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 Discover and describe the many kinds of periodic motion.
 Explain periodic motion of a spring using Hooke’s Law.
 Use the equations of simple harmonic motion to explain that motion.
 Relate simple harmonic motion to everyday events.

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 F_{s} =  k x
 F_{s} is the spring force
 k is the spring constant
 It is a measure of the stiffness of the spring
 A large k indicates a stiff spring and a small k indicates a soft
spring
 x is the displacement of the object from its equilibrium position
 The negative sign indicates that the force is always directed opposite
to the displacement

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 The force always acts toward the equilibrium position
 It is called the restoring force
 The direction of the restoring force is such that the object is being
either pushed or pulled toward the equilibrium position

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 When x is positive (to the right), F is negative (to the left)
 When x = 0 (at equilibrium), F is 0
 When x is negative (to the left), F is positive (to the right)

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 Assume the object is initially pulled to x = A and released from rest
 As the object moves toward the equilibrium position, F and a decrease,
but v increases
 At x = 0, F and a are zero, but v is a maximum
 The object’s momentum causes it to overshoot the equilibrium position
 The force and acceleration start to increase in the opposite direction
and velocity decreases
 The motion continues indefinitely

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 Motion that occurs when the net force along the direction of motion is a
Hooke’s Law type of force
 The force is proportional to the displacement and in the opposite
direction
 The motion of a spring mass system is an example of Simple Harmonic
Motion

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 Not all periodic motion over the same path can be considered Simple
Harmonic motion
 To be Simple Harmonic motion, the force needs to obey Hooke’s Law

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 Amplitude, A
 The amplitude is the maximum position of the object relative to the
equilibrium position
 In the absence of friction, an object in simple harmonic motion will
oscillate between ±A on each side of the equilibrium position

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 The period, T, is the time that it takes for the object to complete one
complete cycle of motion
 From x = A to x =  A and back to x = A
 The frequency, ƒ, is the number of complete cycles or vibrations per
unit time

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 Newton’s second law will relate force and acceleration
 The force is given by Hooke’s Law
 F =  k x = m a
 The acceleration is a function of position
 Acceleration is not constant and therefore the uniformly accelerated
motion equation cannot be applied

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 Acceleration can be used to define simple harmonic motion
 An object moves in simple harmonic motion if its acceleration is
directly proportional to the displacement and is in the opposite
direction

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 A compressed spring has potential energy
 The compressed spring, when allowed to expand, can apply a force to an
object
 The potential energy of the spring can be transformed into kinetic
energy of the object

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 The energy stored in a stretched or compressed spring or other elastic
material is called elastic potential energy
 The energy is stored only when the spring is stretched or compressed
 Elastic potential energy can be added to the statements of Conservation
of Energy and WorkEnergy

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 A block sliding on a frictionless system collides with a light spring
 The block attaches to the spring

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 The block is moving on a frictionless surface
 The total mechanical energy of the system is the kinetic energy of the
block

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 The spring is partially compressed
 The energy is shared between kinetic energy and elastic potential energy
 The total mechanical energy is the sum of the kinetic energy and the
elastic potential energy

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 The spring is now fully compressed
 The block momentarily stops
 The total mechanical energy is stored as elastic potential energy of the
spring

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 When the block leaves the spring, the total mechanical energy is in the
kinetic energy of the block
 The spring force is conservative and the total energy of the system
remains constant

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 Conservation of Energy allows a calculation of the velocity of the
object at any position in its motion
 Speed is a maximum at x = 0
 Speed is zero at x = ±A
 The ± indicates the object can be traveling in either direction

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 A ball is attached to the rim of a turntable of radius A
 The focus is on the shadow that the ball casts on the screen
 When the turntable rotates with a constant angular speed, the shadow
moves in simple harmonic motion

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 Period
 This gives the time required for an object of mass m attached to a
spring of constant k to complete one cycle of its motion
 Frequency
 Units are cycles/second or Hertz, Hz

