Simple Harmonic Motion

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# Simple Harmonic Motion - PowerPoint PPT Presentation

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

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