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Physics 1025F Vibrations & Waves. OSCILLATIONS. Dr. Steve Peterson [email protected]. Chapter 11: Vibrations and Waves. Periodic motion occurs when an object vibrates or oscillates back and forth over the same path. Periodic Motion. - PowerPoint PPT Presentation
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1UCT PHY1025F: Vibrations & Waves
Physics 1025FVibrations & Waves
Dr. Steve [email protected].
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OSCILLATIONS
2UCT PHY1025F: Vibrations & Waves
Chapter 11: Vibrations and Waves
Periodic motion occurs when an object vibrates or oscillates back and forth over the same path
3UCT PHY1025F: Vibrations & Waves
Periodic motion, processes that repeat, is one of the important kinds of behaviours in Physics
Periodic Motion
4UCT PHY1025F: Vibrations & Waves
Equilibrium position – position where net force is zeroRestoring force – force acting to restore equilibriumOscillation – periodic motion governed by a restoring force
Equilibrium and Oscillation
5UCT PHY1025F: Vibrations & Waves
A graph or motion that has the form of a sine or cosine function is called sinusoidal. A sinusoidal oscillation is called simple harmonic motion (SHM)
Equilibrium and Oscillation
6UCT PHY1025F: Vibrations & Waves
SHM is characterised by…
Amplitude A: maximum distance of object from equilibrium position
Period T: time it takes for object to complete one complete cycle of motion; e.g. from x = A to x = −A and back to x = A
Frequency ƒ: number of complete cycles or vibrations per unit time
Displacement x: is the distance measuredfrom the equilibrium point
Simple Harmonic Motion
𝑇=1𝑓
7UCT PHY1025F: Vibrations & Waves
SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s Law- (i.e. force proportional to displacement and always directed
towards equilibrium position)
Not all periodic motion over the same path can be classified as SHMInitially, we will look at the horizontal mass-spring system as a representative example of SHM
Simple Harmonic Motion
8UCT PHY1025F: Vibrations & Waves
x is the displacement of the mass m from its equilibrium position (x = 0 at the equilibrium position)The negative sign indicates that the force is always directed opposite to displacement (i.e. restoring force towards equilibrium)
Hooke’s Law Review
k is the spring constant
spring force
sF kx
9UCT PHY1025F: Vibrations & Waves
A prosthetic leg contains a spring to absorb shock as the person is walking. If an 80 kg man compresses the spring by 5 mm when standing with his full weight on the prosthetic, what is the spring constant (k)?
How far would the spring compress for a 100 kg man?
Example: Hooke’s Law
10UCT PHY1025F: Vibrations & Waves
From Newton II, for a mass-spring system:
For a horizontal mass-spring system & all other cases of SHM, acceleration depends on positionSince acceleration is not constant in SHM standard “equations of motion” cannot be applied
Horizontal Mass on a Spring
netF kx maka xm
11UCT PHY1025F: Vibrations & Waves
V&S Example 13.2: A 0.350-kg object attached to a spring of force constant 1.30 x 102 N/m is free to move on a frictionless horizontal surface. If the object is released from rest at x = 0.10 m, find the force on it and its acceleration at x = 0.10 m, x = 0.05 m, x = 0 m, x = -0.05 m, and x = -0.10 m.
Example: SHM
12UCT PHY1025F: Vibrations & Waves
SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s LawFor a pendulum, the restoring force is
Does this motion qualify as simple harmonic motion?A. YesB. No
The Simple Pendulum
sinmgFnet
13UCT PHY1025F: Vibrations & Waves
A pendulum only exhibits SHM if it is restricted to small-angle oscillations (< 10°). For such small angles (in radians), we get the small-angle approximation, where
The Simple Pendulum
sin
14UCT PHY1025F: Vibrations & Waves
Using the small-angle approximation, the restoring force becomes
The pendulum displacement (the arclength s) is proportional to the anglegiving
The Simple Pendulum
mgmgFnet sin
Ls
sLmg
LsmgFnet
Linear restoring force
15UCT PHY1025F: Vibrations & Waves
The potential energy of a spring (Section 6-4):
The kinetic energy of the mass (Section 6-3):
Therefore the total energy of the spring-mass system is:
This total energy is conserved (assuming no friction, etc…)
Energy in a Mass-Spring System
16UCT PHY1025F: Vibrations & Waves
• Energy is all PE when
• Total energy is• Energy is all KE when
• Total energy is conserved, so
Energy in Simple Harmonic Motion
17UCT PHY1025F: Vibrations & Waves
A 4.0 kg mass attached to a horizontal spring with stiffness 400 N/m is executing simple harmonic motion. When the object is 0.1 m from equilibrium position it moves with 2.0 m/s. • Calculate the amplitude of the oscillation
• Calculate the maximum velocity of the oscillation
Example: Energy of Spring
18UCT PHY1025F: Vibrations & Waves
Conservation of energy allows the calculation of the velocity of an object attached to a spring at any position in its motion:
Energy in Simple Harmonic Motion
19UCT PHY1025F: Vibrations & Waves
• The velocity of the rotating object is equal to the maximum velocity of the object in SHM.
