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Presented by CHITRA JOSHI PGT (PHYSICS) KV OFD RAIPUR DEHRADUN SIMPLE HARMONIC MOTION Simple harmonic motion is a special type of periodic motion, in which a particle moves to and fro repeatedly about a mean(equilibrium position under a restoring force, which is always directed towards the mean position and whose magnitude at any instant is directly propotional to the displacement of the particle from the mean position at that instant. GENERAL CHARACTERS OF SIMPLE HARMONIC MOTION Displacement-The displacement of a particle executing s.h.m. at any instant is defined as the distance of the particle from the mean position at that 1 instant. y=asinθ=asinωt AMPLITUDE The maximum displacement on either side of mean position is called amplitude of motion.as the maximum value of sinθ or cosθ=1 VELOCITY The velocity of the particle executing s.h.m. at any instant, is defined as the time rate of change of its displacement at that instant. Thus, the velocity in s.h.m. at the instant t is, v=wacos(ωt+ø0) =wa √(1-sin2(ωt+ø)) V0 =wa =velocity amplitude ACCELERATION The acceleration of the particle executing s.h.m. at any instant is defined as the time rate of change of its velocity at that instant. OSCILLATIONS PERIODIC MOTION Periodic motion of a body is that motion which is repeated identically after a fixed interval of time.The fixed interval of time after which the motion is repeated is called periodic motion. E.g.-The revolution of earth around the sun is a periodic motion.Its period of revolution is one year. OSCILLATORY MOTION Oscillatory or vibratory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point (called mean position or equilibrium position) in a definite interval of time. E.g-The motion of the pendulum of a wall clock is oscillatory motion. HARMONIC OSCILLATION Harmonic oscillation is that oscillation which can be expressed in terms of single harmonic function(i.e.sine function or cosine function). “A harmonic oscillation of constant amplitude and of single frequency is called simple harmonic oscillation. Mathematically, y=asineωt=asin2πt/T y=acosωt=acos2πt/T Here y=displacement of body from mean of position at any instant t. a=maximum displacement ω=angular frequency=2πv=2π/T v,Tare frequency and time period of harmonic oscillation. NON HARMONIC OSCILLATION Non-harmonic oscillation is that oscillation which cannot be expressed in terms of single harmonic function. Mathematically, y=asinωt+bsinωt Or y=asin2πt + bsin4πt T T PERIODIC FUNCTIONS Periodic functions are those functions which are used to represent periodic motion. A function f(t) is said to be periodic, if f(t)=f(t+T)=f(t+2t) PHASE Phase of a vibrating particle, at any instant is a physical quantity which completely expresses the position and direction of motion of the particle at that instant with respect to its mean position. it is denoted by ø. ø=2πt/T+ø0 SIMPLE HARMONIC MOTION Simple harmonic motion is a special type of periodic motion, in which a particle moves to and fro repeatedly about a mean(i.e. equilibrium) position under a restoring force, which is always directed towards the mean position and whose magnitude at any instant is directly propotional to the displacement of the particle from the mean position at that instant. GENERAL CHARACTERS OF SIMPLE HARMONIC MOTION Displacement- The displacement of a particle executing s.h.m. at any instant is defined as the distance of the particle from the mean position at that 1 instant. y=asinθ=asinωt AMPLITUDE The maximum displacement on either side of mean position is called amplitude of motion. As the maximum value of sinθ or cosθ=1 Velocity-The velocity of the particle executing s.h.m. at any instant, is defined as the time rate of change of its displacement at that instant.Thus, the velocity in s.h.m. at the instant t is, v=wacos(ωt+ø0) =wa√1-sin2(ωt+ø0) =wa√1-(y2/a2) v=w√a2-y2 ACCELERATION The acceleration of the particle executing S.H.M at any instant is defined as the time rate of change of its velocity at that instant. Time period- it is defined as the time taken by the particle executing S.H.M.to complete one vibration. A=ω2/y Or ω=√A/y Time period,T=2π/ω=2π√y/A T=2π√displacement/acceleration TOTAL ENERGY IN S.H.M. A particle executing S.H.M. possesses two types of energy. Potential energy-This is an account of the displacement of the particle from its mean position. Kinetic energy-This is an account of the velocity of the particle. POTENTIAL ENERGY Consider a particle of mass m, executing linear S.H.M.with amplitude a and constant angular frequency ω.suppose t second after starting from the mean position, the displacement of the particle is y. y=asinωt 1 Therefore velocity of the particle at instant t,v=dy/dt=awcosωt 2 Acceleration of the particle at this instant, A=dv/dt=-aw2sinωt=-ω2y 3 Restoring force, f=mass x acceleration f=-mw2y=-ky f=-ky mw2=k force constant The work done dw=-fdy =-(-ky)dy dw=kydy Total work done for displacing the particle from the mean position to a position of displacement y will be W = 0∫y kydy =1/2 ky2 This work done appears as potential energy U of the given instant. U=1/2 ky2 4 U=1/2 mw2y2=1/2mw2a2sin2wt U=1/2mw2a2sin2wt 