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Periodic MotionPeriodic Motion• defined: motion that
repeats at a constant rate• equilibrium position: forces
are balanced
Periodic MotionPeriodic Motion• For the spring example,
the mass is pulled down to y = -A and then released.
• Two forces are working on the mass: gravity (weight) and the spring.
Periodic MotionPeriodic Motion• Damping: the effect of
friction opposing the restoring force in oscillating systems
Periodic MotionPeriodic Motion• Restoring force (Fr): the
net force on a mass that always tends to restore the mass to its equilibrium position
Simple Harmonic Motion
Simple Harmonic Motion
• defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position
Simple Harmonic Motion
Simple Harmonic Motion
• The restoring force in SHM is described by:
Fr x = -kΔx• Δx = displacement from
equilibrium position
Simple Harmonic Motion
Simple Harmonic Motion
• Table 12-1 describes relationships throughout one oscillation
Simple Harmonic Motion
Simple Harmonic Motion
• Amplitude: maximum displacement in SHM
• Cycle: one complete set of motions
Simple Harmonic Motion
Simple Harmonic Motion
• Period: the time taken to complete one cycle
• Frequency: cycles per unit of time• 1 Hz = 1 cycle/s = s-1
Simple Harmonic Motion
Simple Harmonic Motion
• Frequency (f) and period (T) are reciprocal quantities.
f =T1
T =f1
Reference CircleReference Circle• Circular motion has many
similarities to SHM.• Their motions can be
synchronized and similarly described.
Reference CircleReference Circle• The period (T) for the
spring-mass system can be derived using equations of circular motion:
T = 2πkm
Reference CircleReference Circle• This equation is used for
Example 12-1.• The reciprocal of T gives
the frequency.
T = 2πkm
Overview Overview • The periods of both
pendulums and spring-mass systems in SHM are independent of the amplitudes of their initial displacements.
Pendulum Motion Pendulum Motion • An ideal pendulum has a
mass suspended from an ideal spring or massless rod called the pendulum arm.
• The mass is said to reside at a single point.
Pendulum Motion Pendulum Motion • l = distance from the
pendulum’s pivot point and its center of mass
• center of mass travels in a circular arc with radius l.
Pendulum Motion Pendulum Motion • forces on a pendulum at
rest:• weight (mg)• tension in pendulum arm
(Tp)• at equilibrium when at rest
Restoring Force Restoring Force • When the pendulum is not
at its equilibrium position, the sum of the weight and tension force vectors moves it back toward the equilibrium position.
Fr = Tp + mg
Restoring Force Restoring Force • Centripetal force adds to
the tension (Tp):
Tp = Tw΄+ Fc , where:
Tw΄ = Tw = |mg|cos θ
Fc = mvt²/r
Restoring Force Restoring Force • Total acceleration (atotal) is
the sum of the tangential acceleration vector (at) and the centripetal acceleration.
• The restoring forces causes this atotal.
Restoring Force Restoring Force • A pendulum’s motion does
not exactly conform to SHM, especially when the amplitude is large (larger than π/8 radians, or 22.5°).
Small Amplitude Small Amplitude • defined as a displacement
angle of less than π/8 radians from vertical
• SHM is approximated
Small Amplitude Small Amplitude • The mass of the pendulum
does not affect the period of the swing.
T = 2π|g|l
Small Amplitude Small Amplitude • This formula can even be
used to approximate g (see Example 12-2).
T = 2π|g|l
Physical Pendulums Physical Pendulums • mass is distributed to some
extent along the length of the pendulum arm
• can be an object swinging from a pivot
• common in real-world motion
Physical Pendulums Physical Pendulums • The moment of inertia of an
object quantifies the distribution of its mass around its rotational center.
• Abbreviation: I • A table is found in
Appendix F of your book.
Damped OscillationsDamped Oscillations• Resistance within a
spring and the drag of air on the mass will slow the motion of the oscillating mass.
Damped OscillationsDamped Oscillations• Damped harmonic
oscillators experience forces that slow and eventually stop their oscillations.
