PHY 102 Waves and Oscillations Lecture Notes PDF

PHY 102 (ELECTROMAGNETIC WAVES AND OSCILLATION) WEEK 9 LECTURE NOTE Waves and Oscillation Waves A wave is any disturbance which travels through a medium in such a way that kinetic energy is transferred from one point to another without any net medium of the medium itself. Mechanical and Electromagne...

PHY 102 (ELECTROMAGNETIC WAVES AND OSCILLATION) WEEK 9 LECTURE NOTE Waves and Oscillation Waves A wave is any disturbance which travels through a medium in such a way that kinetic energy is transferred from one point to another without any net medium of the medium itself. Mechanical and Electromagnetic waves There are basically two classifications of waves depending on their transfer mode Mechanical waves: They require material medium for their propagation e.g sound wave Electromagnetic waves: they require no medium of propagation. They can travel through vacuum e.g. gamma ray, X-rays, radio waves, ultraviolent, light waves etc Transverse and Longitudinal waves The direction of propagation of a wave may either be the same as that of the wave or perpendicular to the vibration. Transverse wave: Here, the direction of propagation is perpendicular to the vibration Longitudinal wave: the direction of vibration of the wave is the same (i.e parallel) as the direction of propagation. Terms used in wave 1. Amplitude (A) : this is the maximum displacement from the rest or equilibrium position. The unit is meter. 2. Period (T): This is the time required for the wave to complete one cycle. The unit is the second 3. Frequency (f): This is the number of complete oscillation made in one second. Its unit is cycle per second or simply Hertz (Hz) 4. Wavelength (λ): This is the distance between successive crest of successive trough. It is measured in meter 5. Velocity of the wave (v): It is the distance which the wave covered in one second. Relation between T and f, λ and V The frequency F and period T are related as : 1 𝑥 𝜆 𝐹 = 𝑇 ; 𝑊ℎ𝑒𝑟𝑒 𝑓, 𝜆 𝑎𝑛𝑑 𝑣 𝑎𝑟𝑒 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑏𝑦 𝑉 = 𝑓𝜆 = =𝑇 𝑡 Example: A wave travels a distance of 60cm in 3s. The distance between successive crests of the wave is 4cm. What is the frequency?  Solution 𝑥 60 𝑉= ; 𝑣 = = 20, 𝜆 = 4𝑐𝑚 𝑡 3 𝑉 20 𝑓= = = 5𝐻𝑧 𝜆 4 Some characteristics/properties of waves Reflection, Refraction, Interference, Diffraction, Polarization Oscillation Motion is important in everyday living. One of the most important types of motion encountered in nature is oscillatory or vibrational motion. A particle is oscillating when it moves periodically about an equilibrium position. Periodic Motion Periodic motion is the type of motion which repeats itself at a regular time interval. 1.The periodic motion in which there is existence of a restoring force and the body moves along the same path to and fro about a definite point called equilibrium position/mean position, is called oscillatory motion. 2. In all type of oscillatory motion one thing is common i.e each body (performing oscillatory motion) is subjected to a restoring force that increases with increase in displacement from mean position. 3. Types of oscillatory motion:- It is of two types such as linear oscillation and circular oscillation. Example of linear oscillation:- 1. Oscillation of mass spring system. 2. Oscillation of fluid column in a U-tube. 3. Oscillation of floating cylinder. 4. Oscillation of body dropped in a tunnel along earth diameter. 5. Oscillation of strings of musical instruments. Example of circular oscillation:- 1. Oscillation of simple pendulum. 2. Oscillation of solid sphere in a cylinder (If solid sphere rolls without slipping). 3. Oscillation of a circular ring suspended on a nail. 4. Oscillation of balance wheel of a clock. 5. Rotation of the earth around the sun. Oscillatory system:- 1. The system in which the object exhibit to & fro motion about the equilibrium position with a restoring force is called oscillatory system. 2. Oscillatory system is of two types such as mechanical and non- mechanical system. 3. Mechanical oscillatory system:-  In this type of system body itself changes its position.  For mechanical oscillation two things are especially responsible i.e Inertia & Restoring force.  