5 Kinetic energy – K.E. of the particle at the instant t, is given by K=1/2mv2 K=1/2m(awcoswt)2 K=1/2mw2a2cos2wt 6 K=1/2ka2cos2wt K=1/2ka2(1-sin2wt) K=1/2ka2(1-y2/a2) =1/2k(a2-y2) K=1/2mw2(a2-y2) 7 Total energy of the particle at the instant it will be E=U+K=1/2ky2+1/2k(a2-y2) E=1/2ka2=1/2mw2a2 E=1/2m(2 תa)2 W=2 תa E=2m ת2v2a2 Waves Wave motion : A wave motion is a means of transferring energy and momentum frm one point to another without any actual transportation of matter between these points. Types of waves We usually come across three types of waves-They are Mechanical waves Electromagnetic waves Matter waves Mechanical waves-These are not familiar.for e.g. waves on water surface, waves on string, sound waves e.t.c. These waves are governed by Newton’s laws of motion. ELECTROMAGNETIC WAVES Electromagnetic waves,which require no material medium for their production and propagation i.e. they can pass through vaccum and other material medium. MATTER WAVES(DE-BROGLIE WAVE ) These waves are associated with moving particles of matter, like electrons, protons, neutrons, atoms, molecules e.t.c. Types of mechanical wave motion-The mechanical waves are of two types1)Transverse wave motion 2)Longitudinal wave motion Transverse wave motion-A transverse motion is that wave motion in which individual particles of the medium execute simple harmonic motion about their mean position in a direction perpendicular to the direction of propagation of wave motion. A E C λ Normal level partical λ B D LONGITUDNAL WAVE MOTION A longitudnal wave motion is that motion in which individual particles of the madium execute simple harmonic motion about their mean position along the same direction along which the wave is propagated. For e.g.1)Sound waves travel through air in the form of longitudnal waves. 2)Vibrations of air column in organ pipes are longitudnel. RELATION BETWEEN PARTICLE VELOCITY AND WAVE VELOCITY-The equation of a plane progressive wave travelling with a velocity v along the direction of xaxis. Y(x,t)=r sin[2π/λ(vt-x)+ø0 ] 1 If initial phase,ø0=0 then 2 Y(x,t)=r sin[2π/λ(vt-x)] At any position x, velocity of particle is the rate of change of displacement of the particle with time; it is represented by u(x,t), u(x,t)=d/dt[y(x,t) =d/dt[r sin{2π/λ(vt-x)}] 3 u(x,t)=r cos{2π/λ(vt-x)}*2πv/λ Also,d/dt[y(x,t)=d/dt[r sin{2π/λ(vt-x)}] =r cos{2π/λ(vt-x)}(-2π/λ) 4 Dividing eqn 3 and 4 U(x,t) =-v d/dx{y(x,t)} Or u(x,t)=-v d/dx{y(x,t)} Some important terrms connected with wave motionAMPLITUDE-The amplitude of a wave is the magnitude of maximum displacement of the element from their equilibrium position,as the wave passes through them.itis represented by r. WAVE LENGTH-It is equal to the distance travelled by the wave during the time, any one particle of the medium completes one vibration about its mean position. ANGULAR WAVE NUMBER Angular wave number of a wave is 2π times of the no. of waves that can be commodated per unit length. k=2π/λ S.i. unit of k=radian m-1 FREQUENCY-It may be define as frequency of a wave as the number of complete wave lengths traversed by the wave in one second.it is represented by v and is measured in heartz. Angular frequency of the wave is 2π times the frequency of the wave.it is represented by ω and is measured in rad s-1 ω=2πv TIME PERIOD Time period of a wave is equal to the time taken by the wave to travel a distance equal to one wave length. It is represented as T. T=I/v DOPPLER EFFECT IN SOUND According to Doppler’s effect, whenever there is a relative motion between a source of sound and listener, the apparent frequency of sound heard by the listener is different from the actual frequency of sound emitted by the source. EXPRESSION FOR APPARENT FREQUENCY Suppose medium, source and listener move in same direction then apparent wavelength becomes λ’={(v+vm)-vs) ᶹ These waves of wave length λ’ travel towards the listener. Relative velocity of sound waves with respect to listener=(v+vm)-Vl .This is the distance available in one second to waves of wave length λ’. Apparent frequency of sound waves heard by listener is ᶹ’ =(v+vm)-Vl λ’ putting value of λ’ In case the medium is stationary vm=0 ᶹ ’=(v-vL)ᶹ (v-vs) Specials cases1)if the source is moving towards the listener but the listener is at rest,then vs is positive and vL=0, ᶹ’= ( v) ᶹ v-vs i.e ᶹ’> ᶹ 2)if the source is moving away from the listener is at rest, then vs is negative and vL=0 ᶹ’= ( v) ᶹ = ( v) ᶹ v-(-vs) v+vs i.e. ᶹ’<ᶹ 3)if the source is at rest and listener is moving away from the source, then vs=0 and vL=positive ᶹ’=(v-vL) ᶹ v i.e. ᶹ’<ᶹ If the source is at rest and listener is moving towards the source, then vs=0 vL=negetive. ᶹ’={v-(-vL)} ᶹ v i.e. ᶹ’>ᶹ 4)if the source and listener are approaching each othert, then vs is positive and Vl is neg. ᶹ’={v-(-vL)} ᶹ =(v+vL) ᶹ v-vs (v-vs) i.e. ᶹ’>ᶹ 6)if the source and listener are moving away from each other, then vs is nag and vL is posi ᶹ’=(v-vL) ᶹ. (v+vs) i.e. ᶹ’<ᶹ 7)if the source and listener are both in motion in the same direction and with same velocity, then vs=vL = v’ i.e. ᶹ’=ᶹ 8)if the source and listener move at right angles to the direction of wave propagation.vs cos 90 °=0 and vL cos 90°=0 and ᶹ’=ᶹ THANK YOU