• The circle circumference is and the rotation time is , thus
• From energy, we have: • Combining them gives:
OR
SHM and Uniform Circular Motion
20UCT PHY1025F: Vibrations & Waves
Simple Harmonic Motion• The position, velocity and acceleration are all sinusoidal• The frequency does not depend on the amplitude• The object’s motion can be written as
2 ( ) cos ty t AT
max2 ( ) sinytv t v
T
max2 ( ) cosyta t a
T
21UCT PHY1025F: Vibrations & Waves
Giancoli Example 11-7: The displacement of an object is described by the following equation, where x is in meters and t is in seconds: .Determine the oscillating object’s (a) amplitude, (b) frequency, (c) period, (d) maximum speed, and (e) maximum acceleration.
Example: SHM
22UCT PHY1025F: Vibrations & Waves
Using the small-angle approximation, the restoring force becomes
The pendulum displacement (the arclength s) is proportional to the angle giving
The Simple Pendulum (Review)
mgmgFnet sin
sLmg
LsmgFnet
23UCT PHY1025F: Vibrations & Waves
Simple harmonic motion is based on the restoring force obeying Hooke’s Law, so let’s compare the pendulum force to Hooke’s law.
If we take , then our frequency equation becomes:
And the period equation becomes:
Frequency of Simple Pendulum
12
gfL
netmgF sL
12
kfm
netF kx
2 LTg
24UCT PHY1025F: Vibrations & Waves
Two observations:– The frequency and period of oscillation
depend on physical properties of the oscillator.• Spring: Mass & Spring Constant• Pendulum: Length
– They do not depend on the amplitude of the oscillation.• Pendulum frequency does not depend on mass
The pendulum depends only on and making it a useful timing device
Frequency and Period
25UCT PHY1025F: Vibrations & Waves
• Damped harmonic motion happens when energy is removed (by friction, or design) from the oscillating system.
• Resonance occurs when energy is added to an oscillator at the natural frequency of the oscillator.
Damping & Resonance
26UCT PHY1025F: Vibrations & Waves
All systems have a natural frequency, the frequency at which a system will oscillate if left by itself.
Natural Frequency
27UCT PHY1025F: Vibrations & Waves
Resonance occurs when energy is added to an oscillator at the natural frequency of the oscillator.
If an external force of this frequency is applied, the resulting SHM has huge amplitude!
Resonance
28UCT PHY1025F: Vibrations & Waves
The basic properties of waves (the wave model) cover aspects of wave behaviour common to all waves.A wave is the motion of a disturbance.
The Wave Model
Waves carry energy & momentum without the physical transfer of material.A traveling wave is an organized disturbance with a well-defined wave speed.
29UCT PHY1025F: Vibrations & Waves
Mechanical Waves… require some source of disturbance and a medium that can be disturbed with some physical connection or mechanism through which adjacent portions can influence each other (e.g. waves on a string, sound, water waves)
Two Types of Waves: Mechanical
30UCT PHY1025F: Vibrations & Waves
Electromagnetic Waves... don’t require a medium and can travel in a vacuum (e.g. visible light, x-rays etc)
Two Types of Waves: Electromagnetic
31UCT PHY1025F: Vibrations & Waves
A wave pulse can be created with a single ‘snap’ on a rope • Energy is transmitted from one point on the rope to the
next
A periodic (continuous) wave can be created by wiggling the rope up and down continuously• Energy is continuously being transmitted along the rope
Making a wave
32UCT PHY1025F: Vibrations & Waves
Types of Mechanical Travelling WavesTransverse waves:
Longitudinal waves:
In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion.
In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave. A longitudinal wave is also called a
compression wave.
33UCT PHY1025F: Vibrations & Waves
• crests and troughs are the high and low points of a wave• amplitude, , is the height of a crest (depth of a trough)• wavelength, , is the distance between crests (troughs)• frequency, , is the number of cycles per unit time• period, , is the length of a cycle• wave velocity, , is the velocity the wave crest travels
Some definitions…
𝒗=𝝀𝑻 =𝝀 𝒇
34UCT PHY1025F: Vibrations & Waves
Waves on a string (transverse waves) are propagated by the difference in directions of the tensions.Sounds waves (longitudinal waves) are pressure waves.
