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•Chapter 5
•Section 2 Energy
Kinetic Energy
• The energy of an object that is due to the object’s
motion is called kinetic energy.
• Kinetic energy depends on speed and mass.
1
KE  mv 2
2
1
2
kinetic energy =  mass   speed
2
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•Chapter 5
•Section 2 Energy
Potential Energy
• Potential Energy is the stored energy associated
with an object because of the position, shape, or
condition of the object.
• Gravitational potential energy is the energy an
object has because of its position in a gravitational
field.
• Gravitational potential energy depends on height
from a zero level and the mass of the object.
PEg = mgh
gravitational PE = mass  free-fall acceleration  height
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•Chapter 5
•Section 2 Energy
Potential Energy, continued
•
Elastic potential energy is the energy available for
use when a deformed elastic object returns to its
original configuration.
1 2
PEelastic  kx
2
elastic PE =
1
 spring constant  (distance compressed or stretched)
2
•
The symbol k is called the spring constant, a
parameter that measures the spring’s resistance to
being compressed or stretched.
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2
•Chapter 5
•Section 3 Conservation of
Energy
Mechanical Energy
• Mechanical energy is the sum of kinetic energy and
all forms of potential energy associated with an object
or group of objects.
ME = KE + ∑PE
• Mechanical energy is often conserved.
MEi = MEf
initial mechanical energy = final mechanical energy
(in the absence of friction)
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Chapter 11
Section 1 Simple Harmonic
Motion
Simple Harmonic Motion
• Simple harmonic motion describes any regular
vibrations or oscillations that repeat the same
movement on either side of the equilibrium position.
• Every simple harmonic motion is a back-and-forth
motion over the same path. For example,
pendulums & springs
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Measures of Simple Harmonic Motion
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Amplitude, Period, and Frequency in SHM
• In SHM, the maximum displacement from equilibrium
is defined as the amplitude of the vibration.
– A pendulum’s amplitude can be measured by the angle
between the pendulum’s equilibrium position and its
maximum displacement.
– For a mass-spring system, the amplitude is the maximum
amount the spring is stretched or compressed from its
equilibrium position.
• The SI units of amplitude are the radian (rad) or
degrees and the meter (m).
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Amplitude, Period, and Frequency in SHM
• The period (T) is the time that it takes a complete
cycle to occur.
– The SI unit of period is seconds (s).
• The frequency (f) is the number of cycles or
vibrations per unit of time.
– The SI unit of frequency is hertz (Hz) or
cycles/sec.
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Amplitude, Period, and Frequency in SHM,
continued
• Period and frequency are inversely related:
1
1
f  or T 
T
f
• Thus, any time you have a value for period or
frequency, you can calculate the other value.
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Measures of Simple Harmonic Motion
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Period of a Simple Pendulum in SHM
• The period of a simple pendulum depends on the
length and on the free-fall acceleration.
L
T  2
ag
2


4

2
T 

 g 
length
period  2
free-fall acceleration
• The period does not depend on the mass of the bob
or on the amplitude (for small angles).
• Lab results
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Chapter 11
Section 1 Simple Harmonic
Motion
SHM in springs
• The direction of the
force acting on the
mass (Felastic) is
opposite the direction
of the mass’s
displacement from
equilibrium (x = 0).
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Chapter 11
Section 1 Simple Harmonic
Motion
SHM in springs
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.
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Chapter 11
Section 1 Simple Harmonic
Motion
Simple Harmonic Motion
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Chapter 11
Section 1 Simple Harmonic
Motion
Force and Energy in Simple Harmonic Motion
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Chapter 11
Section 1 Simple Harmonic
Motion
Hooke’s Law
• 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.
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Chapter 11
Section 1 Simple Harmonic
Motion
Spring Constant
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Chapter 11
Section 1 Simple Harmonic
Motion
Practice Questions
1. In pinball games, the force exerted by a compressed spring is
used to release a ball. If the distance the spring is compressed
is doubled, how will the force change? If the spring is replaced
with one that is half as stiff, how will the force acting on the ball
change?
2. 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?
3. Suppose the spring from above is replaced with a spring that
stretches 36 cm from its equilibrium position.
• What is the spring constant?
• Is this spring stiffer or less stiff?
4. How much force is required to pull a spring 3.0 cm from its
equilibrium position if the spring constant is 2700 N/m?
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Chapter 11
Section 1 Simple Harmonic
Motion
Section Review
•
Which of these periodic motions are simple harmonic?
– a child swinging on a playground swing (θ = 45)
– a CD rotating in a player
– an oscillating clock pendulum (θ = 10)
•
A pinball machine uses a spring that is compressed 4.0 cm to launch
a ball. If the spring is 13 N/m, what is the force on the ball?
•
How does the restoring force acting on a pendulum bob change as the
bob swings toward the equilibrium position? How do the bob’s
acceleration (along the direction of motion) and velocity change?
•
When an acrobat reaches the equilibrium position, the net force acting
along the direction of motion is zero. Why does the acrobat swing past
the equilibrium position?
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Period of a Mass-Spring System in SHM
• The period of an ideal mass-spring system
depends on the mass and on the spring constant.
m
T  2
k
mass
period  2
spring constant
• The period does not depend on the amplitude.
• This equation applies only for systems in which the
spring obeys Hooke’s law.
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Practice: Mass-Spring System
• A 125 N object vibrates with a period of 3.5 s when
hanging from a spring. What is the spring constant of
the spring?
• A spring of 30.0 N/m is attached to different masses,
and the system is set in motion. Find the period and
frequency of vibration for masses of 2.3 kg and 15 g.
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Chapter 11
Section 2 Measuring Simple
Harmonic Motion
Section Review
• A child swings on a playground swing with a 2.5 m
long chain.
– What is the period and frequency of the child in
motion
• A 0.75 kg mass attached to a vertical spring stretches
the spring 0.30 m.
– What is the spring constant?
– The mass-spring system is now placed on a
horizontal surface and set vibrating, What is the
period of the vibration?
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