Topic 4: Oscillations and waves Flashcards
Describe examples of oscillations
- Mass moving between two horizontal springs
- Mass moving on a vertical spring
- Simple pendulum
- Buoy bouncing up and down in water
- An oscillating ruler as a result of one end being displaced while the other is fixed
Describe where the kinetic energy and potential energy store of a mass moving between two horizontal springs is.
Kinetic energy: Moving mass
Potential energy: Elastic potential energy in the springs
Describe where the kinetic energy and potential energy store of a mass moving on a vertical spring is.
Kinetic energy: Moving mass
Potential energy: Elastic potential energy in the springs and gravitational potential energy
Describe where the kinetic energy and potential energy store of a simple pendulum is.
Kinetic energy: Moving pendulum bob
Potential energy: Gravitational potential energy of bob
Describe where the kinetic energy and potential energy store of a buoy bouncing up and down in water is.
Kinetic energy: Moving buoy
Potential energy: Gravitational potential energy of buoy and water
Describe where the kinetic energy and potential energy store of an oscillating ruler as a result of one end being displaced while the other is fixed is.
Kinetic energy: Moving sections of the ruler
Potential energy: Elastic potential energy of the bent ruler
Define: displacement (in terms of SHM).
The instantaneous distance of the moving object from its mean position in a specified direction.
What is the standard index measurement for displacement?
metres, m
What is the symbol for displacement (in terms of SHM)?
x
Define: amplitude (in terms of SHM).
The maximum displacement from the mean position.
What is the standard index measurement for amplitude?
metres, m
What is the symbol for amplitude (in terms of SHM)?
A
Define: frequency (in terms of SHM).
The number of oscillations completed per unit time.
What is the standard index measurement for frequency?
number of cycles per second or Hertz, Hz
What is the symbol for frequency (in terms of SHM)?
f
What is the defining equation for frequency?
Where:
f is the frequency in Hz
T is the period in seconds
Define: period (in terms of SHM)
The time taken for one complete oscillation.
What is the standard index measurement for period?
seconds, s
What is the symbol for period (in terms of SHM)?
T
What is the defining equation for period (SHM)?
Where:
T is period in seconds
f is frequency in Hz
Define: phase difference
A measure of how in phase different particles are.
When are particles said to be in phase?
If they are moving together.
When are particles said to be out of phase?
If they are not moving together.
What is the standard index measurement for phase difference?
degrees, °
or
radians, rad
What is the symbol for phase difference?
phi, ϕ
When are particles said to be completely out of phase?
at 180º or π rad
Define: simple harmonic motion
The motion that takes place when the acceleration, a, of an object is always directed towards, and is proportional to, its displacement from a fixed point. This acceleration is caused by a restoring force that must always be pointed towards the mean position and also proportional to the displacement from the mean position.
What is the defining equation for SHM?
a = -ω2x
Where:
- a* is acceleration in m s-2
- ω* is the angular frequency in rad s-1
- x* is the displacement in m
- ω2 is the gradient of the line
The negative sign in the equation shows that acceleration always occurs in the direction towards the mean position.
What is the defining equation for the angular frequency?
T = period of P = time taken to turn 360º (2π radians)
f = frequency of P = 1 / T
ω = angular velocity of P = (angle turned through) over (time taken)
Therefore,
ω = 2π/T = 2πf
Describe the interchange between kinetic energy and potential energy during SHM
In SHM, the total energy is interchanged between kinetic energy and potential energy. If no resistive forces that dissipate energy act on the motion, the total energy is constant and the oscillation is said to be undamped.
Potential energy increases as we move away from the equilibrium position and kinetic energy decreases. As we come closer to the equilibrium position its vice versa. Potential energy can be expressed as a sine curve, kinetic energy as a cosine curve.
Define: damping
Involves a frictional force that is always in the opposite direction to the direction of motion of the oscillating particle. As the particle oscillates, it does work against this resistive (or dissipative) force and so the particle loses energy.
total energy of the particle ∝ (amplitude)<span>2</span>
This means that the amplitude decreases exponentially with time.
Define: light damping
The resistive force is small, so a small fraction of the total energy is removed with each cycle and hence the amplitude decreases. The time period of the oscillations is not affected and the oscillations continue for a significant number of cycles. The time taken for the oscillation to die out can be long.
Define: critical damping
Involves an intermediate value for resistive force such that the time taken for the particle to return to zero displacement is a minimum. There is no overshoot. Examples = electric meters with moving pointers and door closing mechanisms.
Define: heavy damping
Involves large resistive forces and can completely prevent oscillations from taking place. Time taken for the particle to return to zero displacement can be long.
Define: natural frequency of vibration
When the system is temporarily displaced from its equilibrium position and the system oscillates as a result.
Give an example of natural frequency of vibration.
When the rim of a wine glass is tapped.
Define: forced oscillations.
It is possible to force a system to oscillate at any frequency by subjecting it to a changing force that varies with the chosen frequency. This periodic driving frequency must be provided from outside the system. When the driving frequency is first applied, a combination of natural and forced oscillations take place, producing complex transient oscillations. Once the amplitude of the transient oscillations dies down, a steady condition is achieved.
What are the conditions of forced oscillations?
- the system oscillates at the driving frequency
- the amplitude of the forced oscillations is fixed. Each cycle, energy is dissipated as a result of damping and the driving force does not work on the system. The overall result is that the energy of the system remains constant.
- the amplitude of the forced oscillations depends on:
- the comparative values of the natural frequency and the driving frequency
- the amount of damping present in the system.
Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency of vibration.
The amplitude of the forced oscillation depends on comparative values of the natural frequency and the driving frequency. In addition it also depends on the amount of damping present.
