6 7 Mechanical oscillations and waves 2022.pptx

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1. 26.10.2022 1 Mechanical oscillations 1. According to oscillating object : mechanical and electromagnetic oscillations ; 2. According to the type of motion : translational and rotational oscillations ; 3. According to the external action : free and forced oscillations ; 4. Non-damped and damped oscillations ( according to time ). Period of the oscillations ( T , s); frequency ( ; Hz = s -1 ).

2. Simple Harmonic Motion Repetitive motion, usually called periodic motion , is important in science, engineering , and daily life. Common examples of objects in periodic motion are windshield wipers on a car and the pendulum on a grandfather clock. However , periodic motion is also involved in the alternating current that powers the electronic grid of modern cities, atomic vibrations in molecules, and your own heartbeat and circulatory system. Simple harmonic motion is a particular type of repetitive motion, which is displayed by a pendulum or by a massive object on a spring . October 26, 2022 2

3. October 26, 2022 Chapter 14 3 Simple Harmonic Motion W e have talked about the energy stored in springs. L et’s talk about the motion of a mass on a spring. Stretch a spring and let it go. This up and down motion is called simple harmonic motion . Let’s look at a video study of simple harmonic motion: Put video frames put next to each other.

4. 26.10.2022 4 Simple harmonic motion (SHM) -A A X t T A -A A - p hase of oscillation ; - a ngular frequency .

5. 26.10.2022 5 Displacement , velocity and acceleration x t T /2 -A A 3T /2 t = 0: x = A  v t v m -v m a t -a m a m

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7. Quick Check Identical blocks oscillate on the end of a vertical spring, one on Earth and one on the Moon. Where is the period of the oscillations greater? on Earth on the Moon same on both Earth and Moon cannot be determined from the information given October 26, 2022 7

8. Quick Check A block of mass m oscillates on a horizontal spring with period T = 2.0 s . If a second identical block is glued to the top of the first block, the new period will be 1.0 s 1.4 s 2.0 s 2.8 s 4.0 s

9. A block of mass m oscillates on a horizontal spring with period T = 2.0 s . If a second identical block is glued to the top of the first block, the new period will be 1.0 s 1.4 s 2.0 s 2.8 s 4.0 s Quick Check

10. Test question 4.1. A mass oscillates on a horizontal spring with period T = 2.0 s . If the amplitude of the oscillation is doubled, the new period will be 1.0 s 1.4 s 2.0 s 2.8 s 4.0 s

11. Test question 4.1. A mass oscillates on a horizontal spring with period T = 2.0 s . If the amplitude of the oscillation is doubled, the new period will be 1.0 s 1.4 s 2.0 s 2.8 s 4.0 s

12. 26.10.2022 12 Mathematical pendulum

13. 26.10.2022 13 Physical pendulum  C

14. Quick Check The solid disk The circular hoop Both have the same period. There’s not enough information to tell. A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation?

15. The solid disk The circular hoop Both have the same period. There’s not enough information to tell. A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation? Quick Check

16. 26.10.2022 16 Energy of oscillation

17. Energy of Mass on a Spring October 26, 2022 17

18. 26.10.2022 18 W=W K +W P W P (t) W K (t) kA 2 /2 kA 2 /4 T/2 T W W = W K + W P = const

19. Quick Check A block oscillates on a very long horizontal spring. The graph shows the block’s kinetic energy as a function of position. What is the spring constant? 1 N/m 2 N/m 4 N/m 8 N/m I have no idea.

20. A block oscillates on a very long horizontal spring. The graph shows the block’s kinetic energy as a function of position. What is the spring constant? 1 N/m 2 N/m 4 N/m 8 N/m I have no idea. Quick Check

21. A 2.00-kg mass attached to a spring is displaced 8.00 cm from the equilibrium position. It is released and then oscillates with a frequency of 4.00 Hz. What is the energy of the motion when the mass passes through the equilibrium position? What is the speed of the mass it is 2.00 cm from the equilibrium position? 26.10.2022 21

22. 26.10.2022 22 Combination of oscillations x  

23. 26.10.2022 23 Combination of two SHM with equal frequencies x  x 1 x 2

24. 26.10.2022 24 I. A 1 = A 2 . - not a SHM anymore !

25. 26.10.2022 25 A: 2A 1 -2A 1 T p /2 3T p /2 t

26. 26.10.2022 26 II. A 1 +A 2 A 1 -A 2 -(A 1 - A 2 ) -(A 1 +A 2 ) - amplitude modulation

27. 26.10.2022 27 Combination of perpendicular oscillations A 2 A 1 y x I. vai

28. 26.10.2022 28 II. y x          - Lissajous figures .