Waves on a String and in Air
35UCT PHY1025F: Vibrations & Waves
Both waves on a string and sound waves require a medium and the properties of the medium determine the speed of the wave.For wave on a string, the speed is given by:where is the tension in the string and is the linear mass density:
Observations:- Wave speed increases with increasing tension- Wave speed decreases with increasing linear density
Wave Speed: String
sTv
36UCT PHY1025F: Vibrations & Waves
Two travelling waves can meet and pass through each other without being destroyed or even altered.
The Principle of Superposition
Principle of Superposition- when two waves pass
through the same point, the displacement is the sum of the individual displacements
Pulses are unchanged after the interference.
37UCT PHY1025F: Vibrations & Waves
Constructive:
Two waves, 1 and 2, have the same frequency and amplitude and are “in phase.”
The combined wave, 3, has the same frequency but a greater amplitude.
Constructive Interference
38UCT PHY1025F: Vibrations & Waves
Destructive:
Two waves, 1 and 2, have the same amplitude and frequency but one is inverted relative to the other (i.e. they are 180° “out of phase”)
When they combine, the waveforms cancel.
Destructive Interference
39UCT PHY1025F: Vibrations & Waves
Just like light reflects off water or an echo bounces off a cliff, a wave pulse on a string will reflect at a boundary.Whenever a traveling pulse reaches a boundary, some or all of the pulse is reflected.There are two types of boundaries:- Fixed end- Loose end
Wave Pulse Reflection
40UCT PHY1025F: Vibrations & Waves
When a pulse is reflected from a fixed end, the pulse is inverted, but the shape and amplitude remains the same.
Reflection of Pulses – Fixed End
Think about Newton’s 3rd law at the boundary point.
41UCT PHY1025F: Vibrations & Waves
When reflected from a free end, the pulse is not inverted, again the shape and amplitude remains the same.
Reflection of Pulses – Free End
Think about Newton’s 3rd law at the boundary point.
42UCT PHY1025F: Vibrations & Waves
Pulse Refection at a DiscontinuityA discontinuity can act like a fixed or a free end depending on how the medium changes.
Low to high linear mass density acts like fixed end
High to low linear mass density acts like free end
43UCT PHY1025F: Vibrations & Waves
When a travelling wave reflects back on itself, it creates travelling waves in both directions.The wave and its reflection interfere according to the Principle of Superposition.
The wave appears to stand still, producing a standing wave.
Standing Waves
44UCT PHY1025F: Vibrations & Waves
A simple example of a standing wave is a wave on a string, like you will see in Vibrating String practical. The mechanical oscillator creates a traveling wave that is reflected off the fixed end and interferes with itself.The result is a series of nodes and antinodes, with the exact number depending on the oscillating frequency.
Standing Waves on a String
45UCT PHY1025F: Vibrations & Waves
Nodes are points where the amplitude is 0. (destructive interference)
Anti-nodes are points where the amplitude is maximum. (constructive interference)
Distance between two successive nodes is ½ λ.
Standing Waves on a String
46UCT PHY1025F: Vibrations & Waves
The figure shows the “n = 2” standing wave mode.
The red arrows indicate the direction of motion of the parts of the string.
All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion.
Standing Waves on a String
47UCT PHY1025F: Vibrations & Waves
There are restrictions to a standing wave on a string.1. Two ends of the string are fixed,
so and must be nodes.2. Standing waves spacing is
between nodes, so the nodes must be equally spaced.
Standing Wave on a String
As a result, standing waves will only form at particular modes, which have numbers, i.e. , , etc.
48UCT PHY1025F: Vibrations & Waves
Each mode has a specific wavelength.
Standing Wave on a String
For , the wavelength is: .
In general, the wavelength for a standing wave on a string is: for
Note: The mode number () is equal to the number of anti-nodes.
49UCT PHY1025F: Vibrations & Waves
The standing wave on a string can exist only if it has one of these wavelengths: .
Standing Wave on a String
We can also calculate the frequency of the standing wave:
for
50UCT PHY1025F: Vibrations & Waves
Standing Wave on a StringThe first mode is called the fundamental frequency: . All other modes have a frequency that are multiples of this fundamental frequency: .
The fundamental frequency () is known as the first harmonic, is the second harmonic, is the third harmonic, etc …