11-Modelling Oscillations Flashcards
What is the time for one complete swing of an oscillatior?
The time period
What does the displacement of an oscillator vary about?
It varies about an equilibrium position
What is the amplitude of an oscillation?
The greatest displacement from an equilibrium position
What is the crucial property of a pendulum?
That the time period of a complete swing does not depend on the amplitude of the oscillations
What property is shared by simple harmonic oscillators?
That the swings of a pendulum are approximately the same time
How does the displacement of a simple harmonic oscillator vary?
It varies sinusoidally (like a sine wave). It varies like a sine wave
What is angular frequency expressed in?
Radians per second
What letter represents angular frequency?
w
What is the equation of angular frequency?
Angular frequency = 2 pi / Time period
In a time period T, how much does the phase change by?
2 pi
In a time period 2T, how much does the phase change by?
4 pi
As Time period = 1/frequency, how can we also write an equation for angular frequency?
As 2 X pi X frequency
What 2 equation can be used to find x?
x = A cos2XpiXfrequencyXtime or x=A sin 2 X pi X frequency X time
What is the acceleration of a simple harmonic oscillator proportional to?
The displacement from the equilibrium postion
Where is the acceleration of a simple harmonic oscillator always directed towards?
The equilibrium postion
As the acceleration is proportional to -x, what equation can we use to find the acceleration of a simple harmonic oscillator?
a = -kx
What is the general relationship for all simple harmonic oscillators for acceleration?
a = -w squared times x a= -4pi squared X f squared X x
In a simple harmonic oscillator, when the dipacement is at its maximum, what is the velocity value?
O
In a simple harmonic oscillator, when the dipacement is at its maximum, what is the acceleration value?
At a maximum
How is an displacment-time graph and an acceleration-time graph for a simple harmonic oscillator related?
They are compleately out of phase, there are a flip image of one another
In a displacement-time graph, what is the value of the displacement when the velocity is at a maximum?
0
As the velocity of an osillator is the derivative of the displacement, what is the equation of the velocity?
V = - 2 X pi X f X A X sin2 pi f T
When a mass is placed between two rigid walls by a pair of springs and is displaced to the right, where does the force point?
Large force to the left
When a mass is placed between two rigid walls by a pair of springs and is displaced to the left, where does the force point?
Large force to the right
When a mass is placed between two rigid walls by a pair of springs and is displaced to the right, where does the acceleration point?
To the left
As acceleration = F / m, and F=-kx, what does the acceleration also equal?
a = -kx / m
What does the time period of an oscillator equal in terms of mass and spring constant?
T = 2 pi root(m/k)
What is T squared proportional to?
The mass
In a pendulum, what direction does the restoring force act?
In the direction of the equilibrium position
For small displacements, what is the restoring force?
The horizontal component of the tenstion
As the restoring force F = -Tx / L, what does the acceleration equal?
F = -mgx / L a = -gx / L
For small displacements in a pendulum, what is the accleration proportional to?
negative displacement, so the pendulum will oscillate with simple harmonic motion
What does the time period of a pendulum equal in terms of L and g?
Time period = 2 pi Root L/g
What is constant in a free oscillator?
They have a CONSTANT amplitude
What will a freely oscillating pendulum do?
Swing at a natural frequency with the same amplitude all the time
What is resonance?
A large amplitude oscillation produced when the driving frequency matches the natural frequency of the system
What does the elastic potential energy of an oscillator equal?
1/2 k A^2, where A is the maximum amplitude of the oscillation
When the displacement of an oscillator is at a maximum, where is all of its energy?
Stored as elastic potential energy 1/2kA^2
When the displacement of an oscillator is 0, where is all of its energy stored?
As potential energy
At any time, what is the total energy of an oscillator equal to?
The elastic potential energy + The kinetic energy
Why do the oscillators we observe(e.g. pendulum) not oscillate indefinitely?
Because energy leaks away from them due to damping
What is damping?
The action of forces such as friction and air resistance
What is light damping?
When the maximum displacement of the oscillation is reduced in each oscillation, but the time period is roughly constant
What is critical damping?
When the oscillator stops at the equilibrium position without completing a cycle
When does resonance occur?
When the frequency of a driving oscillation matches the natural frequency, increasing the amplitude of the oscillations until the energy losses per cycle are equal to the energy provided by the driver
In a frequency-displacement graph, what does low damping do to the curve?
It causes a large maximum respsonse with a sharp resonance peak and a narrow range at 1/2 peak response
In a frequency-displacement graph, what does large damping do to the curve?
Smaller maximum response, with a broader resonance peak at 1/2 peak response
When will a system exhibit simple harmonic motion?
If the acceleration is proportional to minus the displacement, then a system will exhibit SHM
For example, mass on spring oscillator F = ma = - kx, where x is displacement and a is acceleration
will exhibit SMH as a = -(k/m) x
When sound waves of a particular frequency are directed at a wine glass, the glass shatter, why does this happen?
All objects/systems have a set of natural frequencies, the frequency at which the object will oscillate freely if set in motion. When the object is oscillated at the natural frequency by some external source of energy, resonance occurs: energy is absorbed very efficiently and the amplitude of the oscillations increases considerably.. In the wine glass case, when the speaker frequency (driving frequency) matches one of the natural frequencies of the glass, energy is absorbed very efficiently, large amplitude oscillations result which eventually lead to the glass shattering
