IB – PHYSICS (Core)– Oscillation and Wave
Topic 4.1 Kinematics of simple harmonic motionOscillations occur when a system is disturbed from a position of stable equilibrium. This displacement from equilibrium changes periodically over time. Thus, Oscillations are said to be periodic, and display periodic motion.. Some examples of oscillations are shown in figure 1 below.Fig 1 : Some basic examples of OscillationsNotice that for a system to be oscillating, the shape of the displacement - time graph does not matter. The only Property that matters is that the motion is periodic.Basic properties of Oscillating Systems Amplitude of the oscillation: The amplitude of the oscillation is the parameter that varies with time and this resides on the y-axis of the oscillation graphs. In figure 1, the amplitude of the oscillation is the maximum displacement of the object from its equilibrium position. Time Period (T) of the oscillation: The time period of the oscillation is simply the time taken for the oscillation to repeat itself. That is, it is the time between successive oscillations of the system.Frequency: The frequency of the oscillating system is simply the number of compete oscillations that happen in 1 second. So,The units of frequency are cycles per second which are given the name Hertz Angular FrequencyThe Angular Frequency of a system is the rotational analogue to frequency. It is given the symbol ω and is measured in radians per second (rads-1). It is defined by the equationbut,and so is related to frequency byPhaseThe Phase of an oscillation is the amount the oscillation lags behind, or leads in front of a reference oscillation. For example, take a sine oscillation of maximum amplitude, A, and angular frequency, ω, and also a cosine oscillation of maximum amplitude, A, and angular frequency, ω as in figure 3.Figure 3 : Diagram to show the phase of two oscillationsNow, here in figure 3 we can take the sine wave to be our reference oscillation. It can be seen from the diagram that the cosine wave lags behind the sine wave by π/2 (1/4 of a wavelength). So, we can say that the two waves are out of phase by π/2 or that there is a phase difference of π/2. Oscillations can have phase differences of any multiple of π. However, if they have a phase difference of either 0 or 2π they are said to be in phase.Harmonic OscillatorA harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:where k is a positive constant.If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with constant amplitude and a constant frequency (which does not depend on the amplitude).In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation, its frequency, f, the reciprocal of the period f?=?1?T (i.e. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system. Overall then, the equation describing simple harmonic motion isx = Acosωt.Alternatively a sine can be used in place of the sine with the phase shifted by π?2.The general differential force equation for an object of mass m experiencing SHM is:where k is the spring constant which relates the displacement of the object to the force applied to the object. The general solution for this equation is given above.Angular frequency in circular motion is the rate of change of angle. It is measured in radians per. second. Since 2π radians is equivalent to one complete rotation in time period T:The time period of an oscillation is the time taken to repeat the pattern of motion once. In general:However, depending on the type of oscillation, the value of ω changes. For a mass on a spring:The frequency of the oscillations given by:.Velocity and Acceleration of a simple harmonic oscillator.The displacement of a simple harmonic oscillator is:x = AcosωtVelocity is the rate of change of displacement, so:Acceleration is the rate of change of velocity, so:The velocity and acceleration oscillate with a quarter and half a period delay from the displacement. The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.4.2 Energy in simple harmonic motionThe kinetic energy K of the system at time t isand the potential energy isThe total mechanical energy of the system therefore has the constant valueExamples of Simple harmonic motion:Simple PendulumA pendulum is a weight suspended from a pivot so it can swing freely.When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period? Figure : The Simple PendulumIf a pendulum of mass m attached to a string of length L is displaced by an angle from the vertical (see figure below), it experiences a net restoring force due to gravity: F = - mgsin . For small angles, sin , providing is expressed in radians (try it on your calculator for = 0.1,0.5,1.0 radians). In terms of radians, θ=sl radians where s is the arc length and L is the length of the string. Thus, for small displacements, s , the restoring force can be written: F=-mglsSince the restoring force is proportional to the displacement, the pendulum is a simple harmonic oscillator with ``spring constant'' k = mg/L . The period of a simple pendulum is therefore: ? T=2π√mk or, T=2π√lgNote: In this small angle approximation, the amplitude of the pendulum has no effect on the period. This is what makes pendulums such good time keepers. As they inevitably lose energy due to frictional forces, their amplitude decreases, but the period remains constant. A Particle In A Bowl:Consider a particle of mass m, placed inside a frictionless of radius r. It is in the equilibrium when at its bottom most position of the bowl. When the particle is displaced from the bottom most position to a position P it starts executing SHM. The forces on the particle at P are shown in figure.The force trying to bring the particle back in its position is given by,F =ma = -mg sinθFrom which, a= -g sinθ, using