1. Chapter 9 Oscillatory Motion

2. Main Points of chapter 9
The kinematics of simple harmonic motion
Connection to circular motion
The dynamics of simple harmonic motion s
Energy
Simple and physical pendulums
Damped and driven harmonic motion

3. 9-1 The Kinematics of Simple Harmonic Motion
Any motion that repeats itself at regular intervals is called periodic motion
Examples: circular motion, oscillatory 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
This oscillation is called Simple Harmonic 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 .
SI: rad/s
amplitude A: The maximum distance of
displacement to the equilibrium point
phase wt+f, phase angle (constant) f
The value of A and f depend on the displacement and velocity of the particle at time t = 0 (the initial conditions)

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

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

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

8. They are in phase They have a phase difference of p o A1 -A1 A2 - A2x1 x2 T t x x o A1
-A1 A2
- A2 x1 x2 T t
They are in phase
They have a phase
difference of p

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

10. a(t) is phase –shifted to the left from x(t) by p
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.

11. 9-2 A Connection to Circular Motion
A reference particle P´ moving in a reference circle of radius A with steady angular velocity w . Its projection P on the x axis executes simple harmonic motion.
Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs.
demo

12. ACT A mass oscillates up & down on a spring
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. 9-3 Springs and Simple Harmonic Motion
The block–spring system forms a linear simple harmonic oscillator k x m
F = -kx a
Hooke’s law
Combining with Newton’s second law
a differential equation for x(t)
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).

14. The period of the motion is independent of the amplitude
Solution
The period of the motion is independent of the amplitude
The initial conditions
t=0；x=x0，v=v0

15. X=0
X=A
X=-A
X=A; v=0; a=-amax
X=0; v=-vmax; a=0
X=-A; v=0; a=amax
X=0; v=vmax; a=0
X=A; v=0; a=-amax

16. Another solution is ok
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 vm of the oscillating block, and where is the block when it occurs? (d)  What is the magnitude am of the maximum acceleration of the block? (e)  What is the phase constant f for the motion? (f)  What is the displacement function x(t) for the spring–block system?

18. Solution
At equilibrium point

19. Correct phase constant is1550
Example At t = 0, the displacement x(0) of the block in a linear oscillator is cm. The block's velocity v(0) then is m/s, and its acceleration a(0) is m/s2. (a)  What is the angular frequency w of this system? (b)  What are the phase constant f and amplitude A?
Solution
155
-25
Correct phase constant is1550

20. Additional Constant Forces
Solution
Simple harmonic motion with the same frequency, but equilibrium point is shifted from x=0 to x=x1

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

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?
(a) v(t) = -vmax sin(wt) a(t) = -amax cos(wt)
(b) v(t) = vmax sin(wt) a(t) = amax cos(wt) k y
(c) v(t) = vmax cos(wt) a(t) = -amax cos(wt) d
t = 0 m
(both vmax and amax are positive numbers)

23. 9-4 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. They are all positive. U(t) and K(t) peak twice during every period
(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 xm. For x = 0 the energy is all kinetic, and for x = ±xm it is all potential. The mechanical energy is conserved

25. Note
The potential energy and the kinetic energy
peak twice during every period
The mechanical energy is conserved for a linear
harmonic oscillator
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
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 Same for both!

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

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 H2 molecule looks something like this: U x

29. since x0 is minimum of potential
If we do a Taylor expansion of this function about the minimum, we find that for small displacements, the potential is quadratic:
since x0 is minimum of potential U x x0 U
x 
then
Restoring force

30. Identifying SHM
c, c’ positive constant

31. He gets back 84 minutes later, at 1:24 p.m.
Transport Tunnel
A straight tunnel with a frictionless interior is dug through the Earth. A student jumps into the hole at noon. What time does he get back? x
g = 9.81 m/s2
RE = 6.38 x 106 m
He gets back 84 minutes later, at 1:24 p.m.

32. 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.
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.
g = 2R
9.81 = 2 6.38(10)6 m
 = s-1
so T = = 5067 s
84 min

33. Example The potential energy of a diatomic molecule whose two atoms have the same mass, m, and are separated by a distance r is given by the formula
where A and e are positive constants. (a) Find the equilibrium separation of the two atoms. (b) Show that if the atoms are slightly displaced, then they will undergo simple harmonic motion about the equilibrium position. Calculate the angular frequency of the harmonic motion.
Solution
(a) The equilibrium separation occurs where the potential energy is a minimum, so we set

34. (b) We do a Taylor expansion of this U(r) function about the equilibrium separation
This has the form of the elastic potential energy, so the motion will be simple harmonic
The spring constant
The reduced mass
The angular frequency is

35. 9-5 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 T mg
If q is small

36. Solution
with
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

37. The energy of a simple pendulum:
For small q
The total energy is conserved

38. ACT
You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T1.
Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T2. Which of the following is true:
(a) T1 = T2
(b) T1 > T2
(c) T1 < T2
You make a pendulum shorter, it oscillates faster (smaller period)

39. “Effective g” is larger when accelerating upward
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)

40. 9-6 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 q is small T Mg

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

42. 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? (ICM = mR2 for a hoop)
(a)
(b)
(c)
pivot (nail) D

43. 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 L0 between the pivot point O of the stick and the center of oscillation of the stick?
Solution

44. 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.

45. y Solution Choose the equilibrium position as the origin mg F O
T is independent of the board’s length

46. Example A block of mass m is attached to a spring of constant k through a disk of mass M which is free to rotate about its fixed axis. Find the period of small oscillations M
Solution
Choose the equilibrium position as the origin T’ T T’ T o x k m mg

47. 9-7 Damped Harmonic Motion
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.
Look at drag force that is proportional to velocity; b is the damping coefficient:
Then the equation of motion is:

48. Damping factor
If a is small
Solution
Natural frequency

49. The larger the value of t ,the slower the exponential
Life time
The larger the value of t ,the slower the exponential

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

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

52. Example For the damped oscillator: 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?

53. Solution

54. 9-8 Driven Harmonic Motion
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 driving force
Equation of motion becomes:
solution

55. After long times

56. The condition for the maximum of A
If b=0
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 maximum. This effect is called resonance

57. For small b, the total width at half maximum
peak becomes broader as b increases:

58. The role played by the frequency of a driving force is a critical one
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
demo

59. Summary of chapter 9
Simple Harmonic Motion
Springs

60. Summary of chapter 9 Cont.
Energy
Simple and physical pendulums

61. Summary of chapter 9 Cont.
Damped and driven harmonic motion
If b=0
resonance