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Simple Harmonic Motion. Periodic Motion. defined: motion that repeats at a constant rate equilibrium position: forces are balanced . Periodic Motion. For the spring example, the mass is pulled down to y = -A and then released.

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Presentation Transcript
slide2

Periodic Motion

  • defined: motion that repeats at a constant rate
  • equilibrium position: forces are balanced
slide3

Periodic 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.
slide4

Periodic Motion

  • for the spring:

ΣF = Fw + Fs

ΣFy = mgy + (-kΔy)

slide5

Periodic Motion

  • Damping: the effect of friction opposing the restoring force in oscillating systems
slide6

Periodic Motion

  • Restoring force (Fr): the net force on a mass that always tends to restore the mass to its equilibrium position
slide7

Simple Harmonic Motion

  • defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position
slide8

Simple Harmonic Motion

  • The restoring force in SHM is described by:

Fr x = -kΔx

  • Δx = displacement from equilibrium position
slide9

Simple Harmonic Motion

  • Table 12-1 describes relationships throughout one oscillation
slide10

Simple Harmonic Motion

  • Amplitude: maximum displacement in SHM
  • Cycle: one complete set of motions
slide11

Simple Harmonic Motion

  • Period: the time taken to complete one cycle
  • Frequency: cycles per unit of time
    • 1 Hz = 1 cycle/s = s-1
slide12

1

1

f =

T =

T

f

Simple Harmonic Motion

  • Frequency (f) and period (T) are reciprocal quantities.
slide13

Reference Circle

  • Circular motion has many similarities to SHM.
  • Their motions can be synchronized and similarly described.
slide14

m

T = 2π

k

Reference Circle

  • The period (T) for the spring-mass system can be derived using equations of circular motion:
slide15

m

T = 2π

k

Reference Circle

  • This equation is used for Example 12-1.
  • The reciprocal of T gives the frequency.
slide17

Overview

  • Galileo was among the first to scientifically study pendulums.
slide18

Overview

  • The periods of both pendulums and spring-mass systems in SHM are independent of the amplitudes of their initial displacements.
slide19

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.
slide20

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.
slide21

Pendulum Motion

  • forces on a pendulum at rest:
    • weight (mg)
    • tension in pendulum arm (Tp)
  • at equilibrium when at rest
slide22

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

slide23

Restoring Force

  • Centripetal force adds to the tension (Tp):

Tp = Tw΄+ Fc , where:

Tw΄ = Tw = |mg|cosθ

Fc = mvt²/r

slide24

Restoring Force

  • Total acceleration (atotal) is the sum of the tangential acceleration vector (at) and the centripetal acceleration.
  • The restoring forces causes this atotal.
slide25

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°).
slide26

Small Amplitude

  • defined as a displacement angle of less than π/8 radians from vertical
  • SHM is approximated
slide27

l

T = 2π

|g|

Small Amplitude

  • For small initial displacement angles:
slide28

l

T = 2π

|g|

Small Amplitude

  • Longer pendulum arms produce longer periods of swing.
slide29

l

T = 2π

|g|

Small Amplitude

  • The mass of the pendulum does not affect the period of the swing.
slide30

l

T = 2π

|g|

Small Amplitude

  • This formula can even be used to approximate g (see Example 12-2).
slide31

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
slide32

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.
slide33

I

T = 2π

|mg|l

Physical Pendulums

  • period of a physical pendulum:
slide35

Damped Oscillations

  • Resistance within a spring and the drag of air on the mass will slow the motion of the oscillating mass.
slide36

Damped Oscillations

  • Damped harmonic oscillators experience forces that slow and eventually stop their oscillations.
slide37

Damped Oscillations

  • The magnitude of the force is approximately proportional to the velocity of the system:

fx = -βvx

β is a friction proportionality constant

slide38

Damped Oscillations

  • The amplitude of a damped oscillator gradually diminishes until motion stops.
slide39

Damped Oscillations

  • An overdamped oscillator moves back to the equilibrium position and no further.
slide40

Damped Oscillations

  • A critically damped oscillator barely overshoots the equilibrium position before it comes to a stop.
slide41

Driven Oscillations

  • To most efficiently continue, or drive, an oscillation, force should be added at the maximum displacement from the equilibrium position.
slide42

Driven Oscillations

  • The frequency at which the force is most effective in increasing the amplitude is called the natural oscillation frequency (f0).
slide43

Driven Oscillations

  • The natural oscillation frequency (f0) is the characteristic frequency at which an object vibrates.
  • also called the resonant frequency
slide44

Driven Oscillations

  • terminology:
    • in phase
    • pulses
    • driven oscillations
    • resonance
slide45

Driven Oscillations

  • A driven oscillator has three forces acting on it:
    • restoring force
    • damping resistance
    • pulsed force applied in same direction as Fr
slide46

Driven Oscillations

  • The Tacoma Narrows Bridge demonstrated the catastrophic potential of uncontrolled oscillation in 1940.
slide48

Waves

  • defined: oscillations of extended bodies
  • medium: the material through which a wave travels
slide49

Waves

  • disturbance: an oscillation in the medium
  • It is the disturbance that travels; the medium does not move very far.
slide50

Graphs of Waves

Waveform graphs

Vibration graphs

slide51

Types of Waves

  • longitudinal wave: disturbance that displaces the medium along its line of travel
  • example: spring
slide52

Types of Waves

  • transverse wave: disturbance that displaces the medium perpendicular to its line of travel
  • example: the wave along a snapped string
slide53

Longitudinal Waves

  • Any physical medium can carry a longitudinal wave.
  • Rarefaction zone: molecules are spread apart and have lower density and pressure
  • Compression zone: molecules are pushed together and have higher density and pressure
slide54

Longitudinal Waves

  • travel faster in solids than gases
  • water waves have both longitudinal and transverse components—a “combination” wave
slide55

Periodic Waves

  • carry information and energy from one place to another
slide56

Periodic Waves

  • amplitude (A): the greatest distance a wave displaces a particle from its average position

A = ½(ypeak - ytrough)

A = ½(xmax - xmin)

slide57

Periodic Waves

  • wavelength (λ): the distance from one peak (or compression zone) to the next, or from one trough (or rarefaction zone) to the next
slide58

Periodic 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
slide59

Periodic Waves

  • wave speed (v): the speed of the disturbance
  • for periodic waves:

λf = v

slide60

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
slide61

Sound Waves

  • travel faster through solids than liquids, and faster through liquids than gases
  • have three characteristics:
slide62

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
slide63

Loudness

  • a sound must be ten times as intense to be perceived as twice as loud
  • sound is measured in decibels (dB)
slide64

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
slide65

Quality

  • results from combinations of waves of several frequencies
  • fundamental and harmonics
  • why a trumpet sounds different than an oboe
slide66

Sound Waves

  • All three characteristics affect the way sound is perceived.
slide67

Doppler Effect

  • related to the relative velocity of the observer and the sound source
  • 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
slide68

Mach Speed

  • measurement is dependent on the composition and density of the atmosphere
  • speed of sound changes with altitude
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