θ = x/r we get,a= -gsinxrThe particle starts oscillating but the oscillation is not simple harmonic. Assuming θ to be small we approximate sinθ = θ and the expression for acceleration changes to, a=-gxr or a=-ω2x, where ω2=gr, So the particle executes SHM only for very small amplitude. The time period is given by,T=2πrgDamped SHMIn physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator. In real oscillators friction, or damping, slows the motion of the system. Depending on the friction coefficient, the system can:Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors. (critically damped)Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator).4.3 Forced Oscillations and Resonance:If an oscillator is displaced and then released it will begin to vibrate. If no more external forces are applied to the system it is a free oscillator and its frequency is called its natural frequency. If a force is continually or repeatedly applied to keep the oscillation going, it is a forced oscillator.The to and fro motion of a body about a mean position is called oscillation.? When a body execute oscillations under the action of external periodic force, Those oscillations are called forced oscillations.When one of the two bodies of same natural frequency is set into vibration, the other body also vibrates with larger amplitude under the influence of the first one. This phenomenon is called Resonance.Take two hollow boxed A and B, open it one side. Place two boxes with their opened sides facing each other with some distance apart. Mount two tuning forks of same natural frequencies one each on the two boxes. Vibrate the tuning fork on the box A, the tuning fork on box B also begins to vibrate due to resonance.? When A is vibrated, the air inside the box A vibrates and these vibrations are transferred into the box B and in turn vibrated the tuning fork on B.Examples and Applications on Forced Oscillations and ResonanceWhen solders are crossing a suspension bridge, they are asked to break their steps. This is because, when the frequency of the marching coincides with the natural frequency of the bridge, the bridge vibrates with larger amplitude and collapses due to resonance.???????A radio is tuned to obtain a clear sound, such that the frequency of the radio has to coincide with the frequency of the incoming electro-magnetic waves.We can find the velocity of sound using resonance phenomenon with resonating air-column experiment.4.4 Wave Characteristics:In mathematics and science, a wave is a disturbance that travels through space and time, usually by transference of energy. Waves are described by a wave function that can take on many forms depending on the type of wave. A mechanical wave is a wave that propagates through a medium due to restoring forces produced upon its deformation.Waves travel and transfer energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist instead of oscillations or vibrations around almost fixed locations. Imagine a cork on rippling water, it would bob up and down staying in about the same place while the wave itself moves outward. When we say that a wave carries energy but not mass, we are referring to the fact that even as a wave travels outward from the center (carrying energy of motion), the medium itself does not flow with it.Categories of WavesWaves come in many shapes and forms. While all waves share some basic characteristic properties and behaviors, some waves can be distinguished from others based on some observable (and some non-observable) characteristics. It is common to categorize waves based on these distinguishing characteristics.?Longitudinal versus Transverse Waves versus Surface WavesOne way to categorize waves is on the basis of the direction of movement of the individual particles of the medium relative to the direction which the waves travel. Categorizing waves on this basis leads to three notable categories: transverse waves, longitudinal waves, and surface waves.A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction which the wave moves. Transverse waves are always characterized by particle motion being perpendicular to wave motion.Crest and Trough The section of the wave that rises above the undisturbed position is called the crest. That section which lies below the undisturbed position is called the trough. These sections are labeled in the following diagram: Wavelength - Distance between two successive crest or trough and is represented as lambda λ.Frequency – Number of complete waves generated per second and its unit is Hertz (HZ).Amplitude – Height of a crest or the depth of a trough measured from the undisturbed position of what is carrying the wave.A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction which the wave movesLongitudinal waves are always characterized by particle motion being parallel to wave motion.A sound wave traveling through air is a classic example of a longitudinal wave.The compressions are regions of high air pressure while the rarefactions are regions of low air pressure. The diagram below depicts a sound wave created by a tuning fork and propagated through the air in an open tube. The compressions and rarefactions are labeled.Wave Fronts and RaysWaves, whether it be electomagnetic or other, are conveniently described in terms of wave fronts. A wave front is the line or surface defined by adjacent protions of a wave that are in phase.