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 The angular frequency is related to the frequency

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 Use of a reference circle allows a description of the motion
 x = A cos (2πƒt)
 x is the position at time t
 x varies between +A and A

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 When x is a maximum or minimum, velocity is zero
 When x is zero, the velocity is a maximum
 When x is a maximum in the positive direction, a is a maximum in the
negative direction

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 This experiment shows the sinusoidal nature of simple harmonic motion
 The spring mass system oscillates in simple harmonic motion
 The attached pen traces out the sinusoidal motion

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 The simple pendulum is another example of simple harmonic motion
 The force is the component of the weight tangent to the path of motion

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 In general, the motion of a pendulum is not simple harmonic
 However, for small angles, it becomes simple harmonic
 In general, angles < 15° are small enough
 sin θ = θ
 F =  m g θ
 This force obeys Hooke’s Law

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 This shows that the period is independent of of the amplitude
 The period depends on the length of the pendulum and the acceleration of
gravity at the location of the pendulum

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 Only ideal systems oscillate indefinitely
 In real systems, friction retards the motion
 Friction reduces the total energy of the system and the oscillation is
said to be damped

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 Damped motion varies depending on the fluid used
 With a low viscosity fluid, the vibrating motion is preserved, but the
amplitude of vibration decreases in time and the motion ultimately
ceases
 This is known as underdamped oscillation

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 With a higher viscosity, the object returns rapidly to equilibrium after
it is released and does not oscillate
 The system is said to be critically damped
 With an even higher viscosity, the piston returns to equilibrium without
passing through the equilibrium position, but the time required is
longer
 This is said to be over damped

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 Plot a shows a critically damped oscillator
 Plot b shows an overdamped oscillator

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 A wave is the motion of a disturbance
 Mechanical waves require
 Some source of disturbance
 A medium that can be disturbed
 Some physical connection between or mechanism though which adjacent
portions of the medium influence each other
 All waves carry energy and momentum

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 In a transverse wave, each element that is disturbed moves
perpendicularly to the wave motion

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 In a longitudinal wave, the elements of the medium undergo displacements
parallel to the motion of the wave
 A longitudinal wave is also called a compression wave

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 The red curve is a “snapshot” of the wave at some instant in time
 The blue curve is later in time
 A is a crest of the wave
 B is a trough of the wave

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 A longitudinal wave can also be represented as a sine curve
 Compressions correspond to crests and stretches correspond to troughs

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 Amplitude is the maximum displacement of string above the equilibrium
position
 Wavelength, λ, is the distance between two successive points that
behave identically

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 v = ƒ λ
 Is derived from the basic speed equation of distance/time
 This is a general equation that can be applied to many types of waves

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 The speed on a wave stretched under some tension, F
 The speed depends only upon the properties of the medium through which
the disturbance travels

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 Two traveling waves can meet and pass through each other without being
destroyed or even altered
 Waves obey the Superposition Principle
 If two or more traveling waves are moving through a medium, the
resulting wave is found by adding together the displacements of the
individual waves point by point
 Actually only true for waves with small amplitudes

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 Two waves, a and b, have the same frequency and amplitude
 The combined wave, c, has the same frequency and a greater amplitude

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 Two pulses are traveling in opposite directions
 The net displacement when they overlap is the sum of the displacements
of the pulses
 Note that the pulses are unchanged after the interference

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 Two waves, a and b, have the same amplitude and frequency
 They are 180° out of phase
 When they combine, the waveforms cancel

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 Two pulses are traveling in opposite directions
 The net displacement when they overlap the displacements of the pulses
subtract
 Note that the pulses are unchanged after the interference

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 Whenever a traveling wave reaches a boundary, some or all of the wave is
reflected
 When it is reflected from a fixed end, the wave is inverted

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 When a traveling wave reaches a boundary, all or part of it is reflected
 When reflected from a free end, the pulse is not inverted

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