 # Simple Harmonic Motion

## Presentation on theme: "Simple Harmonic Motion"— Presentation transcript:

Simple Harmonic Motion

Periodic Motion defined: motion that repeats at a constant rate
equilibrium position: forces are balanced These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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. These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

Periodic Motion for the spring: ΣF = Fw + Fs ΣFy = mgy + (-kΔy)
These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

Periodic Motion Damping: the effect of friction opposing the restoring force in oscillating systems These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

Periodic Motion Restoring force (Fr): the net force on a mass that always tends to restore the mass to its equilibrium position These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

Simple Harmonic Motion
defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position We must realize that we do this—we pre-suppose. There must be some starting point.

Simple Harmonic Motion
The restoring force in SHM is described by: Fr x = -kΔx We must realize that we do this—we pre-suppose. There must be some starting point. Δx = displacement from equilibrium position

Simple Harmonic Motion
Table 12-1 describes relationships throughout one oscillation We must realize that we do this—we pre-suppose. There must be some starting point.

Simple Harmonic Motion
Amplitude: maximum displacement in SHM Cycle: one complete set of motions We must realize that we do this—we pre-suppose. There must be some starting point.

Simple Harmonic Motion
Period: the time taken to complete one cycle Frequency: cycles per unit of time 1 Hz = 1 cycle/s = s-1 We must realize that we do this—we pre-suppose. There must be some starting point.

Simple Harmonic Motion
Frequency (f) and period (T) are reciprocal quantities. We must realize that we do this—we pre-suppose. There must be some starting point. f = T 1 T = f 1

Reference Circle Circular motion has many similarities to SHM.
Their motions can be synchronized and similarly described. These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

Reference Circle The period (T) for the spring-mass system can be derived using equations of circular motion: These ?’s are totally beyond the realm of science. Are they important? How do we answer them? T = 2π k m

Reference Circle This equation is used for Example 12-1.
The reciprocal of T gives the frequency. These ?’s are totally beyond the realm of science. Are they important? How do we answer them? T = 2π k m

Periodic Motion and the Pendulum

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 Fr = Tp + mg
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 Tp = Tw΄+ Fc , where: Tw΄ = Tw = |mg|cos θ Fc = mvt²/r
Centripetal force adds to the tension (Tp): Tp = Tw΄+ Fc , where: Tw΄ = Tw = |mg|cos θ 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

Small Amplitude For small initial displacement angles: T = 2π |g| l

Small Amplitude Longer pendulum arms produce longer periods of swing.

Small Amplitude The mass of the pendulum does not affect the period of the swing. T = 2π |g| l

Small Amplitude This formula can even be used to approximate g (see Example 12-2). T = 2π |g| l

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.

Physical Pendulums period of a physical pendulum: T = 2π |mg|l I

Oscillations in the Real World

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.

β is a friction proportionality constant
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

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 actual sound emitted by the object does not change 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

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