PHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

Transcription

1 PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 15 Lecture RANDALL D. KNIGHT

2 Chapter 15 Oscillations IN THIS CHAPTER, you will learn about systems that oscillate in simple harmonic motion. Slide 15-2

3 Chapter 15 Preview Slide 15-3

4 Chapter 15 Preview Slide 15-4

5 Chapter 15 Preview Slide 15-5

6 Chapter 15 Preview Slide 15-6

7 Chapter 15 Preview Slide 15-7

8 Chapter 15 Preview Slide 15-8

9 Oscillatory Motion Objects that undergo a repetitive motion back and forth around an equilibrium position are called oscillators. The time to complete one full cycle, or one oscillation, is called the period T. The number of cycles per second is called the frequency f, measured in Hz: 1 Hz = 1 cycle per second = 1 s 1 Slide 15-9

10 Simple Harmonic Motion A particular kind of oscillatory motion is simple harmonic motion. In the figure an air-track glider is attached to a spring. The glider s position measured 20 times every second. The object s maximum displacement from equilibrium is called the amplitude A of the motion. Slide 15-10

11 Simple Harmonic Motion The top image shows position versus time for an object undergoing simple harmonic motion. The bottom image shows the velocity versus time graph for the same object. The velocity is zero at the times when x = ± A; these are the turning points of the motion. The maximum speed v max is reached at the times when x = 0. Slide 15-11

12 Simple Harmonic Motion If the object is released from rest at time t = 0, we can model the motion with the cosine function: Cosine is a sinusoidal function. ω is called the angular frequency, defined as ω = 2π/T The units of ω are rad/s: ω = 2πf Slide 15-12

13 Simple Harmonic Motion The position of the oscillator is Using the derivative of the position function, we find the velocity: The maximum speed is v max = ωa Slide 15-13

14 Example 15.1 A System in Simple Harmonic Motion Slide 15-14

15 Example 15.1 A System in Simple Harmonic Motion Slide 15-15

16 Example 15.1 A System in Simple Harmonic Motion Slide 15-16

17 Example 15.2 Finding the Time Slide 15-17

18 Simple Harmonic Motion and Circular Motion Figure (a) shows a shadow movie of a ball made by projecting a light past the ball and onto a screen. As the ball moves in uniform circular motion, the shadow moves with simple harmonic motion. The block on a spring in figure (b) moves with the same motion. Slide 15-18

19 The Phase Constant What if an object in SHM is not initially at rest at x = A when t = 0? Then we may still use the cosine function, but with a phase constant measured in radians. In this case, the two primary kinematic equations of SHM are: Slide 15-19

20 The Phase Constant Oscillations described by different values of the phase constant. Slide 15-20

21 Example 15.3 Using the Initial Conditions Slide 15-21

22 Example 15.3 Using the Initial Conditions Slide 15-22

23 Example 15.3 Using the Initial Conditions Slide 15-23

24 Example 15.3 Using the Initial Conditions Slide 15-24

25 Energy in Simple Harmonic Motion An object of mass m on a frictionless horizontal surface is attached to one end of a spring of spring constant k. The other end of the spring is attached to a fixed wall. As the object oscillates, the energy is transformed between kinetic energy and potential energy, but the mechanical energy E = K + U doesn t change. Slide 15-25

26 Energy in Simple Harmonic Motion Energy is conserved in Simple Harmonic Motion: Slide 15-26

27 Frequency of Simple Harmonic Motion In SHM, when K is maximum, U = 0, and when U is maximum, K = 0. K + U is constant, so K max = U max : So Earlier, using kinematics, we found that So Slide 15-27

28 Example 15.4 Using Conservation of Energy Slide 15-28

29 Example 15.4 Using Conservation of Energy Slide 15-29

30 Example 15.4 Using Conservation of Energy Slide 15-30

31 Simple Harmonic Motion Motion Diagram The top set of dots is a motion diagram for SHM going to the right. The bottom set of dots is a motion diagram for SHM going to the left. At x = 0, the object s speed is as large as possible, but it is not changing; hence acceleration is zero at x = 0. Slide 15-31

32 Acceleration in Simple Harmonic Motion Acceleration is the timederivative of the velocity: In SHM, the acceleration is proportional to the negative of the displacement. Slide 15-32

