Simple Harmonic Motion. Periodic Motion defined: motion that repeats at a constant rate equilibrium position: forces are balanced

Simple Harmonic Motion


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• for the spring:

ΣF = Fw + FsΣFy = mgy + (-kΔy)

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



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



• The restoring force in SHM is described by:

Fr x = -kΔx• Δx = displacement from

equilibrium position



• Table 12-1 describes relationships throughout one oscillation



• Amplitude: maximum displacement in SHM

• Cycle: one complete set of motions



• Period: the time taken to complete one cycle

• Frequency: cycles per unit of time• 1 Hz = 1 cycle/s = s-1



• 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

Periodic Motion and the Pendulum

Periodic Motion and the Pendulum

Overview Overview • Galileo was among the first

to scientifically study pendulums.

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 • For small initial

displacement angles:

T = 2π|g|l

Small Amplitude Small Amplitude • Longer pendulum arms

produce longer periods of swing.

T = 2π|g|l

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.

Physical Pendulums Physical Pendulums • period of a physical

pendulum:

T = 2π|mg|lI

Oscillations in the Real WorldOscillations in the Real World

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.

WavesWaves

• 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

Graphs of WavesGraphs of Waves

Waveform graphs

Vibration graphs

• 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

• carry information and energy from one place to another

Periodic WavesPeriodic Waves

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


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


• 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


• wave speed (v): the speed of the disturbance

• for periodic 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

Sound Waves Sound Waves • All three characteristics

affect the way sound is perceived.

• 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

• measurement is dependent on the composition and density of the atmosphere

• speed of sound changes with altitude

Mach SpeedMach Speed

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Simple Harmonic Motion. Periodic Motion defined: motion that repeats at a constant rate equilibrium position: forces are balanced