Oscillations
Oscillating Systems
Oscillating systems often undergo periodic motion, where the motion of the object will repeat at regular intervals. One such example of this is simple harmonic motion.
Simple Harmonic Motion
This is where a force acting on an object is proportional to the position of the object, relative to some equilibrium position (where this force is also directed such a
position). In the case of a spring, this force can be quantified withThe acceleration of a particle undergoing SHM is not constant. However, the
.acceleration of the mass can be determined using Newton’s second law:
It is this acceleration equation which defines a system undergoing SHM. From this equation, it is clear that the acceleration is proportional to the displacement of the object. Furthermore, the direction of this acceleration is opposite to the
displacement from the equilibrium point. Note that when a block completes one full oscillation, it has moved 4A, where A is the amplitude of the oscillation. This is
because it must move from max displacement, to neutral, to minimum displacement, and then all the way back to max displacement.
When a block is hung from a vertical spring, its weight will cause the spring to stretch to some equilibrium point. If the resting position of the spring is defined as y=0, then it is clear that:
When representing SHM mathematically, it is useful to choose the x‐axis as the one in which the oscillation occurs. Applying Newton’s second law:
Furthermore, if
, then2 where f is the frequency of the oscillations.
Clearly, the period and frequency of the motion are very dependent upon the mass of the particle and the force constant of the spring. Evidently, the frequency is proportional to the spring constant, but inversely proportional to the mass.
In representing SHM graphically, an equation which can define SHM is:
.cos
.sin
Note that in the above equations:
•
2 , 2
•
is the angular frequency (units of radian/s)• A is the amplitude of the motion, or the maximum displacement the particle achieves
•
is the phase constant, or initial phase angleNote that the phase of the motion is given by
.cos
Since
, it is possible to determine the value of A and
byapplying the condition t=0 to the equation.
Because of the nature of the sine and cosine functions oscillating between ‐1 and 1, the maximum values of velocity and acceleration are given by:
.
Another important fact is that the velocity function is typically 90° out of phase with the displacement function, while the acceleration is typically 180° out of phase with the displacement function.
Energy of Collisions
Assuming that the spring‐mass system is moving on a frictionless surface, the kinetic energy can be found by:
. sin
. sin ,
The elastic potential energy can be found by:
. cos
Hence the total energy of the system can be given by:
.
Note that this total energy given above remains constant, and that it is proportional to the square of the amplitude. The energy is being continuously transferred
between the potential and kinetic energy of the block.
Molecular Model of Simple Harmonic Motion
If the atoms in a molecule do not move too far apart, the force between them can be modelled as if there were springs between the atoms. Hence the potential energy
Uniform Circular Motion
There exists a clear link between simple harmonic motion and circular motion. If a disk with a knob is rotated and viewed from above, the knob appears to move back and forth as though it were undergoing simple harmonic motion. This becomes evident in the diagram below:
• The particle moves along the circle with a constant angular velocity ω.
• OP means an angle θ with the x‐axis.
• At time t, the angle θ ill bew
•
.cos .sin
.sin
• Hence
which t equirement for SHMQu evidently, simple h ne can be represented by
he r
ite armonic motion along a straight li
a projection of uniform circular motion along the diameter of a reference circle. This allows uniform circular motion to be considered a combination of 2 simple harmonic motions:
• One along the x‐axis
• The other along the y‐axis Where the two differ in phase by 90°.
Pendulums
The motion of a simple pendulum in the vertical plane is driven by gravitational force. This motion is very similar to that of particles undergoing simple harmonic
re the angle of the pendulum is small.
motion, whe
From the diagram above, the forces acting on the bob are tension (T) and weight force (mg). The tangential component of gravitational force is a restoring force, given by
sin
. The arc length i given bys
.Hence
and
• In the tangential direction,
sin
• Hence the motion is simple harmonic with
• The function θcan be writtenas:
cos
• The angular frequency is given by:
• The period is given by:
2
is longer or is in a field of lower gravitational plete 1 period.
