Presentation on theme: "Physics 1025F Vibrations & Waves"— Presentation transcript:
1Physics 1025F Vibrations & Waves OSCILLATIONSDr. Steve Peterson
2Chapter 11: Vibrations and Waves Periodic motion occurs when an object vibrates or oscillates back and forth over the same path
3Periodic MotionPeriodic motion, processes that repeat, is one of the important kinds of behaviours in Physics
4Equilibrium and Oscillation Equilibrium position – position where net force is zeroRestoring force – force acting to restore equilibriumOscillation – periodic motion governed by a restoring force
5Equilibrium 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)
6Simple 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 point𝑇= 1 𝑓
7Simple 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 SHMInitially, we will look at the horizontal mass-spring system as a representative example of SHM
8Hooke’s Law Reviewspring forcek is thespring constantx 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)
9Example: Hooke’s LawA 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?
10Horizontal 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 positionSince acceleration is not constant in SHM standard “equations of motion” cannot be applied
11Example: SHMV&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.
12The Simple PendulumSHM 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. NoSin theta is not a linear relationship.
13The Simple PendulumA 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
14Linear restoring force The Simple PendulumUsing the small-angle approximation, the restoring force becomes The pendulum displacement (the arclength s) is proportional to the angle givingLinear restoring force
15Energy 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…)
16Energy in Simple Harmonic Motion Energy is all PE when 𝒙=𝑨Total energy isEnergy is all KE when 𝒙=𝟎Total energy is conserved, so
17Example: 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 oscillationCalculate the maximum velocity of the oscillation
18Energy 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:
19SHM 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, thus𝑣 𝑚𝑎𝑥 = 2𝜋𝐴 𝑇 =2𝜋𝐴𝑓From energy, we have: 𝑣 𝑚𝑎𝑥 = 𝑘 𝑚 𝐴Combining them gives:𝑇=2𝜋 𝑚 𝑘 OR 𝑓= 1 2𝜋 𝑘 𝑚
20Simple Harmonic Motion The position, velocity and acceleration are all sinusoidalThe frequency does not depend on the amplitudeThe object’s motion can be written as
21Example: SHMGiancoli 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.
22The 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
23Frequency 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:
24The pendulum depends only on 𝐿 and 𝑔 making it a useful timing device Frequency and PeriodTwo observations:The frequency and period of oscillation depend on physical properties of the oscillator.Spring: Mass & Spring ConstantPendulum: LengthThey do not depend on the amplitude of the oscillation.Pendulum frequency does not depend on massThe pendulum depends only on 𝐿 and 𝑔 making it a useful timing device
25Damping & ResonanceDamped 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.
26Natural FrequencyAll systems have a natural frequency, the frequency at which a system will oscillate if left by itself.
27ResonanceResonance 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!
28The Wave ModelThe 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.
29Two 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)
30Two Types of Waves: Electromagnetic Electromagnetic Waves ... don’t require a medium and can travel in a vacuum (e.g. visible light, x-rays etc)
31Making a waveA wave pulse can be created with a single ‘snap’ on a ropeEnergy is transmitted from one point on the rope to the nextA periodic (continuous) wave can be created by wiggling the rope up and down continuouslyEnergy is continuously being transmitted along the rope
32Types 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.
33Some definitions… 𝒗= 𝝀 𝑻 =𝝀𝒇 crests and troughs are the high and low points of a waveamplitude, 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 timeperiod, 𝑇, is the length of a cyclewave velocity, 𝑣, is the velocity the wave crest travels𝒗= 𝝀 𝑻 =𝝀𝒇
34Waves 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.
35Wave Speed: StringBoth 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 and𝜇 is the linear mass density: 𝜇= 𝑚 𝐿Observations:Wave speed increases with increasing tensionWave speed decreases with increasing linear density
36The Principle of Superposition Two travelling waves can meet and pass through each other without being destroyed or even altered.Principle of Superpositionwhen two waves pass through the same point, the displacement is the sum of the individual displacementsPulses are unchanged after the interference.
37Constructive 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.
38Destructive 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.
39Wave Pulse ReflectionJust 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 endLoose end
40Reflection 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.
41Reflection 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.
42Pulse 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 endHigh to low linear mass density acts like free end
43Standing WavesWhen 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.
44Standing 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.
45Standing 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.
46Standing 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.
47Standing 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.
48Standing 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.
49Standing 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 2𝐿for 𝑚=1, 2, 3, 4, …
50Standing 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 …