adapted from holt book on physics
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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. Simple Harmonic Motion. - PowerPoint PPT PresentationTRANSCRIPT
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Simple harmonic motionChapter 10
Adapted from Holt book on physics
Section 1 Simple Harmonic Motion
Section 2 Springs
Section 3 Pendulum
Section 4 Review
Physics 101: Lecture 22, Pg 2
Simple Harmonic MotionSimple Harmonic Motion
Period = T (seconds per cycle)
Frequency = f = 1/T (cycles per second)
Angular frequency = = 2f = 2/T
Defn: any periodic motion that is the result of a restoring force
that is proportional to displacement.
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Chapter 10
Simple Harmonic Motion
Section 1 Simple Harmonic Motion
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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 (= radians)
– For a mass-spring system, the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position. (x = meters)
Section 2 Measuring Simple Harmonic Motion
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Chapter 10
Measures of Simple Harmonic Motion
Section 2 Measuring Simple Harmonic Motion
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Chapter 11
Amplitude of a pendulum
Section 2 Measuring Simple Harmonic Motion
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Chapter 11
Simple Harmonic Motion
Section 1 Simple Harmonic Motion
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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).
Section 1 Simple Harmonic Motion
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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.
Section 1 Simple Harmonic Motion
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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.
Section 1 Simple Harmonic Motion
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Chapter 11
Spring Constant
Section 1 Simple Harmonic Motion
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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?
Section 1 Simple Harmonic Motion
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Chapter 11
Sample Problem, continued
Section 1 Simple Harmonic Motion
Unknown:
k = ?
1. DefineGiven: m = 0.55 kg x = –2.0 cm = –0.20 m g = 9.81 m/s2
Diagram:
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Chapter 11
Sample Problem, continued
Section 1 Simple Harmonic Motion
2. PlanChoose 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
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Chapter 11
Sample Problem, continued
Section 1 Simple Harmonic Motion
2. Plan, continued Rearrange the equation to isolate the unknown:
kx mg 0
kx mg
k mg
x
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Chapter 11
Sample Problem, continued
Section 1 Simple Harmonic Motion
3. Calculate Substitute the values into the equation and
solve:
k mg
x
(0.55 kg)(9.81 m/s2 )
–0.020 m
k 270 N/m
4. Evaluate The value of k implies that 270 N of force is
required to displace the spring 1 m.
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Chapter 11
The Simple Pendulum
• A simple pendulum consists of a mass called a bob, which is attached to a fixed string.
Section 1 Simple Harmonic Motion
The forces acting on the bob at any point are the force exerted by the string and the gravitational force.
• At any displacement from equilibrium, the weight of the bob (Fg) can be resolved into two components.
• The x component (Fg,x = Fg sin ) is the only force acting on the bob in the direction of its motion and thus is the restoring force.
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Chapter 11
The Simple Pendulum, continued
• The magnitude of the restoring force (Fg,x = Fg sin ) is proportional to sin .
• When the maximum angle of displacement is relatively small (<15°), sin is approximately equal to in radians.
Section 1 Simple Harmonic Motion
• 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.