--If an arc is drawn along one of the crests of a circular water wave moving out from a point source, all the particles on the line will be in phase. The curvature of a short segment of a spherical or circular wave front is small. The segment may be approximated as a linear wave front or a plane wave front, just as we assume the surface of the Earth to be locally flat. In a uniform medium wave fronts propagate outward from the source at a wave speed characteristic of the medium. For example, the speed of light travels fastest in a vacuum: The geometrical description of a wave in terms of wave fronts tends to neglect the fact that the wave is actually sinusoidal. The concept of a ray simplifies the wave description even further. A ray is a line drawn perpendicular to to a series of wave fronts and pointing in the direction of propagation. A beam of light can be simplified and represented by a group of parallel rays or just a sinlge ray. 4.5 Wave properties:Reflection of wave:Waves carry energy and momentum, and whenever a wave encounters an obstacle, they are reflected by the obstacle. This reflection of waves is responsible for echoes, radar detectors, and for allowing standing waves which are so important to sound production in musical instruments.At a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change).At a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave. Refraction of Wave:Refraction of waves involves a change in the direction of waves as they pass from one medium to another. Refraction, or the bending of the path of the waves, is accompanied by a change in speed and wavelength of the waves. Speed of a wave is dependent upon the properties of the medium through which the waves travel. So if the medium (and its properties) are changed, the speed of the waves is changed. The most significant property of water which would affect the speed of waves traveling on its surface is the depth of the water. Water waves travel fastest when the medium is the deepest. Thus, if water waves are passing from deep water into shallow water, they will slow down. The decrease in speed will also be accompanied by a decrease in wavelength. So as water waves are transmitted from deep water into shallow water, the speed decreases, right0the wavelength decreases, and the direction changes.This boundary behavior of water waves can be observed in a ripple tank if the tank is partitioned into a deep and a shallow section. If a pane of glass is placed in the bottom of the tank, one part of the tank will be deep and the other part of the tank will be shallow. Waves traveling from the deep end to the shallow end can be seen to refract (i.e., bend), decrease wavelength (the wavefronts get closer together), and slow down (they take a longer time to travel the same distance). When traveling from deep water to shallow water, the waves are seen to bend in such a manner that they seem to be traveling more perpendicular to the surface. If traveling from shallow water to deep water, the waves bend in the opposite direction.Diffraction of wave:Diffraction involves a change in direction of waves as they pass through an opening or around a barrier in their path. Water waves have the ability to travel around corners, around obstacles and through openings. This ability is most obvious for water waves with longer wavelengths. Diffraction can be demonstrated by placing small barriers and obstacles in a ripple tank and observing the path of the water waves as they encounter the obstacles. The waves are seen to pass around the barrier into the regions behind it; subsequently the water behind the barrier is disturbed. The amount of diffraction (the sharpness of the bending) increases with increasing wavelength and decreases with decreasing wavelength. In fact, when the wavelength of the waves are smaller than the obstacle, no noticeable diffraction occurs.Diffraction of water waves is observed in a harbor as waves bend around small boats and are found to disturb the water behind them. The same waves however are unable to diffract around larger boats since their wavelength is smaller than the boat. Diffraction of sound waves is commonly observed; we notice sound diffracting around corners, allowing us to hear others who are speaking to us from adjacent rooms.Interference of Waves:Wave interference is the phenomenon which occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape which results from the net effect of the two individual waves upon the particles of the medium. To begin our exploration of wave interference, consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units. The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green.This type of interference is sometimes called constructive interference. Constructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement which is greater than the displacement of the two interfering pulses. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward. This is shown in the diagram below for two downward displaced pulses.In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units.Destructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. This is depicted in the diagram below.In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. Conditions for sustained interference:(1) The sources must be coherent (i.e., they must maintain aconstant phase relationship with one another).(2) The sources must be monochromatic (i.e., of a singlewavelength).(3) The linear superposition principle is applicable.Young's Double-Slit Experiment?Slits S1 and S2 serve as a pair of coherent light sources.?A visible pattern of bright and dark parallel bands (calledfringes) are produced on screen C.?Bright bands: produced by constructive interference. Dark bands: produced by destructive interference.Path difference of two waves: ??= r2 - r1 = d sinlefttopθCondition for constructive interference:Path?difference???d sinθ??m?λ(m = 0, ?1, ?2, …)Condition for destructive interference:Path?difference???d sinθ??(m??1/2)λ (m = 0, ?1, ?2, …) ................
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