1. Adapted from Holt book on physics
Chapter 10
Simple harmonic motion
Adapted from Holt book on physics
Section 1 Simple Harmonic Motion
Section 2 Springs
Section 3 Pendulum
Section 4 Review

2. Simple Harmonic Motion
Defn: any periodic motion that is the result of a restoring force that is proportional to displacement.
Period = T (seconds per cycle)
Frequency = f = 1/T (cycles per second)
Angular frequency =  = 2f = 2/T

3. Simple Harmonic Motion
Section 1 Simple Harmonic Motion
Chapter 10
Simple Harmonic Motion

4. Chapter 10 Amplitude in SHM
Section 2 Measuring Simple Harmonic Motion
Chapter 10
Amplitude in SHM
In SHM, the maximum displacement from equilibrium is defined as the amplitude of the vibration.
A pendulum’s amplitude = swing angle from the vertical (Dq= radians)
For a mass-spring system, the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position. (Dx = meters)

5. Measures of Simple Harmonic Motion
Section 2 Measuring Simple Harmonic Motion
Chapter 10
Measures of Simple Harmonic Motion

6. Amplitude of a pendulum
Section 2 Measuring Simple Harmonic Motion
Chapter 11
Amplitude of a pendulum

7. Simple Harmonic Motion
Section 1 Simple Harmonic Motion
Chapter 11
Simple Harmonic Motion

8. Section 1 Simple Harmonic Motion
Chapter 11
Hooke’s Law
One type of periodic motion is the motion of a mass attached to a spring.
The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0).

9. Chapter 11 Hooke’s Law, continued At equilibrium:
Section 1 Simple Harmonic Motion
Chapter 11
Hooke’s Law, continued
At equilibrium:
The spring force and the mass’s acceleration become zero.
The speed reaches a maximum.
At maximum displacement:
The spring force and the mass’s acceleration reach a maximum.
The speed becomes zero.

10. spring force = –(spring constant  displacement)
Section 1 Simple Harmonic Motion
Chapter 11
Hooke’s Law, continued
Measurements show that the spring force, or restoring force, is directly proportional to the displacement of the mass.
This relationship is known as Hooke’s Law:
Felastic = –kx
spring force = –(spring constant  displacement)
The quantity k is a positive constant called the spring constant.

11. Section 1 Simple Harmonic Motion
Chapter 11
Spring Constant

12. Chapter 11 Sample Problem Hooke’s Law
Section 1 Simple Harmonic Motion
Chapter 11
Sample Problem
Hooke’s Law
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

13. Sample Problem, continued
Section 1 Simple Harmonic Motion
Chapter 11
Sample Problem, continued
1. Define
Given:
m = 0.55 kg
x = –2.0 cm = –0.20 m
g = 9.81 m/s2
Diagram:
Unknown:
k = ?

14. Sample Problem, continued
Section 1 Simple Harmonic Motion
Chapter 11
Sample Problem, continued
2. Plan
Choose an equation or situation: When the mass is attached to the spring,the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass.
Fnet = 0 = Felastic + Fg
Felastic = –kx
Fg = –mg
–kx – mg = 0

15. Sample Problem, continued
Section 1 Simple Harmonic Motion
Chapter 11
Sample Problem, continued
2. Plan, continued
Rearrange the equation to isolate the unknown:

16. Sample Problem, continued
Section 1 Simple Harmonic Motion
Chapter 11
Sample Problem, continued
3. Calculate
Substitute the values into the equation and solve:
4. Evaluate
The value of k implies that 270 N of force is required to displace the spring 1 m.

17. Chapter 11 The Simple Pendulum
Section 1 Simple Harmonic Motion
Chapter 11
The Simple Pendulum
A simple pendulum consists of a mass called a bob, which is attached to a fixed string.
At any displacement from equilibrium, the weight of the bob (Fg) can be resolved into two components.
The x component (Fg,x = Fg sin q) is the only force acting on the bob in the direction of its motion and thus is the restoring force.
The forces acting on the bob at any point are the force exerted by the
string and the
gravitational force.

18. The Simple Pendulum, continued
Section 1 Simple Harmonic Motion
Chapter 11
The Simple Pendulum, continued
The magnitude of the restoring force (Fg,x = Fg sin q) is proportional to sin q.
When the maximum angle of displacement q is relatively small (<15°), sin q is approximately equal to q in radians.
As a result, the restoring force is very nearly proportional to the displacement.
Thus, the pendulum’s motion is an excellent approximation of simple harmonic motion.