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Chapter 11
Restoring Force and Simple Pendulums
Section 1 Simple Harmonic Motion
Physics 101: Lecture 22, Pg 20
PendulumPendulum
Lg
gL
T 2
2
For “small oscillation”, period does not depend on
•mass
•amplitude Demos:
M,A,L dependence
Physics 101: Lecture 22, Pg 21
Concept QuestionConcept 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
Lg
gL
T 2
2
CORRECT
Physics 101: Lecture 22, Pg 22
Concept QuestionConcept Question
A pendulum is hanging vertically from the ceiling of an elevator. Initiallythe elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be1. greater than T 2. equal to T3. less than T
Lg
gL
T 2
2
CORRECT
“Effective g” is larger when accelerating upward
(you feel heavier)
Physics 101: Lecture 22, Pg 23
Concept QuestionConcept Question
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 halved3. The period of the oscillation would stay the same
+2A
t
-2A
x
CORRECT
Physics 101: Lecture 22, Pg 24
Potential Energy of a SpringPotential Energy of a Spring
0x
2S kx
21
PE
Where x is measured fromthe equilibrium position PES
m
xx=0
Physics 101: Lecture 22, Pg 25
Same thing for a vertical spring:Same thing for a vertical spring:
0y
PES
m
y
2S ky
21
PE
Where y is measured fromthe equilibrium position
y=0
Physics 101: Lecture 22, Pg 27
Chapter 11
Force and Energy in Simple Harmonic MotionForce and Energy in Simple Harmonic Motion
Section 1 Simple Harmonic Motion
Physics 101: Lecture 22, Pg 28
Concept QuestionConcept 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 12. Case 23. Same
CORRECT
Physics 101: Lecture 22, Pg 29
Concept QuestionConcept Question
PE = 1/2kx2 KE = 0
x=0 x=+Ax=-A x=0 x=+Ax=-A
samefor both
PE = 0 KE = KEMAX
same for both
Physics 101: Lecture 22, Pg 31
Simple Harmonic Motion:Simple Harmonic Motion:Quick ReviewQuick Review
x(t) = [A]cos(t)
v(t) = -[A]sin(t)
a(t) = -[A2]cos(t)
x(t) = [A]sin(t)
v(t) = [A]cos(t)
a(t) = -[A2]sin(t)
xmax = A
vmax = A
amax = A2
Period = T (seconds per cycle)
Frequency = f = 1/T (cycles per second)
Angular frequency = = 2f = 2/T
OR
Physics 101: Lecture 22, Pg 32
Review: Period of a SpringReview: Period of a Spring
For simple harmonic oscillator
= 2f = 2/T
For mass M on spring with spring constant k
k
m2 T
m
k
Demos:
A,m,k dependence
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Multiple Choice
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 leftB. to the rightC. to the left or to the right depending on whether the
spring is stretched or compressedD. perpendicular to the motion of the mass
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Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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/mG. –2.0 101 N/m J. 2.0 101 N/m
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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/mG. –2.0 101 N/m J. 2.0 101 N/m
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Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 energyB. gravitational potential energyC. kinetic energyD. a combination of two or more of the above
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 energyB. gravitational potential energyC. kinetic energyD. a combination of two or more of the above
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 energyG. gravitational potential energyH. kinetic energyJ. a combination of two or more of the above
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 energyG. gravitational potential energyH. kinetic energyJ. a combination of two or more of the above
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 vibrationD. All of the above affect the period.
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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 vibrationD. All of the above affect the period.
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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. 8 s H. s G. 4 s J. s
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
Standardized Test PrepChapter 11
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. 8 s H. s G. 4 s J. s
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Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
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 bobC. the component of the bob’s weight perpendicular to the
motion of the bobD. the elastic force of the stretched string
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
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 bobC. the component of the bob’s weight perpendicular to the
motion of the bobD. the elastic force of the stretched string
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
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 bobH. the free-fall acceleration at the pendulum’s locationJ. All of the above affect the period.
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
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 bobH. the free-fall acceleration at the pendulum’s locationJ. All of the above affect the period.
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
A pendulum bob hangs from a string and moves with simple harmonic motion.
9. If the pendulum completes exactly 12cycles in 2.0 min, what is the frequency ofthe pendulum?
A. 0.10 Hz B. 0.17 Hz C. 6.0 Hz D. 10 Hz
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
A pendulum bob hangs from a string and moves with simple harmonic motion.
9. If the pendulum completes exactly 12cycles in 2.0 min, what is the frequency ofthe pendulum?
A. 0.10 Hz B. 0.17 Hz C. 6.0 Hz D. 10 Hz
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ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
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. 1.76 H. 106G. 21.6 J. 239
Copyright © by Holt, Rinehart and Winston. All rights reserved.
ResourcesChapter menu
Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
Standardized Test PrepChapter 11
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. 1.76 H. 106G. 21.6 J. 239