Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering...
-
date post
19-Dec-2015 -
Category
Documents
-
view
225 -
download
0
Transcript of Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering...
![Page 1: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/1.jpg)
Oscillations about Equilibrium
![Page 2: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/2.jpg)
Forces and elastic materials
Elastic material• Capable of recovering shape after deformation• Rubber ball versus lump of clay
Spring forces1. Applied force proportional to distance spring is
compressed or stretched2. Internal restoring force arises, returning spring to
original shape3. Restoring force also proportional to stretched or
compressed distance
![Page 3: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/3.jpg)
Forces and vibrations
• Vibration - repetitive back and forth motion
• At the equilibrium position, spring is not compressed
• When disturbed from equilibrium position, restoring force acts toward equilibrium
• Carried by inertia past equilibrium to other extreme
• Example of “simple harmonic motion”
![Page 4: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/4.jpg)
Simple Harmonic MotionA spring exerts a restoring force that is proportional to the displacement from equilibrium:
![Page 5: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/5.jpg)
Periodic Motion
Period: time required for one cycle of periodic motion
Frequency: number of oscillations per unit time
This unit is called the Hertz:
![Page 6: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/6.jpg)
Describing vibrations
• Amplitude - maximum extent of displacement from equilibrium
• Cycle - one complete vibration
• Period - time for one cycle• Frequency - number of
cycles per second (units = hertz, Hz)
• Period and frequency inversely related
![Page 7: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/7.jpg)
Simple Harmonic Motion
If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time:
It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect.
![Page 8: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/8.jpg)
Connections between Uniform Circular Motion and Simple Harmonic Motion
An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:
![Page 9: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/9.jpg)
The Period of a Mass on a Spring
The period is
![Page 10: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/10.jpg)
Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:
Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:
![Page 11: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/11.jpg)
Energy Conservation in Oscillatory Motion
This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.
![Page 12: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/12.jpg)
The Pendulum
A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).
The angle it makes with the vertical varies with time as a sine or cosine.
![Page 13: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/13.jpg)
The Pendulum
Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
![Page 14: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/14.jpg)
The Pendulum
However, for small angles, sin θ and θ are approximately equal.
![Page 15: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/15.jpg)
The Pendulum
Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:
![Page 16: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/16.jpg)
Driven Oscillations and ResonanceAn oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.
![Page 17: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/17.jpg)
Driven Oscillations and Resonance
If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.
![Page 18: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/18.jpg)
Waves
• Periodic (traveling) disturbances transporting energy• Causes
– Periodic motion disturbing surroundings
– Pulse disturbance of short duration
• Mechanical waves– Require medium for propagation
– Waves move through medium
– Medium remains in place
![Page 19: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/19.jpg)
Kinds of waves
Longitudinal waves
• Vibration direction parallel to wave propagation direction
• Particles in medium move closer together/farther apart
• Example: sound waves
• Gases and liquids - support only longitudinal waves
![Page 20: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/20.jpg)
Kinds of waves, cont.Transverse waves• Vibration direction perpendicular to
wave propagation direction• Example: plucked stringSolids - support both longitudinal and
transverse wavesSurface water waves• Combination of both• Particle motion = circular
![Page 21: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/21.jpg)
Waves in air
• Longitudinal waves only• Large scale - swinging
door creates macroscopic currents
• Small scale - tuning fork creates sound waves
• Series of condensations (overpressures) and rarefactions (underpressures)
![Page 22: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/22.jpg)
Types of Waves
Water waves are a combination of transverse and longitudinal waves.
![Page 23: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/23.jpg)
Describing waves
Graphical representation• Pure harmonic waves = sines
or cosines
Wave terminology• Wavelength
• Amplitude
• Frequency
• Period
Wave propagation speed
![Page 24: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/24.jpg)
Waves on a String
The speed of a wave is determined by the properties of the material through which it propagates.
For a string, the wave speed is determined by:
1. the tension in the string, and
2. the mass of the string.
As the tension in the string increases, the speed of waves on the string increases as well.
![Page 25: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/25.jpg)
Waves on a String
The total mass of the string depends on how long it is; what makes a difference in the speed is the mass per unit length. We expect that a larger mass per unit length results in a slower wave speed.
![Page 26: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/26.jpg)
14-2 Waves on a String
As we can see, the speed increases when the force increases, and decreases when the mass increases.
![Page 27: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/27.jpg)
Waves on a String
When a wave reaches the end of a string, it will be reflected. If the end is fixed, the reflected wave will be inverted:
![Page 28: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/28.jpg)
Waves on a String
If the end of the string is free to move transversely, the wave will be reflected without inversion.
![Page 29: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/29.jpg)
Sound Waves
Sound waves are longitudinal waves, similar to the waves on a Slinky:
Here, the wave is a series of compressions and stretches.
![Page 30: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/30.jpg)
Sound Waves
In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.
![Page 31: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/31.jpg)
Sound Waves
The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.
![Page 32: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/32.jpg)
Sound Waves
Sound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz.
Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic.
Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves.
![Page 33: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/33.jpg)
Sound IntensityThe intensity of a sound is the amount of energy that passes through a given area in a given time.
![Page 34: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/34.jpg)
Sound Intensity
Expressed in terms of power,
![Page 35: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/35.jpg)
14-5 Sound Intensity
Sound intensity from a point source will decrease as the square of the distance.
![Page 36: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/36.jpg)
Sound Intensity
When you listen to a variety of sounds, a sound that seems twice as loud as another is ten times more intense. Therefore, we use a logarithmic scale to define intensity values.
Here, I0 is the faintest sound that can be heard:
![Page 37: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/37.jpg)
Sound Intensity
The quantity β is called a bel; a more common unit is the decibel, dB, which is a tenth of a bel.
The intensity of a sound doubles with each increase in intensity level of 10 dB.
![Page 38: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/38.jpg)
The Doppler Effect
The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.
When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.
![Page 39: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/39.jpg)
The Doppler Effect
The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:
![Page 40: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/40.jpg)
The Doppler Effect
Combining results gives us the case where both observer and source are moving:
The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements.
![Page 41: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/41.jpg)
Superposition and Interference
Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.
![Page 42: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/42.jpg)
Superposition and Interference
If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right).
![Page 43: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/43.jpg)
Superposition and Interference
Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.
![Page 44: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/44.jpg)
Superposition and Interference
Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.
![Page 45: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/45.jpg)
Standing Waves
A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.
![Page 46: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/46.jpg)
Standing Waves
The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.
![Page 47: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/47.jpg)
Standing Waves
There must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible.
Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.
![Page 48: Oscillations about Equilibrium. Forces and elastic materials Elastic material Capable of recovering shape after deformation Rubber ball versus lump of.](https://reader031.fdocuments.in/reader031/viewer/2022032704/56649d405503460f94a19f67/html5/thumbnails/48.jpg)
In order for different strings to have different fundamental frequencies, they must differ in length and/or linear density.
A guitar has strings that are all the same length, but the density varies.
Standing Waves