33 Dynamics of Simple Harmonic Motion Consider a mass m oscillating on a horizontal spring with no friction. The spring force is Since the spring force is the net force, Newton s second law gives Since a x = ω 2 x, the angular frequency must be. Slide 15-33

34 Vertical Oscillations Motion for a mass hanging from a spring is the same as for horizontal SHM, but the equilibrium position is affected. Slide 15-34

35 Example 15.6 Bungee Oscillations Slide 15-35

36 Example 15.6 Bungee Oscillations Slide 15-36

37 Example 15.6 Bungee Oscillations Slide 15-37

38 The Simple Pendulum Consider a mass m attached to a string of length L which is free to swing back and forth. If it is displaced from its lowest position by an angle θ, Newton s second law for the tangential component of gravity, parallel to the motion, is Slide 15-38

39 The Simple Pendulum If we restrict the pendulum s oscillations to small angles (< 10º), then we may use the small angle approximation sin θ θ, where θ is measured in radians. and the angular frequency of the motion is found to be Slide 15-39

40 Example 15.7 The Maximum Angle of a Pendulum Slide 15-40

41 Example 15.7 The Maximum Angle of a Pendulum Slide 15-41

42 The Simple-Harmonic-Motion Model Slide 15-42

43 The Physical Pendulum Any solid object that swings back and forth under the influence of gravity can be modeled as a physical pendulum. The gravitational torque for small angles (θ < 10º) is Plugging this into Newton s second law for rotational motion, τ = Iα, we find the equation for SHM, with Slide 15-43

44 Example 15.9 A Swinging Leg as a Pendulum Slide 15-44

45 Example 15.9 A Swinging Leg as a Pendulum Slide 15-45

46 Example 15.9 A Swinging Leg as a Pendulum Slide 15-46

47 Damped Oscillations An oscillation that runs down and stops is called a damped oscillation. The shock absorbers in cars and trucks are heavily damped springs. The vehicle s vertical motion, after hitting a rock or a pothole, is a damped oscillation. One possible reason for dissipation of energy is the drag force due to air resistance. The forces involved in dissipation are complex, but a simple linear drag model is Slide 15-47

48 Damped Oscillations When a mass on a spring experiences the force of the spring as given by Hooke s Law, as well as a linear drag force of magnitude F drag = bv, the solution is where the angular frequency is given by Here is the angular frequency of the undamped oscillator (b = 0). Slide 15-48

49 Damped Oscillations Position-versus-time graph for a damped oscillator. Slide 15-49

50 Damped Oscillations A damped oscillator has position x = x max cos(ωt + ϕ 0 ), where This slowly changing function x max provides a border to the rapid oscillations, and is called the envelope. The figure shows several oscillation envelopes, corresponding to different values of the damping constant b. Slide 15-50

51 Mathematical Aside: Exponential Decay Exponential decay occurs in a vast number of physical systems of importance in science and engineering. Mechanical vibrations, electric circuits, and nuclear radioactivity all exhibit exponential decay. The graph shows the function: where e = is Euler s number. exp is the exponential function. v 0 is called the decay constant. Slide 15-51

52 Energy in Damped Systems Because of the drag force, the mechanical energy of a damped system is no longer conserved. At any particular time we can compute the mechanical energy from Where the decay constant of this function is called the time constant τ, defined as The oscillator s mechanical energy decays exponentially with time constant τ. Slide 15-52

53 Driven Oscillations and Resonance Consider an oscillating system that, when left to itself, oscillates at a natural frequency f 0. Suppose that this system is subjected to a periodic external force of driving frequency f ext. The amplitude of oscillations is generally not very high if f ext differs much from f 0. As f ext gets closer and closer to f 0, the amplitude of the oscillation rises dramatically. A singer or musical instrument can shatter a crystal goblet by matching the goblet s natural oscillation frequency. Slide 15-53

54 Driven Oscillations and Resonance The response curve shows the amplitude of a driven oscillator at frequencies near its natural frequency of 2.0 Hz. Slide 15-54

55 Driven Oscillations and Resonance The figure shows the same oscillator with three different values of the damping constant. The resonance amplitude becomes higher and narrower as the damping constant decreases. Slide 15-55

56 Chapter 15 Summary Slides Slide 15-56

57 General Principles Slide 15-57

58 General Principles Slide 15-58

59 Important Concepts Slide 15-59

60 Important Concepts Slide 15-60

61 Applications Slide 15-61

62 Applications Slide 15-62