E.g oscillation of mass spring system, oscillation of fluid-column in a U-tube, oscillation of simple pendulum, rotation of earth around the sun, oscillation of body dropped in a tunnel along earth diameter, oscillation of floating cylinder, oscillation of a circular ring suspended on a nail, oscillation of atoms and ions of solids, vibration of swings…etc. 4.Non-mechanical oscillatory system:- In this type of system, body itself doesn‟t change its position but its physical property varies periodically. e.g:-The electric current in an oscillatory circuit, the lamp of a body which is heated and cooled periodically, the pressure in a gas through a medium in which sound propagates, the electric and magnetic waves propagates undergoes oscillatory change. Simple Harmonic Motion:- It is the simplest type of oscillatory motion. A particle is said to execute simple harmonic oscillation (shm) if the restoring force is directed towards the equilibrium position and its magnitude is directly proportional to the magnitude and displacement from the equilibrium position. SHM is the motion of a particle whose acceleration is proportional to the distance from a fixed point and is always towards that point. It is expressed mathematically as; a = -ω2x Consider a spring that is attached to the mass. When the mass is displaced a distance x away its equilibrium position, the spring provides a restoring force on the mass that is proportional to the displacement. If F is the restoring force on the oscillator when its displacement from the equilibrium position is x, then; F α –x Here, the negative sign implies that the direction of restoring force is opposite to that of displacement of body i.e towards equilibrium position. F= -kx …………. (1) Where, k= proportionality constant called force constant. Since F = ma; then Ma=-kx 𝑑2 𝑥 𝑀 𝑑𝑡 2 = −kx 𝑑2 𝑥 𝑀 𝑑𝑡 2 + 𝑘𝑥 = 0 𝑑2 𝑥 𝑘 𝑑2 𝑥 + 𝑀 𝑥 = 0; That is a = 𝑑𝑡 2 𝑑𝑡 2 𝑑2 𝑥 + ω2 𝑥 = 0 (2) 𝑑𝑡 2 𝑘 𝑘 Where ω2 = 𝑀; Here ω = √𝑀 is the angular frequency of the oscillation Equation (2) is called general differential equation of SHM. Similarly, the solution of differential equation can be given as x=Acos(θ+ωt ) ………(3) Here A denotes amplitude of oscillatory system, (θ+ωt) is called phase and is called epoch/initial phase/phase constant/phase angel.  Equation (6) represents displacement of SHM. Velocity in SHM:- x=Asin(θ+ωt ) 𝑑𝑥 = Aωcos(θ + ωt ) 𝑑𝑡 v = Aωcos(θ + ωt )………… (4) The minimum value of v is 0(as minimum value of Asin(θ+ωt )=0 and maximum value is Aω. The maximum value of v is called velocity amplitude. Acceleration in SHM:- a = −A𝜔2 sin(𝜔𝑡 + 𝜃) ) …………. (5) The minimum value of ‘a’is 0 and maximum value is A𝜔2. The maximum value of ‘a’ is called acceleration amplitude. Also, 𝑎 = 𝜔2 𝑥 a α–y is also the condition for SHM. Time period in SHM:- The time required for one complete oscillation is called the time period (T). It is related to the angular frequency( ω) by. 2𝜋 𝑇= ……………… (6) 𝜔 Frequency in SHM:- The number of oscillation per time is called frequency or it is the reciprocal of time period. 1 𝜔 𝑣 = 𝑇 = 2𝜋……………(7) Free, Damped and Forced Oscillation 1. Free oscillations are shm with constant amplitude and period and no external influnnces. 2. Damped oscillations are shm with decreasing amplitude and varying period due to external or internal influnences. 3. Forced oscillation are shm which are driven externally.. TUTORIAL QUESTIONS 1. A boat at anchor is rocked by waves whose crests are 100m apart and whose velocity is 25m/s. At what interval does the wave crest reach the boat? A. 2500s B. 75s  C. 4s D. 0.24s ANSWER C 2. A wave travels a distance of 20cm in 3s. the distance between successive crests of the wave is 4cm. What is the frequency of the wave? A. 0.6Hz B. 1.67Hz C. 8.0Hz D. 26.6Hz ANSWER B 3. The period of vibration of a wave of wavelength 30m moving at a speed of 300m/s is- ------- A. 0.1s B. 10s C. 270s D. 330s ANSWER A 4. ------------ is the number of complete oscillation made per second A. Frequency B. Period C. Time D. Velocity ANSWER A 5. Wave characteristics include the following except A. Refraction B. Reflection C. Diffraction D. Wavefraction ANSWER D 6. ----------- describes the motion of a particle whose acceleration is proportional to the distance from a fixed point and is always towards that point. A. Simple pendulum B. Simple harmonic motion C. Oscillation D. Vibrational body ANSWER B 7. Examples of electromagnetic waves include the following except A. Light waves B. Ultraviolet C. X-rays D. Sound waves ANSWER D 8. ------------ describe the type of motion which repeats itself at a regular time interval. A. Oscillatory motion B. Periodic motion C. Translational motion  D. Rotational motion ANSWER B 9. The type of oscillation which is externally driven is known as A. Random oscillation B. Forced oscillation C. Free oscillation D. Damped oscillation ANSWER D 10. Calculate the period of oscillation of a vibrating particle whose angular frequency is 5 rad/s A. 2.35s B. 1.25s C. 0.45s D. 1s ANSWER B