普通物理 General Physics 15 - Simple Harmonic Motion 郭艷光 Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件：ykuo@cc.ncue.edu.tw 網頁：http://ykuo.ncue.edu.tw

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Outline 15-1 What Is Physics? 15-2 Simple Harmonic Motion 15-3 The Force Law for Simple Harmonic Motion 15-4 Energy in Simple Harmonic Motion 15-5 An Angular Simple Harmonic Oscillator 15-6 Pendulums 15-7 Simple Harmonic Motion and Uniform Circular Motion 15-8 Damped Simple Harmonic Motion 15-9 Forced Oscillations and Resonance 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 2

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15-1 What Is Physics? When a bat hits a baseball, the bat may oscillate enough to sting the batter’s hands or even to break apart. When an airplane is in flight, the turbulence of the air flowing past the wings makes them oscillate, eventually leading to metal fatigue and even failure. When a train travels around a curve, its wheels oscillate horizontally (“hunt” in mechanical engineering terms) as they are forced to turn in new directions (you can hear the oscillations). 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 3

15-2 Simple Harmonic Motion Frequency: f (1 hertz = 1 Hz= 1 oscillation per second = 1 s-1.) Displacement: 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 4

15-2 Simple Harmonic Motion ω is called the angular frequency of the oscillator. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 5

15-2 Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 6

15-2 Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 7

15-3 The Force Law for Simple Harmonic Motion The acceleration of an object undergoing SHM: Newton’s second law: The force can be written as: 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 8

15-3 The Force Law for Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 9

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-1 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. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 10

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-1 (a) What are the angular frequency, the frequency, and the period of the resulting motion? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 11

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-1 (b) What is the amplitude of the oscillation? (c) What is the maximum speed vm of the oscillating block, and where is the block when it has this speed? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 12

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-1 (d) What is the magnitude am of the maximum acceleration of the block? (e) What is the phase constant Φ for the motion? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 13

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-2 At t = 0, the displacement x(0) of the block in a linear oscillator like that of Fig is －8.50 cm. (Read x(0) as “x at time zero.”) The block’s velocity v(0) then is －0.920 m/s, and its acceleration a(0) is + 47.0 m/s2. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 14

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-2 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15

15-4 Energy in Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 16

15-4 Energy in Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 17

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-3 The huge ball that appears in this chapter's opening photograph hangs from four cables and swings like a pendulum when the building sways in the wind. When the building sways－say, eastward －the massive pendulum does also but delayed enough so that as it finally swings eastward, the building is swaying westward. Thus,the pendulum's motion is out of step with the building's motion, tending to counter it. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 18

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-3 Many other buildings have other types of mass dampers, as these anti - sway devices are called Some, like the John Hancock building in Boston, have a large block oscillating at the end of a spring and on a lubricated track. The principle is the same as with the pendulum: The motion of the oscillator is out of step with the motion of the building. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 19

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-3 Suppose the block has mass m = 2.72×105 kg and is designed to oscillate at frequency f = 10.0 Hz and with amplitude xm= 20.0 cm (a) What is the total mechanical energy E of the spring - block system? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 20

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-3 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 21

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-3 (b) What is the block's speed as it passes through the equilibrium point? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 22

15-5 An Angular Simple Harmonic Oscillator Restoring torque: The constant k is called the torsion constant I is the rotational inertia 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 23

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-4 Figure shows a thin rod whose length L is 12.4 cm and whose mass m is 135 g, suspended at its midpoint from a long wire. Its period Ta of angular SHM is measured to be 2.53s. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 24

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-4 An irregularly shaped object, which we call object X, is then hung from the same wire, as in figure and its period Tb is found to be 4.76s. What is the rotational inertia of object X about its suspension axis? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 25

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-4 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 26

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15-6 Pendulums 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 27

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15-6 Pendulums 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 28

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15-6 Pendulums 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 29

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15-6 Pendulums 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 30

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-5 A meter stick swings about a pivot point at one end, at distance h from the stick’s center of mass. (a) What is the period of oscillation T? (b) What is the distance L0 between the pivot point O of the stick and the center of oscillation of the stick? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 31

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 15-6 Pendulums 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 32

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-6 A competition diving board sits on a fulcrum about one-third of the way out from the fixed end of the board (Fig .a). In a running dive, a diver takes three quick steps along the board, out past the fulcrum so as to rotate the board’s free end downward. As the board rebounds back through the horizontal, the diver leaps upward and toward the board’s free end (Fig .b). 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 33

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-6 A skilled diver trains to land on the free end just as the board has completed 2.5 oscillations during the leap. With such timing, the diver lands as the free end is moving downward with greatest speed (Fig .c). The landing then drives the free end down substantially, and the rebound catapults the diver high into the air. 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 34

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-6 Figure d shows a simple but realistic model of a competition board. The board section beyond the fulcrum is treated as a stiff rod of length L that can rotate about a hinge at the fulcrum, compressing an (imaginary) spring under the board’s free end. If the rod’s mass is m = 20.0 kg and the diver’s leap lasts tfl = 0.620 s, what spring constant k is required ofthe spring for a proper landing? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 35

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-6 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 36

15-7 Simple Harmonic Motion and Uniform Circular Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 37

15-8 Damped Simple Harmonic Motion Damping force Fd:Fd = － bv b is called the damping constant. Net force on m is : Fnet=－kx － b 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 38

15-8 Damped Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 39

15-8 Damped Simple Harmonic Motion 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 40

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-7 For the damped oscillator of Figure, m = 250 g, k = 85N/m, and b = 70 g/s. (a) What is the period of the motion? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 41

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-7 For the damped oscillator of Figure, m = 250 g, k = 85N/m, and b = 70 g/s. (b) How long does it take for the amplitude of the damped oscillations to drop to half its initial value? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 42

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 Example 15-7 For the damped oscillator of Figure, m = 250 g, k = 85N/m, and b = 70 g/s. (c) How long does it take for the mechanical energy to drop to one-half its initial value? 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 43

15-9 Forced Oscillations and Resonance 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 44

普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 End of chapter 15! 2018/11/16 普通物理教材-15，國立彰化師範大學物理系/郭艷光教授 45