19. Restoring Force and Simple Pendulums
Section 1 Simple Harmonic Motion
Chapter 11
Restoring Force and Simple Pendulums

20. Pendulum For “small oscillation”, period does not depend on mass
amplitude
Demos:
M,A,L dependence

21. Concept Question
Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should:
1. Make the pendulum shorter
2. Make the pendulum longer
CORRECT

22. “Effective g” is larger when accelerating upward
Concept Question
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
CORRECT
“Effective g” is larger when accelerating upward
(you feel heavier)

23. Concept Question x +2A t -2A
If the amplitude of the oscillation (same block and same spring) was doubled, how would the period of the oscillation change? (The period is the time it takes to make one complete oscillation)
1. The period of the oscillation would double. 2. The period of the oscillation would be halved 3. The period of the oscillation would stay the same
CORRECT x
+2A t
-2A

24. Potential Energy of a Springm x
x=0
Where x is measured from the equilibrium position
PES x

25. Same thing for a vertical spring:y
y=0 m
Where y is measured from the equilibrium position
PES y

26. Etotal = 1/2 Mv2 + 1/2 kx2 = constant
In either case... m y
y=0 m x
x=0
Etotal = 1/2 Mv2 + 1/2 kx2 = constant
KE PE
KEmax = 1/2Mv2max=1/2M2A2 =1/2kA2
PEmax = 1/2kA2
Etotal = 1/2kA2

27. Force and Energy in Simple Harmonic Motion
Section 1 Simple Harmonic Motion
Chapter 11
Force and Energy in Simple Harmonic Motion

28. Concept Question
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 kinetic energy of the mass the biggest?
1. Case 1 2. Case 2 3. Same
CORRECT

29. Concept Question PE = 0 KE = KEMAX PE = 1/2kx2 KE = 0 same for both
x=-A
x=0
x=+A
x=-A
x=0
x=+A

30. Concept Question The same would be true for vertical springs...
PE = 1/2k y2
Y=0
PE = 1/2k y2
Y=0

31. Simple Harmonic Motion: Quick Review
x(t) = [A]cos(t)
v(t) = -[A]sin(t)
a(t) = -[A2]cos(t)
x(t) = [A]sin(t)
v(t) = [A]cos(t)
a(t) = -[A2]sin(t) OR
xmax = A
vmax = A
amax = A2
Period = T (seconds per cycle)
Frequency = f = 1/T (cycles per second)
Angular frequency =  = 2f = 2/T

32. Review: Period of a Spring
For simple harmonic oscillator
 = 2f = 2/T
For mass M on spring with spring constant k
Demos:
A,m,k dependence

33. Chapter 11 Multiple Choice
Standardized Test Prep
Multiple Choice
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
1. In what direction does the restoring force act?
A. to the left
B. to the right
C. to the left or to the right depending on whether the spring is stretched or compressed
D. perpendicular to the motion of the mass

34. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant?
F. –5.0  10–2 N/m H. 5.0  10–2 N/m
G. –2.0  101 N/m J. 2.0  101 N/m

35. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant?
F. –5.0  10–2 N/m H. 5.0  10–2 N/m
G. –2.0  101 N/m J. 2.0  101 N/m

36. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
3. In what form is the energy in the system when the mass passes through the equilibrium point?
A. elastic potential energy
B. gravitational potential energy
C. kinetic energy
D. a combination of two or more of the above

37. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
3. In what form is the energy in the system when the mass passes through the equilibrium point?
A. elastic potential energy
B. gravitational potential energy
C. kinetic energy
D. a combination of two or more of the above

38. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
4. In what form is the energy in the system when the mass is at maximum displacement?
F. elastic potential energy
G. gravitational potential energy
H. kinetic energy
J. a combination of two or more of the above

39. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
4. In what form is the energy in the system when the mass is at maximum displacement?
F. elastic potential energy
G. gravitational potential energy
H. kinetic energy
J. a combination of two or more of the above

40. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
5. Which of the following does not affect the period of the mass-spring system?
A. mass
B. spring constant
C. amplitude of vibration
D. All of the above affect the period.

41. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
5. Which of the following does not affect the period of the mass-spring system?
A. mass
B. spring constant
C. amplitude of vibration
D. All of the above affect the period.

42. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?
F. 8p s H. p s
G. 4p s J. p/2 s

43. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?
F. 8p s H. p s
G. 4p s J. p/2 s

44. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
7. What is the restoring force in the pendulum?
A. the total weight of the bob
B. the component of the bob’s weight tangent to the motion of the bob
C. the component of the bob’s weight perpendicular to the motion of the bob
D. the elastic force of the stretched string

45. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
7. What is the restoring force in the pendulum?
A. the total weight of the bob
B. the component of the bob’s weight tangent to the motion of the bob
C. the component of the bob’s weight perpendicular to the motion of the bob
D. the elastic force of the stretched string

46. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
8. Which of the following does not affect the
period of the pendulum?
F. the length of the string
G. the mass of the pendulum bob
H. the free-fall acceleration at the pendulum’s location
J. All of the above affect the period.

47. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
8. Which of the following does not affect the
period of the pendulum?
F. the length of the string
G. the mass of the pendulum bob
H. the free-fall acceleration at the pendulum’s location
J. All of the above affect the period.

48. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
9. If the pendulum completes exactly 12
cycles in 2.0 min, what is the frequency of
the pendulum?
A Hz
B Hz
C. 6.0 Hz
D. 10 Hz

49. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
9. If the pendulum completes exactly 12
cycles in 2.0 min, what is the frequency of
the pendulum?
A Hz
B Hz
C. 6.0 Hz
D. 10 Hz

50. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
10. If the pendulum’s length is 2.00 m and
ag = 9.80 m/s2, how many complete
oscillations does the pendulum make in
5.00 min?
F H. 106
G J. 239

51. Multiple Choice, continued
Chapter 11
Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
10. If the pendulum’s length is 2.00 m and
ag = 9.80 m/s2, how many complete
oscillations does the pendulum make in
5.00 min?
F H. 106
G J. 239