Define: resonance
Occurs when a system is subject to an oscillating force at exactly the same frequency as the natural frequency of oscillation of the system.
Describe resonance in vibrations in machinery
When in operation, the moving parts of machinery provide regular driving forces on the other sections of the machinery. If the driving frequency is equal to the natural frequency, the amplitude of a particular vibration may get dangerously high (e.g. at a particular engine speed a truck’s rear view mirror can be seen to vibrate).
Describe resonance in quartz oscillators.
A quartz crystal feels a force if placed in an electric field. When the field is removed, the crystal will oscillate. Appropriate electronics are added to generate an oscillating voltage from the mechanical movements f the crystal and this is used to drive the crystal at its own natural frequency. These devices provide accurate clocks for microprocessor systems.
Describe resonance in microwave generators.
Microwave ovens produce electromagnetic waves at a known frequency. The changing electric field is a driving force that causes all charges to oscillate. The driving frequency of the microwaves provides energy, which means that water molecules in particular are provided with kinetic energy and the temperature increases.
Define: wave pulse.
Involves one oscillation.
Define: continuous progressive (travelling wave)
Involves a succession of individual oscillations.
Outline characteristics of wave motion.
- transfer energy from one place to another
- do so without a net motion of the medium through which they travel
- involve oscillations (vibrations) of one sort or another.
- oscillations are often SHM
Define: transverse waves
Have a crest and a trough. The oscillations are at right angles to the direction of energy transfer. Cannot be propagated through fluids.
Define: longitudinal waves
Oscillations are parallel to the direction of energy transfer.
Describe four examples of transverse waves.
- Water ripples - a floating object gains an ‘up and down’ motion
- Light waves - the back of the eye (the retina) is stimulated when light is received.
- Earthquake waves - buildings collapse during an earthquake.
- Waves along a stretched rope - a sideways pulse will travel down a rope that is held taut between two people.
Describe three examples of longitudinal waves.
- Soundwaves - the sound received at an ear makes the eardrum vibrate
- Compression waves down a spring - a compression pulse will travel down a spring that is held taut between two people.
- Earthquake waves - buildings collapse during an earthquake.
Define: wave front
Highlight parts of the waves that are moving together.
Define: ray
Highlight the direction of energy transfer.
Define: crest
The top of the wave.
Define: trough
The bottom of the wave
Define: compression
A high pressure point on the wave
Define: rarefaction
A low pressure point on the wave.
Define: displacement (waves)
Measures the change that has taken place as a result of a wave passing a particular point. Zero displacement refers to the mean position. For mechanical waves, the displacement is the distance that the particle moves from its undisturbed position.
Symbol for displacement (waves)
x
Define: amplitude (waves)
Maximum displacement from the mean position. If the wave does not lose any of its energy, its amplitude is constant.
Symbol for amplitude (waves)
A
Define: period (waves)
Time taken for one complete oscillation. It is the time taken for one complete wave to pass any given point.
Symbol: period (waves)
T
Define: frequency (waves)
Number of oscillations that take place in one second.
Define: wavelength
The shortest distance along the wave between two points that are in phase.
Symbol: wavelength
λ
Define: wave speed
Speed at which the wave fronts pass a stationary observer.
Symbol: wave speed
v
Define: intensity (waves)
Power per unit area that is received by the observer.
intensity ∝ amplitude2
Standard index measurement for intensity (waves)
W m-2
Displacement-time graph transverse and longitudinal wave.
Displacement-position graph transverse and longitudinal wave.
Derive and apply the relationship between wave speed, wavelength and frequency
Time taken for one complete oscillation is the period of the wave, T.
In this time, the wave pattern will have moved on by one wavelength, λ.
This means that the speed of the wave must be: v = λ/T
Since
1/T = f
- v* = λ X 1/T
- v* =* λ* X f
- v = λ f*
Wavelength of gamma rays
10-10 and smaller m
Wavelength of x-rays
10-9 – 10-10 m
Wavelength of ultraviolet rays
10-7 – 10-9 m
Wavelength of visible light
10-6 – 10-7 m
Wavelength of infrared
10-3 – 10-6 m
Wavelength of microwaves
10-1 – 10-3
Wavelength of radio waves
105 – 10-1
What speed do electromagnetic waves travel at in a vacuum?
Same speed
What happens when a wave meets a boundary?
It is partially reflected and partially transmitted.
Diagram for reflection of light
State Snell’s Law.
The ratio (sin i) / (sin r) is a constant for a given frequency.
Equation for refractive index.
Define: refractive index
A measure of the change in speed of the wave as it passes between two mediums.
Diagram for refraction.
What happens to a light ray when it moves from a less dense medium to a more dense one?
Wave speed is greatest in the less dense medium
Wavelength is greatest in the less dense medium
Frequency is the same
The reflected pulse becomes inverted when a wave in a less dense medium is heading towards a boundary with a more dense medium
The amplitude of the incident pulse is always greater than the amplitude of the reflected pulse.
Less dense -> more dense: light bends towards the normal
Diffraction diagram where slit is bigger than wavelength
Diffraction diagram where slit is about the same size as the wavelength
Define: superposition.
The overall disturbance at any point and at any time where the waves meet is the vector sum of the disturbances that would have been produced by each of the individual waves.
Explain constructive interference.
Explain destructive interference.
What is the path difference in constructive interference?
n λ
What is the phase difference in constructive interference?
Zero
What is the path difference in destructive interference?
(n + ½ ) λ
What is the phase difference in destructive interference?
180° / π radians
Apply the principle of superposition to determine the resultant of two waves
Add two waves.