Damped Oscillations
Based on the above, if the pendulum force, it will take longer to com
In many real systems, non‐conservative forces are present, such as friction, air resistance and viscosity. In such cases, the mechanical energy of the system will diminish over time, and hence the motion is described as being damped. Below is the graph of a damped oscillation:
Based on this diagram, the amplitude of the oscillation decrease with time, where the blue line is representative of the envelope of the motion.
An example of damped motion is when an object is attached to a spring and submerged in a viscous liquid. The retarding force can be expressed as
,where b is a positive constant, known as the damping constant. From Newton’s second law,There are three types of damping, shown by the graph below of position
versus time:
.
(a) Is an underdamped oscillator (b) Is a critically damped oscillator (c) Is an overdamped oscillator
Forced Oscillations
Because of the concept of damped oscillations, the process of forced oscillations was developed. This is where loss of energy in a damped system is compensated for by applying an external force, for instance periodically pushing a pendulum. The amplitude of the motion remains constant if the energy input per cycle exactly equals the mechanical energy lost by the system in each cycle from resistive forces.
Generally speaking:
sin
sin
W here xtis the transient force, and xssisth steady state force.e
While the motion of an object undergoing simple harmonic motion is given by
.cos
, theamplitude o the driven oscillation is given by: f ⁄
Where
is the natural frequency of the undamped oscillator, given by .
Quite clearly, the amplitude f the motion is dependent on the frequency.o Resonance
When th frequency of the driving force is near the natural frequency, (oe r
),
there will be an increase in amplitude. This dramatic increase is referred to as resonance, while the natural frequency of the system is also referred to as the resonance frequency of the system.
The maximum peak of resonance occurs when the driving frequency equals the natural frequency. The amplitude then increases with decreased damping.
Consequently, the curve (shown below) will broaden as damping increases. The shape of this curve is hence dependent on b.
Wave Motion
ropagation of a Disturbance P
There are two major types of waves. They are mechanical waves (where some medium must be disturbed and the wave propagates through this medium) and electromagnetic waves. The energy of any disturbance is transferred over distance, but not the matter (the particles move back and forth or up and down).
A pulse on a rope can be created by flicking a rope under tension in direct This pulse can then travel through the rope, causing the pulse to have a definite height and speed of propagation (which is typically unique for each medium).
ion.
If the ope is
Longitudinal waves are those where the travelling wave causes the particles to move parallel to the direction of the wave.
Complex waves can also be formed, such as water waves. These can exhibit a combination of the properties of transverse and longitudinal waves.
r continuously flicked, a periodic disturbance can be formed as a wave.
Waves can travel either parallel or perpendicular to the motion of the
disturbance. Transverse waves are comparable to a sine curve, and the particles are displaced perpendicular to the direction of propagation.
Travelling Waves
A travelling wave or pulse can be represented by an equation which gives the disturbance as a function of displacement and time.
w:
In the diagram belo
Evidently, the shape of the pulse at t=0 can be represented by
,0
. Thisdescribes the transverse position, y, of the element of the string located at each value of x at t=0. The speed of the pulse is v and in time t, will travel a distance of vt.
At time=t, the shape is represented by
. This is b ecause the pulse moves down the rope and isn’t always sinusoidal. Hence, to achieve the original curve, the displacement must be subtracted.Consequently, for a curve travel ling to the right,
,
.Meanwhile, for a curve travelling to the left,
,
. As such, y(x,t) is the wave function, representing the y‐coordinate of any element located at position x at any time t. For a fixed t, it is called the waveform, asit defines the curve t anya specific time.Properties of Waves
The amplitude of a wave is the maximum displacement from the equilibrium position.
The wavelength is the distance between any 2 identical points on adjacent waves.
ncy is the number of waves per second.
The period is the time taken for one wavelength.
