1 / 68

# 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Simple Harmonic Motion' - dinah

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

• defined: motion that repeats at a constant rate

• equilibrium position: forces are balanced

• 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.

• for the spring:

ΣF = Fw + Fs

ΣFy = mgy + (-kΔy)

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

• 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

1

f =

T =

T

f

Simple Harmonic Motion

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

• Circular motion has many similarities to SHM.

• Their motions can be synchronized and similarly described.

T = 2π

k

Reference Circle

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

T = 2π

k

Reference Circle

• This equation is used for Example 12-1.

• The reciprocal of T gives the frequency.

• Galileo was among the first to scientifically study pendulums.

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

• 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.

• l = distance from the pendulum’s pivot point and its center of mass

• center of mass travels in a circular arc with radius l.

• forces on a pendulum at rest:

• weight (mg)

• tension in pendulum arm (Tp)

• at equilibrium when at rest

• 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

• Centripetal force adds to the tension (Tp):

Tp = Tw΄+ Fc , where:

Tw΄ = Tw = |mg|cosθ

Fc = mvt²/r

• Total acceleration (atotal) is the sum of the tangential acceleration vector (at) and the centripetal acceleration.

• The restoring forces causes this atotal.

• A pendulum’s motion does not exactly conform to SHM, especially when the amplitude is large (larger than π/8 radians, or 22.5°).

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

• SHM is approximated

T = 2π

|g|

Small Amplitude

• For small initial displacement angles:

T = 2π

|g|

Small Amplitude

• Longer pendulum arms produce longer periods of swing.

T = 2π

|g|

Small Amplitude

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

T = 2π

|g|

Small Amplitude

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

• 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

• 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.

T = 2π

|mg|l

Physical Pendulums

• period of a physical pendulum:

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

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

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

fx = -βvx

β is a friction proportionality constant

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

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

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

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

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

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

• also called the resonant frequency

• terminology:

• in phase

• pulses

• driven oscillations

• resonance

• A driven oscillator has three forces acting on it:

• restoring force

• damping resistance

• pulsed force applied in same direction as Fr

• 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

• disturbance: an oscillation in the medium

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

Waveform graphs

Vibration graphs

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

• example: spring

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

• example: the wave along a snapped string

• 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

• travel faster in solids than gases

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

• carry information and energy from one place to another

• 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

• 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

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

• have three characteristics:

• 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

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

• sound is measured in decibels (dB)

• 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

• results from combinations of waves of several frequencies

• fundamental and harmonics

• why a trumpet sounds different than an oboe

• All three characteristics affect the way sound is perceived.

• 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

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

• speed of sound changes with altitude