1. Physics 1025F Vibrations & Waves
OSCILLATIONS
Dr. Steve Peterson

2. Chapter 11: Vibrations and Waves
Periodic motion occurs when an object vibrates or oscillates back and forth over the same path

3. Periodic Motion
Periodic motion, processes that repeat, is one of the important kinds of behaviours in Physics

4. Equilibrium and Oscillation
Equilibrium position – position where net force is zero
Restoring force – force acting to restore equilibrium
Oscillation – periodic motion governed by a restoring force

5. Equilibrium and Oscillation
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)

6. Simple Harmonic Motion
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 measured from the equilibrium point
𝑇= 1 𝑓

7. Simple Harmonic Motion
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 SHM
Initially, we will look at the horizontal mass-spring system as a representative example of SHM

8. Hooke’s Law Review
spring force
k is the
spring constant
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)

9. Example: Hooke’s Law
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?

10. Horizontal Mass on a Spring
From Newton II, for a mass-spring system:
For a horizontal mass-spring system & all other cases of SHM, acceleration depends on position
Since acceleration is not constant in SHM standard “equations of motion” cannot be applied

11. Example: SHM
V&S Example 13.2: A 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 = m, and x = m.

12. The Simple Pendulum
SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s Law For a pendulum, the restoring force is Does this motion qualify as simple harmonic motion? A. Yes B. No
Sin theta is not a linear relationship.

13. The Simple Pendulum
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

14. Linear restoring force
The Simple Pendulum
Using the small-angle approximation, the restoring force becomes The pendulum displacement (the arclength s) is proportional to the angle giving
Linear restoring force

15. Energy in a Mass-Spring System
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…)

16. Energy in Simple Harmonic Motion
Energy is all PE when 𝒙=𝑨
Total energy is
Energy is all KE when 𝒙=𝟎
Total energy is conserved, so

17. Example: Energy of Spring
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

18. Energy in Simple Harmonic Motion
Conservation of energy allows the calculation of the velocity of an object attached to a spring at any position in its motion:

19. SHM and Uniform Circular Motion
The velocity of the rotating object is equal to the maximum velocity of the object in SHM.
The circle circumference is 2𝜋𝐴 and the rotation time is 𝑇, thus
𝑣 𝑚𝑎𝑥 = 2𝜋𝐴 𝑇 =2𝜋𝐴𝑓
From energy, we have: 𝑣 𝑚𝑎𝑥 = 𝑘 𝑚 𝐴
Combining them gives:
𝑇=2𝜋 𝑚 𝑘 OR 𝑓= 1 2𝜋 𝑘 𝑚

20. 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

21. Example: SHM
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: 𝑥= 0.30 m cos 8.0𝑡 . Determine the oscillating object’s (a) amplitude, (b) frequency, (c) period, (d) maximum speed, and (e) maximum acceleration.

22. The Simple Pendulum (Review)
Using the small-angle approximation, the restoring force becomes The pendulum displacement (the arclength s) is proportional to the angle 𝒔=𝑳𝜽 giving

23. Frequency of Simple Pendulum
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 𝑘= 𝑚𝑔 𝐿 , then our frequency equation becomes: And the period equation becomes:

24. The pendulum depends only on 𝐿 and 𝑔 making it a useful timing device
Frequency and Period
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 𝐿 and 𝑔 making it a useful timing device

25. Damping & Resonance
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.

26. Natural Frequency
All systems have a natural frequency, the frequency at which a system will oscillate if left by itself.

27. Resonance
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!

28. The Wave Model
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.
Waves carry energy & momentum without the physical transfer of material.
A traveling wave is an organized disturbance with a well- defined wave speed.

29. Two Types of Waves: Mechanical
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)

30. Two Types of Waves: Electromagnetic
Electromagnetic Waves ... don’t require a medium and can travel in a vacuum (e.g. visible light, x-rays etc)

31. Making a wave
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

32. Types of Mechanical Travelling Waves
Transverse waves:
In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion.
Longitudinal waves:
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.

33. Some definitions… 𝒗= 𝝀 𝑻 =𝝀𝒇
crests and troughs are the high and low points of a wave
amplitude, 𝐴, is the height of a crest (depth of a trough)
wavelength, 𝜆, is the distance between crests (troughs)
frequency, 𝑓, is the number of cycles per unit time
period, 𝑇, is the length of a cycle
wave velocity, 𝑣, is the velocity the wave crest travels
𝒗= 𝝀 𝑻 =𝝀𝒇

34. Waves on a String and in Air
Waves on a string (transverse waves) are propagated by the difference in directions of the tensions. Sounds waves (longitudinal waves) are pressure waves.

35. Wave Speed: String
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 𝑇 𝑠 is the tension in the string and
𝜇 is the linear mass density: 𝜇= 𝑚 𝐿
Observations:
Wave speed increases with increasing tension
Wave speed decreases with increasing linear density

36. The Principle of Superposition
Two travelling waves can meet and pass through each other without being destroyed or even altered.
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.

37. Constructive Interference
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.

38. Destructive Interference
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.

39. Wave Pulse Reflection
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

40. Reflection of Pulses – Fixed End
When a pulse is reflected from a fixed end, the pulse is inverted, but the shape and amplitude remains the same.
Think about Newton’s 3rd law at the boundary point.

41. Reflection of Pulses – Free End
When reflected from a free end, the pulse is not inverted, again the shape and amplitude remains the same.
Think about Newton’s 3rd law at the boundary point.

42. Pulse Refection at a Discontinuity
A 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

43. Standing 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.

44. Standing Waves on a String
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.

45. Standing Waves on a String
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.

46. Standing Waves on a String
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.

47. Standing Wave on a String
There are restrictions to a standing wave on a string.
Two ends of the string are fixed, so 𝑥=0 and 𝑥=𝐿 must be nodes.
Standing waves spacing is 𝜆 2 between nodes, so the nodes must be equally spaced.
As a result, standing waves will only form at particular modes, which have numbers, i.e. 𝑚=1, 𝑚=2, etc.

48. Standing Wave on a String
Each mode has a specific wavelength.
For 𝑚=1, the wavelength is: 𝜆 1 =2𝐿.
In general, the wavelength for a standing wave on a string is:
𝝀 𝒎 = 𝟐𝑳 𝒎 for 𝑚=1, 2, 3, 4, …
Note: The mode number (𝑚) is equal to the number of anti-nodes.

49. Standing Wave on a String
The standing wave on a string can exist only if it has one of these wavelengths: 𝜆 𝑚 = 2𝐿 𝑚 .
We can also calculate the frequency of the standing wave:
𝑓 𝑚 = 𝑣 𝜆 𝑚 = 𝑣 2𝐿 𝑚 =𝑚 𝑣 2𝐿
for 𝑚=1, 2, 3, 4, …

50. Standing Wave on a String
The first mode is called the fundamental frequency: 𝑓 1 = 𝑣 2𝐿 . All other modes have a frequency that are multiples of this fundamental frequency: 𝑓 𝑚 =𝑚 𝑓 1 .
The fundamental frequency ( 𝑓 1 ) is known as the first harmonic, 𝑓 2 is the second harmonic, 𝑓 3 is the third harmonic, etc …