Simple Harmonic Motion
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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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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.

  • Two forces are working on the mass: gravity (weight) and the spring.


Periodic Motion

  • for the spring:

ΣF = Fw + Fs

ΣFy = mgy + (-kΔy)


Periodic Motion

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


Periodic Motion

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


Simple Harmonic Motion

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


Simple Harmonic Motion

  • The restoring force in SHM is described by:

Fr x = -kΔx

  • Δx = displacement from equilibrium position


Simple Harmonic Motion

  • Table 12-1 describes relationships throughout one oscillation


Simple Harmonic Motion

  • Amplitude: maximum displacement in SHM

  • Cycle: one complete set of motions


Simple Harmonic Motion

  • Period: the time taken to complete one cycle

  • Frequency: cycles per unit of time

    • 1 Hz = 1 cycle/s = s-1


1

1

f =

T =

T

f

Simple Harmonic Motion

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


Reference Circle

  • Circular motion has many similarities to SHM.

  • Their motions can be synchronized and similarly described.


m

T = 2π

k

Reference Circle

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


m

T = 2π

k

Reference Circle

  • This equation is used for Example 12-1.

  • The reciprocal of T gives the frequency.



Overview

  • Galileo was among the first to scientifically study pendulums.


Overview

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


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

  • 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

  • forces on a pendulum at rest:

    • weight (mg)

    • tension in pendulum arm (Tp)

  • at equilibrium when at rest


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

  • Centripetal force adds to the tension (Tp):

Tp = Tw΄+ Fc , where:

Tw΄ = Tw = |mg|cosθ

Fc = mvt²/r


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

  • 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

  • defined as a displacement angle of less than π/8 radians from vertical

  • SHM is approximated


l

T = 2π

|g|

Small Amplitude

  • For small initial displacement angles:


l

T = 2π

|g|

Small Amplitude

  • Longer pendulum arms produce longer periods of swing.


l

T = 2π

|g|

Small Amplitude

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


l

T = 2π

|g|

Small Amplitude

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


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

  • 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.


I

T = 2π

|mg|l

Physical Pendulums

  • period of a physical pendulum:



Damped Oscillations

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


Damped Oscillations

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


Damped Oscillations

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

fx = -βvx

β is a friction proportionality constant


Damped Oscillations

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


Damped Oscillations

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


Damped Oscillations

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


Driven Oscillations

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


Driven Oscillations

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


Driven Oscillations

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

  • also called the resonant frequency


Driven Oscillations

  • terminology:

    • in phase

    • pulses

    • driven oscillations

    • resonance


Driven Oscillations

  • A driven oscillator has three forces acting on it:

    • restoring force

    • damping resistance

    • pulsed force applied in same direction as Fr


Driven Oscillations

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



Waves

  • defined: oscillations of extended bodies

  • medium: the material through which a wave travels


Waves

  • disturbance: an oscillation in the medium

  • It is the disturbance that travels; the medium does not move very far.


Graphs of Waves

Waveform graphs

Vibration graphs


Types of Waves

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

  • example: spring


Types of Waves

  • transverse wave: disturbance that displaces the medium perpendicular to its line of travel

  • example: the wave along a snapped string


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


Longitudinal Waves

  • travel faster in solids than gases

  • water waves have both longitudinal and transverse components—a “combination” wave


Periodic Waves

  • carry information and energy from one place to another


Periodic Waves

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

A = ½(ypeak - ytrough)

A = ½(xmax - xmin)


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


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


Periodic Waves

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

  • for periodic waves:

λf = v


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

  • travel faster through solids than liquids, and faster through liquids than gases

  • have three characteristics:


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

  • a sound must be ten times as intense to be perceived as twice as loud

  • sound is measured in decibels (dB)


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

  • results from combinations of waves of several frequencies

  • fundamental and harmonics

  • why a trumpet sounds different than an oboe


Sound Waves

  • All three characteristics affect the way sound is perceived.


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


Mach Speed

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

  • speed of sound changes with altitude


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