1. Chapter 13 periodic motion
Collapse of the Tacoma Narrows
suspension bridge in America in 1940
(p 415)

2. oscillation SHM Energy Damped oscillation Forced oscillation
kinematics
dynamics
Kinematics
equation
Dynamic
equation
Circle of
reference
Energy
Superposition
of shm

3. Key terms:
periodic motion / oscillation
restoring force
amplitude
cycle
period
frequency
angular frequency
simple harmonic motion
harmonic oscillator
circle of reference
phasor
phase angle
simple pendulum

4. physical pendulum
Damping
Damped oscillation
Critical damping
overdamping
underdamping
driving force
forced oscillation
natural angular frequency
resonance
chaotic motion
chaos

5. §1 Dynamic equation
1)　dynamic equation
Ideal model:
A) spring mass system

6. B) The Simple Pendulum
Small angle approximation sin

7. C) physical pendulum

8. Example: Tyrannosaurus rex and physical pendulum
the walking speed of tyrannosaurus rex can be estimated from its leg length L and its stride length s

9. Conclusion:Equation of SHM
Solution:

10. Example:A particle dropped down a hole that extends from one side of the earth, through its center, to the other side. Prove that the motion is SHM and find the period.
Solution:

11. Example:An astronaut on a body mass measuring device (BMMD),designed for use on orbiting space vehicles,its purpose is to allow astronauts to measure their mass in the ‘weight-less’ condition in earth orbit.
The BMMD is a spring mounted chair,if M is mass of astronaut and m effective mass of the BMMD,which also oscillate, show that

13. Example:the system is as follow,prove the block
will oscillate in SHM
Solution:
We have

14. Take a derivative of y with respect to x
Alternative solution
(1)
(2)
Take a derivative of y with respect to x

15. § 2 kinematic equation
2.1 Equation
Solution:

17. 2.2) 　the basic quantity——amplitude、period,phase
A) Basic quantity:
1) Amplitude (A): the maximum magnitude of displacement from equilibrium.
2) Angular frequency():
Spring-mass:
Simple pendulum:
Caution:
is not angular frequency rather than velocity .it depends on the system

18. Lag in phase Ahead in phase Out of phase In phase
3) Phase angle (  = t+ ): the status of the object.
Lag in phase
Ahead in phase
Out of phase
In phase

19. B) The formula to solve: A, , 
1)  is predetermined by the system.
2) A and  are determined by initial condition:
if t=0, x=x0, v=v0 ,
Caution:
Is fixed by initial condition

20. An object of mass 4kg is attached to a spring of k=100N. m-1
An object of mass 4kg is attached to a spring of k=100N.m-1. The object is given an initial velocity of v0=-5m.s-1 and an initial displacement of x0=1. Find the kinematics equation
Solution:

21. Circle of reference method

22. Angle between OQ and axis-x
Compare SHM with UCM
x(+), v(-), a(-)
In first quadrant 
Angle between OQ and axis-x
Phase
Angular Velocity
Angular Frequency 
Projection
Displacement x
Radius
UCM
Amplitude
SHM A

23. Example:Find the initial phase of the two oscillation
x(m)
x(cm)
0.8 6 3 o 1
t(s) o 1
t(s) 2 1 6 1 3
/3 o x o 3 x 4 2

24. From circle of reference
SHM: x-t graph,find 0 , a , b , and the angular frequency
Solution:
x (m)
From circle of reference 2 a b 1
t (s) -2

25. § 3 Energy in SHM Potential energy: Total energy of the system:
Kinetic energy:
Potential energy:
Total energy of the system:

27. Example:Spring mass system
Example:Spring mass system.particle move from left to right, amplitude A1. when the block passes through its equilibrium position, a lump of putty dropped vertically on to the block and stick to it. Find the kinetic equation suppose t=0 when putty dropped on to the block
Solution: k O X M

28. Example:A wheel is free to rotate about its fixed axle,a spring is attached to one of its spokes a distance r from axle.assuming that the wheel is a hoop of mass m and radius R,spring constant k. a) obtain the angular frequency of small oscillations of this system b) find angular frequency and how about r=R and r=0

30. § 4. Superposition of SHM
4.1 mathematics method

31. x= x1+x2= Acos( t+ ) x x1 =A1cos( t+1 ) x2 =A2cos( t+2 ) A
B) circle of reference
x1 =A1cos( t+1 )
x2 =A2cos( t+2 ) ω M
x= x1+x2= Acos( t+ ) M2 A2 A A2
(2-1) A1 M1  2 1 o x x1 x2 x

32. Example:x1=3cos(2t+)cm, x2=3cos(2t+/2)cm, find the superposition displacement of x1 and x2.
Solution:
Draw a circle of reference,

34. If damping force is relative small
§ 5 Damped Oscillations
5.1 Phenomena
5.2 equation
If damping force is relative small

35. overdamping No oscillation underdamping Critical damping
Amplitude decrease

36. :§6 Forced Oscillations
drive an oscillator with a sinusoidally varying force:
The steady-state solution is
where 0=(k/m)½ is the natural frequency of the system.
The amplitude has a large increase near 0 - resonance

38. Computation physics

39. Projectile motion with air resistance (case study:p147)

40. 2. Tracing problem
A plane moves in constant velocity due eastward,a missile trace it,suppose at anytime the missile direct to plane,speed is u,u>v,draw the path of missile
(X,Y)
(X0,h) v h
(x,y) u x

41. (X,Y) y v h
(x,y) u O x
y(0)=0, x(0)=0
Y=h,X(0)=0

42. 3. Planets trajectory
Example:the orbits of satellites in the gravitational field v
Solution: r ms me

43. We get:
reference :《大学物理》吴锡珑 p 149

44. Solution1: Newton’s laws
3. Planets trajectory
Solution1: Newton’s laws

45. Solution2: conservation of mechanical energy