Damped OscillationsDamped Oscillations• The magnitude of the
force is approximately proportional to the velocity of the system:
fx = -βvx β is a friction proportionality
constant
Damped OscillationsDamped Oscillations• The amplitude of a
damped oscillator gradually diminishes until motion stops.
Damped OscillationsDamped Oscillations• An overdamped
oscillator moves back to the equilibrium position and no further.
Damped OscillationsDamped Oscillations• A critically damped
oscillator barely overshoots the equilibrium position before it comes to a stop.
Driven OscillationsDriven Oscillations• To most efficiently
continue, or drive, an oscillation, force should be added at the maximum displacement from the equilibrium position.
Driven OscillationsDriven Oscillations• The frequency at which
the force is most effective in increasing the amplitude is called the natural oscillation frequency (f0).
Driven OscillationsDriven Oscillations• The natural oscillation
frequency (f0) is the characteristic frequency at which an object vibrates.
• also called the resonant frequency
Driven OscillationsDriven Oscillations• terminology:
• in phase• pulses• driven oscillations• resonance
Driven OscillationsDriven Oscillations• A driven oscillator has
three forces acting on it:• restoring force• damping resistance• pulsed force applied in
same direction as Fr
Driven OscillationsDriven Oscillations• The Tacoma Narrows
Bridge demonstrated the catastrophic potential of uncontrolled oscillation in 1940.
• defined: oscillations of extended bodies
• medium: the material through which a wave travels
WavesWaves
• disturbance: an oscillation in the medium
• It is the disturbance that travels; the medium does not move very far.
WavesWaves
• longitudinal wave: disturbance that displaces the medium along its line of travel
• example: spring
Types of WavesTypes of Waves
• transverse wave: disturbance that displaces the medium perpendicular to its line of travel
• example: the wave along a snapped string
Types of WavesTypes of Waves
• Any physical medium can carry a longitudinal wave.
Longitudinal WavesLongitudinal Waves
• Compression zone: molecules are pushed together and have higher density and pressure
• Rarefaction zone: molecules are spread apart and have lower density and pressure
• travel faster in solids than gases
• water waves have both longitudinal and transverse components—a “combination” wave
Longitudinal WavesLongitudinal Waves
• amplitude (A): the greatest distance a wave displaces a particle from its average position
Periodic WavesPeriodic Waves
A = ½(ypeak - ytrough)A = ½(xmax - xmin)
• wavelength (λ): the distance from one peak (or compression zone) to the next, or from one trough (or rarefaction zone) to the next
Periodic WavesPeriodic Waves
• A wave completes one cycle as it moves through one wavelength.
• A wave’s frequency (f) is the number of cycles completed per unit of time
Periodic WavesPeriodic Waves
• wave speed (v): the speed of the disturbance
• for periodic waves:
Periodic WavesPeriodic Waves
λf = v
Sound Waves Sound Waves • longitudinal pressure
waves that come from a vibrating body and are detected by the ears
• cannot travel through a vacuum; must pass through a physical medium
Sound Waves Sound Waves • travel faster through solids
than liquids, and faster through liquids than gases
• have three characteristics:
Loudness Loudness • the interpretation your
hearing gives to the intensity of the wave
• intensity (Is): amount of power transported by the wave per unit area
• measured in W/m²
Loudness Loudness • a sound must be ten times
as intense to be perceived as twice as loud
• sound is measured in decibels (dB)
Pitch Pitch • related to the frequency • high frequency is
interpreted as a high pitch• low frequency is interpreted
as a low pitch• 20 Hz to 20,000 Hz
Quality Quality • results from combinations
of waves of several frequencies
• fundamental and harmonics• why a trumpet sounds
different than an oboe
• related to the relative velocity of the observer and the sound source
Doppler EffectDoppler Effect
• an approaching object has a higher pitch than if there were no relative velocity
• an object moving away has a lower pitch than if there were no relative velocity
• actual sound emitted by the object does not change