oscillations about equilibrium. forces and elastic materials elastic material capable of recovering...
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Oscillations about Equilibrium
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
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”
Simple Harmonic MotionA spring exerts a restoring force that is proportional to the displacement from equilibrium:
Periodic Motion
Period: time required for one cycle of periodic motion
Frequency: number of oscillations per unit time
This unit is called the Hertz:
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
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.
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:
The Period of a Mass on a Spring
The period is
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:
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.
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.
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).
The Pendulum
However, for small angles, sin θ and θ are approximately equal.
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:
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.
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.
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
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
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
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)
Types of Waves
Water waves are a combination of transverse and longitudinal waves.
Describing waves
Graphical representation• Pure harmonic waves = sines
or cosines
Wave terminology• Wavelength
• Amplitude
• Frequency
• Period
Wave propagation speed
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.
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.
14-2 Waves on a String
As we can see, the speed increases when the force increases, and decreases when the mass increases.
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:
Waves on a String
If the end of the string is free to move transversely, the wave will be reflected without inversion.
Sound Waves
Sound waves are longitudinal waves, similar to the waves on a Slinky:
Here, the wave is a series of compressions and stretches.
Sound Waves
In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.
Sound Waves
The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.
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.
Sound IntensityThe intensity of a sound is the amount of energy that passes through a given area in a given time.
Sound Intensity
Expressed in terms of power,
14-5 Sound Intensity
Sound intensity from a point source will decrease as the square of the distance.
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:
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.
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.
The Doppler Effect
The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:
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.
Superposition and Interference
Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.
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).
Superposition and Interference
Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.
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.
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.
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.
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.
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