29. https://mathcurve.com/courbes2d.gb/lissajous/lissajous.shtml 26.10.2022 29

30. October 26, 2022 30 Quick Check What is the expression relating the net force F net and the acceleration a for a mass m being acted upon by a linear restoring force F s plus a damping force F γ that depends on velocity? F net = F s + F γ = ma F net = F s – F γ = ma F net = F s = ma F net = F γ = ma F net = 0 Newton’s Second Law! F = ma F net = ma = F s + F γ

31. 26.10.2022 31 Damped oscillations and then - e quation of damped oscillations x t d amping coefficient

32. 26.10.2022 32 Energy of damped oscillations :

33. A vertical spring with a spring constant of 2.00 N/m has a 0.3-kg mass attached to it, and the mass moves in a medium with a damping constant of 0.025 kg/s. The mass is released from rest at a position 5.00 cm from the equilibrium position.How long does it take for the amplitude to decrease twice ? How long it take for the total energy to decrease twice ? 26.10.2022 33

34. 26.10.2022 34 Forced oscillations equation for forced oscillations .

35. 26.10.2022 35   A  R A R  1 =0  2 >  1  3 >  2 A

36. Tacoma bridge collapse: https :// www.youtube.com/watch?v=XggxeuFDaDU 26.10.2022 36

37. Waves If you throw a stone in a still pond, it creates circular ripples that travel along the surface of the water, radially outward. If an object is floating on the surface of the pond, it moves up and down as the ripples move outward underneath it, but it does not move outward with the wave. If you tie a rope to a wall, hold one end, and rapidly move your arm up and down, you will create a wave crest that travels down the rope. October 26, 2022 37

38. Waves A wave is an excitation that propagates through space or some medium as a function of time but does not generally transport matter with it. Wave motion is different from anything we have studied. Electromagnetic waves and gravitational waves do not need a medium in which to propagate. Other waves need a medium in which to propagate. Sound travels through air but not through a vacuum. Many example of waves surround us in everyday life. In this chapter, we will study the motion of waves through media. October 26, 2022 38

39. 26.10.2022 39 Mechanical waves T ransverse wave - motion of the matter particles conveying the wave are perpendicular to the direction of propagation of the waves itself : V L ongitudinal wave - motion of the particles is back and forth along the direction of propagation : V

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41. October 26, 2022 41 Quick Check Fans at a football game are so excited that their team is winning that they start “the wave” in celebration. Which of the following 4 statements is true: I. This wave is a traveling wave. II. This wave is a transverse wave. III. This wave is a longitudinal wave. IV. This wave is a combination of a longitudinal and transverse wave . I and II II only III only I and IV I and III

42. 26.10.2022 42 - the wavelength of the wavetrain ; distance between two adjacent points in the wave having the same phase .

43. 1 s 2 s 4 s Not enough information to tell. The period of this wave is Test question 4.3.

44. 1 s 2 s 4 s Not enough information to tell. The period of this wave is A sinusoidal wave moves forward one wavelength (2 m) in one period. Test question 4.3.

45. 26.10.2022 45 Wave equation For spherical wave : At time instant t at r = : at r = r : A = A(r): A 2  Wave kinematical equation :

46. 26.10.2022 46 For plane wave : x

47. A sinusoidal wave travels along a string. If the time for a particular point to move from maximum displacement to zero displacement is 0.17 s, what are (a) the period, and (b) frequency? (c) If the wavelength is 1.4 m, what is the speed of the wave? 26.10.2022 47

48. 26.10.2022 48 Wave energy - energy density .

49. October 26, 2022 49 Wave Power and Intensity Power = Energy/ time: The power radiated by a wave is constant in time (all terms on right hand side are constants). The definition of intensity is the power radiated per cross sectional area is:

50. Intensity and Decibels Human hearing spans an extremely wide range of intensities, from the threshold of hearing at ≈ 1 × 10 – 12 W/m 2 (at midrange frequencies) to the threshold of pain   at ≈ 10 W/m 2 . If we want to make a scale of loudness, it’s convenient and logical to place the zero of our scale at the threshold of hearing. To do so, we define the sound intensity level , expressed in decibels (dB), as where I = 1 × 10 – 12 W/m 2 .

51. Intensity and Decibels

52. 26.10.2022 52 Standing waves in the string n = 1, 2, 3 ... T – tension ; n = 1:  1 – basic frequency n = 2, 3 …. – overtones . Wave speed in a stretched string :