended from one end of an unstretchable, massless string of length L that is fixed at the other end If  is  small

35. Solutionwith The motion of a simple pendulum swinging through only small angles is approximately SHM. The period of small-amplitude pendulum is independent of the amplitude --- the pendulum clock The horizontal displacement

36. The energy of a simple pendulum: For small  The total energy is conserved

37. l You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T 1. l Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T 2. Which of the following is true: (a) T 1 = T 2 (b) T 1 > T 2 (c) T 1 < T 2 ACT You make a pendulum shorter, it oscillates faster (smaller period)

38. ACT A pendulum is hanging vertically from the ceiling of an elevator. Initially the elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be 1. greater than T 2. equal to T 3. less than T “Effective g” is larger when accelerating upward (you feel heavier)

39. ACT Imagine you have been kidnapped by space invaders and are being held prisoner in a room with no windows. All you have is a cheap digital wristwatch and a pair of shoes (including shoelaces of known length). Explain how you might figure out whether this room is on the earth or on the moon

40. Hang a shoelace and time it going back and forth, the moons gravity would be seen in the time it takes to go back and forth. The other day i was actually taken by aliens. They let me go b/c I told them I was taking Physics, and they felt really bad for me. If you're on the moon, you should have an extra 'bounce' in your step. You don't need the shoes or wristwatch. just wait until one of the space invaders comes back in and ask him- I'm sure he's really nice :-) You could make a pendulum with the shoes (and shoelaces of known length), then make the pendulum oscillate with a period of 1 second and solve for gravity. If it is 1/6th that of Earth, then you are on the moon. You could even jump up and down, it would be a lot more entertaining than solving an equation. Sorry Dr. Stelzer. pray to the shoe god using ur watch as a spiritual medium Some answers from students

41. 23-6 More About Pendulums The Physical Pendulum Any object, if suspended and then displaced so the gravitational force does no run through the center of mass, can oscillate due to the torque. If  is  small

42. withSolution The period of a physical pendulum is independent of it’s total mass—only how the mass is distributed matters For a simple pendulum

43. P Point P is called the center of oscillation of the physical pendulum for the given suspension point O O Point O is also the center of oscillation for the suspension point P used to measure the rotational inertia I C,P

44. ACT A pendulum is made by hanging a thin hoola-hoop of diameter D on a small nail. What is the angular frequency of oscillation of the hoop for small displacements? (I CM = mR 2 for a hoop) (a) (b) (c) D pivot (nail)

45. Example In Figure below, a meter stick swings about a pivot point at one end, at distance h from its center of mass. (a) What is its period of oscillation T? (b) What is the distance L 0 between the pivot point O of the stick and the center of oscillation of the stick? Solution

46. Example In Figure below, a penguin (obviously skilled in aquatic sports) dives from a uniform board that is hinged at the left and attached to a spring at the right. The board has length L = 2.0 m and mass m = 12 kg; the spring constant k is 1300 N/m. When the penguin dives, it leaves the board and spring oscillating with a small amplitude. Assume that the board is stiff enough not to bend, and find the period T of the oscillations.

47. SolutionChoose the equilibrium position as the origin mg F T is independent of the board’s length y O

48. 23-7 Damped Harmonic Motion Then the equation of motion is: Look at drag force that is proportional to velocity; b is the damping coefficient: Solution A pendulum does not go on swinging forever. Energy is gradually lost (because of air resistance) and the oscillations die away. This effect is called damping.

49. Damping factor bDamping coefficient If b=0undamped

50. Life time The larger the value of  the slower the exponential

51. Natural frequency As b increases,  ’ decreases When Some systems have so much damping that no real oscillations occur. The minimum damping needed for this is called critical damping

52. Over (heavy) damping t x o critical damping heavy damping light damping (light) damping critical damping The time of the critical damping takes for the displacement to settle to zero is a minimum

53. Example For the damped oscillator of Figure m = 250 g, k = 85 N/m, and b = 70 g/s. (a) What is the period of the motion? (b) How long does it take for the amplitude of the damped oscillations to drop to half its initial value? (c) How long does it take for the mechanical energy to drop to one-half its initial value?

54. Solution

55. 23-8 Driven Harmonic Motion Equation of motion becomes: solution In damped harmonic motion, a mechanism such as friction dissipates or reduces the energy of an oscillating system, with the result that the amplitude of the motion decreases in time. Now, applying a sinusoidal driving force

56. After long times

57. The condition for the maximum of A Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. In the absence of damping, if the frequency of the force matches the natural frequency of the system, then the amplitude of the oscillation reaches a mximum. This effect is called resonance If b=0

58. The total width at half maximum

59. The role played by the frequency of a driving force is a critical one. The matching of this frequency with a natural frequency of vibration allows even a relatively weak force to produce a large amplitude vibration Examples Breaking glass The collapse of the Tacoma Narrows Bridge Turbulent winds set up standing waves in the Tacoma Narrows suspension bridge leading to its collapse on November 7, 1940, just four months after it had been opened for traffic

60. 23-8 Superposition for Simple Harmonic Oscillations We will discuss the effects that result when two harmonic oscillations are superposed (added) in one and two dimensions. Summation of 1-D Oscillations 1. The summation of same frequency oscillations Suppose two oscillations with same frequency along the x axis

61.  x The resultant motion is a harmonic oscillation with the same frequency

62.  x

63. constructive destructive

64. Example The displacements of two individual oscillations are shown as below.Find the resultant displacement as a function of time Solution x A1A1 A2A2 From phasor diagram

65. Suppose two oscillations with same frequency along the x axis 2. The summation of different (but not very) frequency oscillations x O Rotates with angular velocity with respect to

66. A max A min Beats occur when A vary periodical x O

67. Calculation of the Beat frequency The result amplitude oscillates as a function of time in a complicated motion

68. Simple harmonic oscillation with nearly the origin frequency A slowly varying term A rapidly varying term The amplitude (envelop)

69. o x1x1 t o x2x2 t o x=x 1 +x 2 t t1t1 t2t2 t3t3 (a) (b) (c)

70. Summation of Perpendicular Oscillations in a plane Two oscillations one along the x axis and the other along the y axis when added result in a two dimensional motion. The path traced is known as Lissajous figure.

71. x o 1. With Same frequencies y

72. y x o Linear vibration along the diagonal

73. x y o

74. y x o Elliptical (circular) motion whose major axis along y-axis y leads x, clockwise x leads y, counterclockwise

75. (4) Other kinds of figures

76. ωy:ωxωy:ωx 2. With different frequencies