The freque Wave Function
This is given by
, .sin
. It describes the motion of a wave moving in the positive x direction with a speed of v. Conversely,, .sin
describes motion in the negative x direction.Wave Speed
While the transverse speed of a wavecan begiven by
,. .cos
, this waveitself, given by:
is different to the speed of the The speed of a wave on the string can also be given by:
/
Power and Intensity in Wave Motion
The kinetic energy of awave is given by
Δ Δ , whereΔ Δ
.
The total kinetic energy in a wave is given by
. This same equation can also be used to define potential ener gy in a wave. Hence, the total energyof a wave is given by
.
The power associated with a wave is given by:
red, amplitude squared and wave speed.Principle of Superposition
y Hence power is proportional to frequency squa
The principle of superposition involves the energy of a series of waves adding at an given point; hence their combination is the algebraic sum of their values. However, the principle of superposition can only apply to linear waves, where the amplitude is smaller than the wavelength.
The principle states that travelling waves can pass through one another without being destroyed or altered. This results in their combination in a resultant wave known as interference.
Interference of Waves
There two types of interference: constructive and destructive. Constructive
interference is where the displacements caused by the two pulses are in the same direction and hence the amplitude of the resulting wave is greater than either wave.
Destructive interference is where the displacements are in opposite directions and hence the amplitude of the resulting wave is less than either wave. In effect the waves cancel each other out to a certain extent.
Mathematically:
1 .sin
2
.sin
1 2 2
.cos sin
Hence the resulting wave is sinusoidal, with the same frequency and wavelength, but with an amplitude of
2.cos
and a phase of
. Hence during perfect constructivend the wa
interference, the phase is zero (where both waves are in phase everywhere) and perfect destructive interference, the phase is an odd multiple of π (a ves cancel one another out) resulting in an
consequently, the amplitude is 2A. In
amplitude of 0. Generally speaking, if the phase is between 0 and an odd multiple of π the amplitude is between 0 and 2A and the functions will continue to add.
Standing Waves
If 2 waves have the same amplitude, wavelength and frequency, and travel in the opposite direction one another with the equations:
.sin
.sin
Where the different signs indicate the different directions, they will interfere according to the superposition principle described in the equation below:
This is the wave function of a standing wave. Evidently, there is no 2.sincos
component, because the wave is no longer travelling. This becomes obvious when a standing wave is observed, where there is no sense of motion as the particles appear to oscillate around a node.
A node occurs at a point of zero amplitude. In other words:
sin sin
An antinode occurs at a point of maximum displacement (2A). In other words: 0 . This corresponds to when ,
.
sin 1
. This correspondsto where
, where n is a positive odd number.Evidently, the amplitude of an individual wave is A, while the amplitude of dergoing SHM is
2.sin
, the amplitude of a standingany particle un while wave
is 2A. However, the amplitude of any particle in a standing wave is given by the same equation as for SHM.
Key to standing waves isthe fact that at points of maximal displacement, the particles are momentarily stationary, while at zero displacement, the particles have differing instantaneous velocities as s some particles move up while others move down.
For a standing wave to be established, the end points must be nodes abd fixed (hence have zero displacement). This will then result in a set of normal modes (or a series of antinodes), where each normal mode refers to the number of
antinodes. The relationship between normal modes and the wavelength viewed is given by
, where n is the nth normal mode of oscillation. Since
, the
natural frequency is given by
. If the string is under tension,
this can be givenby
, if a freely hanging weight is used. In a standing wave,
the number of nodes is one greater than that of antinodes and it will show symmetry
about the midpoint of the strin
Quantisation
g.
This is where only certain frequencies of oscillation are allowed. This is particularly common where boundary conditions must be met.
es Harmonic Seri
The natural frequency corresponds to
1
, and is the lowest frequency. Thefrequencies of the reamining natural modes are integer multiples of the funamental frequency, and will form a harmonic series. The normal modes may be called
Sound Waves
peed of Sound
he speed of sound is dependent upon the properties of the medium it is travelling S