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12.4 Simple Pendulum A simple pendulum also exhibits periodic motion A simple pendulum consists of an object of mass
m suspended by a light string or rod of length L The upper end of the string is fixed
When the object is pulled to the side and released, it oscillates about the lowest point, which is the equilibrium position
The motion occurs in the vertical plane and is driven by the gravitational force
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Simple Pendulum, 2 The forces acting on
the bob are and is the force exerted
on the bob by the string
is the gravitational force
The tangential component of the gravitational force is a restoring force
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Simple Pendulum, 3 In the tangential direction,
The length, L, of the pendulum is constant, and for small values of
This confirms the form of the motion is SHM
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Small Angle Approximation The small angle approximation states
that sin When is measured in radians When is small
Less than 10o or 0.2 rad The approximation is accurate to within about
0.1% when is than 10o
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Simple Pendulum, 4 The function can be written as
= max cos (t + ) The angular frequency is
The period is
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Simple Pendulum, Summary The period and frequency of a simple
pendulum depend only on the length of the string and the acceleration due to gravity
The period is independent of the mass All simple pendula that are of equal
length and are at the same location oscillate with the same period
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12.5 Physical Pendulum If a hanging object oscillates about a
fixed axis that does not pass through the center of mass and the object cannot be approximated as a particle, the system is called a physical pendulum It cannot be treated as a simple pendulum Use the rigid object model instead of the
particle model
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Physical Pendulum, 2 The gravitational force
provides a torque about an axis through O
The magnitude of the torque is
mgd sin I is the moment of
inertia about the axis through O
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Physical Pendulum, 3 From Newton’s Second Law,
The gravitational force produces a restoring force
Assuming is small, this becomes
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Physical Pendulum,4 This equation is in the form of an object
in simple harmonic motion The angular frequency is
The period is
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Physical Pendulum, 5 A physical pendulum can be used to
measure the moment of inertia of a flat rigid object If you know d, you can find I by measuring
the period If I = md then the physical pendulum is
the same as a simple pendulum The mass is all concentrated at the center
of mass
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12.6 Damped Oscillations In many real systems, nonconservative
forces are present This is no longer an ideal system (the type
we have dealt with so far) Friction is a common nonconservative
force In this case, the mechanical energy of
the system diminishes in time, the motion is said to be damped
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Damped Oscillation, Example One example of damped
motion occurs when an object is attached to a spring and submerged in a viscous liquid
The retarding force can be expressed as where b is a constant b is related to the resistive
force
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Damping Oscillation, Example Part 2 The restoring force is – kx From Newton’s Second Law
Fx = -k x – bvx = max
When the retarding force is small compared to the maximum restoring force, we can determine the expression for x This occurs when b is small
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Damping Oscillation, Example, Part 3 The position can be described by
The angular frequency will be
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Damped Oscillations, cont A graph for a
damped oscillation The amplitude
decreases with time The blue dashed
lines represent the envelope of the motion
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Damping Oscillation, Example Summary When the retarding force is small, the
oscillatory character of the motion is preserved, but the amplitude decays exponentially with time
The motion ultimately ceases Another form for the angular frequency
where 0 is the angular frequency in the absence of the
retarding force
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Types of Damping
is also called the natural frequency of the system
If Rmax = bvmax < kA, the system is said to be underdamped
When b reaches a critical value bc such that bc / 2 m = 0 , the system will not oscillate The system is said to be critically damped
If b/2m > 0, the system is said to be overdamped
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Types of Damping, cont Graphs of position versus
time for (a) an underdamped
oscillator (b) a critically damped
oscillator (c) an overdamped oscillator
For critically damped and overdamped there is no angular frequency
Fig 12.15
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12.7 Forced Oscillations It is possible to compensate for the loss
of energy in a damped system by applying an external force
The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces
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Forced Oscillations, 2 After a driving force F(t)=F0sin(t) on an
initially stationary object begins to act, the amplitude of the oscillation will increase
After a sufficiently long period of time,
Edriving = Elost to internal Then a steady-state condition is reached The oscillations will proceed with constant
amplitude
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Forced Oscillations, 3 The amplitude of a driven oscillation is
0 is the natural frequency of the undamped oscillator
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Resonance When the frequency of the driving force
is near the natural frequency () an increase in amplitude occurs
This dramatic increase in the amplitude is called resonance
The natural frequency is also called the resonance frequency of the system
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Resonance, cont. Resonance (maximum peak)
occurs when driving frequency equals the natural frequency
The amplitude increases with decreased damping
The curve broadens as the damping increases
The shape of the resonance curve depends on b
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1.8 Resonance in Structures A structure can be considered an
oscillator It has a set of natural frequencies,
determined by its stiffness, its mass, and the details of its construction
A periodic driving force is applied by the shaking of the ground during an earthquake
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Resonance in Structures If the natural frequency of the building
matches a frequency contained in the shaking ground, resonance vibrations can build to the point of damaging or destroying the building
Prevention includes Designing the building so its natural frequencies
are outside the range of earthquake frequencies Include damping in the building
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Resonance in Bridges, Example
The Tacoma Narrows Bridge was destroyed because the vibration frequencies of wind blowing through the structure matched a natural frequency of the bridge
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Exercises of chapter 12
7, 11, 18, 25, 29, 32, 35, 36, 40, 47, 53, 59