OSCILLATIONS AND WAVES

Transcription

1 OSCILLATIONS AND WAVES OSCILLATION IS AN EXAMPLE OF PERIODIC MOTION No stories this tie, we are going to get straight to the topic. We say that an event is Periodic in nature when it repeats itself in tie. Soe things are periodic in space too (Do you understand what this eans? Think of soe things that are periodic in space). In nature, we coe across any events that are periodic in tie. Our general philosophy in Physics is always to start with soething very siple, understand it, and then to build on it to ake it ore and ore coplex and to understand each of these coplex phenoenon as we ove on. So we start with the siplest of periodic events which is an oscillation a to and fro otion of a single Newtonian 1 particle. An Oscillation is the to and fro otion of a point ass. A very coon exaple is a siple pendulu. A less siple exaple would be a swing in the playground 2. In our discussion about the topic, we will start with the ost siplest of oscillations which is called a Siple Haronic Oscillation also called in short as SHO. You ay later think that there is nothing siple about the SHO but we call it siple because, it is the siplest of oscillations that we know. This eans that it is coparatively, atheatically siple! Exercise 01: List out 5 periodic events in nature. What differences are there between the? Exercise 02: List out 5 Oscillations that you coe across in nature. What differences are there between the? SIMPLE HARMONIC OSCILLATIONS (SHO) A siple haronic oscillation is the two and fro otion of a particle. For a particle to ove to and fro, it needs to undergo a change in velocity and by the first law of Newton, any object that undergoes a change in velocity is under the influence of a force. The nature of this force is what deterines whether or not the oscillation is Siple Haronic in nature. The siplest and easiest exaple we take is to understand SHO is that of an oscillating ass attached on a spring. F = kx F = a It is an experientally observed fact that if we hang asses in sequential order on a spring (1kg, then 2kg, then 3kg and so on), the displaceent produced on the spring after each ass is proportional to the ass. This is known as Hooke s Law 3 and can be atheatically written as F x where x is the displaceent fro the ean position and F is the force on the spring and is called a RESTORING FORCE. It is appropriately naed so as it wishes to restore the spring to its original position. This is atheatically represented by the negative sign in the equation and therefore indicates that the force and the displaceent is always opposite in direction. To change the proportionality to equality, we introduce a constant called the spring constant k. Hence we have F = kx. The value of spring constant depends on the aterial and geoetry of the spring. The larger the value of k, the ore hard it is to copress or decopress the spring. Let us now assage the Hooke s law s equation further. By Newton s 2 nd law, the su total of external, unbalanced force will equal the ass ties the acceleration. F = a. Keeping this in ind, go through the atheatical discussion on the right side. The final result that we get is x + k d2 x = 0 or dt 2 x(t) + k x(t) = 0 This is known as the equation of SHO. It eans that there is a function x(t), that when differentiated twice with respect to tie, will give to a constant ultiplied by the function itself!!! x = k x The general solution for this equation is given by the function x(t) = A cos(ωt + φ) A is called the aplitude, ω is called the angular frequency and φ is called the phase of the otion. See the ath on the right side to understand how this x(t) = A cos(ωt + φ) kx = a a + kx = 0 a + k x = 0 But, a = dv dt = d dt dx dt = d2 x dt 2 d 2 x dt 2 + k x = 0 x + k x = 0 d [x(t)] = v(t) = x (t) = Aω sin(ωt + φ) dt d 2 dt 2 [x(t)] = a(t) = x (t) = Aω2 cos(ωt + φ) x (t) = ω 2 [Acos(ωt + φ)] x (t) = ω 2 [x(t)] 1 A particle is Newtonian in nature, when it obeys Newton s Law and when we can consider it to be a point ass, with all its ass concentrated at one point called the centre of ass. Reeber our discussions about point object and point ass? Centre of Mass is not a ass; it is an abstract point and a Point Mass is not a point but a ass concentrated to a single point! 2 Do you know why it is less siple? 3 Watch MITOCW801, Lecture 10; 01:20 06:40 1 qed_ak_gesphyxi1211

2 function satisfies the SHO equation. Coparing the final equation with the equation of SHO we get the following result x (t) + ω 2 x(t) = 0 x + ω 2 x = 0 & x + k x = 0 ω2 = k ω = k But ω = 2πf = 2π T T = 2π ω T = 2π k The final result gives us the tie period of oscillation of the SHO. Exercise 03: A particle oscillates with Siple Haronic Motion (SHM) along the x axis and its position varies with tie according to the equation x = (5. 00) cos πt + π 4 (i) Deterine the Aplitude, Angular Frequency, Frequency, Phase (ii) Calculate the velocity and acceleration of the particle at any tie t (iii) What is the position and velocity of the particle at tie t = 0 Exercise 04: A ass of 2 kg attached to a spring of spring constant 8 N/ is pulled out 10c fro its ean position and let go without any initial velocity Deterine the equation that represents its otion in tie. Exercise 05: A ass of 2 kg attached to a spring of spring constant 8 N/ is pulled out 10c fro its ean position and is pushed in with an initial velocity of 1/s. Deterine the equation that represents its otion in tie. 2 qed_ak_gesphyxi1211

3 ENERGY CONSIDERATIONS IN AN SHO We will stick to the exaple of the oscillating spring. The Potential Energy and Kinetic Energy and hence total energy of the oscillating spring is given by the following Equations U = 1 2 kx2 & K = 1 2 v2 & T = U + K It is obvious fro the equations that U and K are functions of tie as x and v are functions of tie. Let us know put in the value of x and v into these equations and see what we get the following result x(t) = A cos(ωt + φ) & v(t) = x (t) = Aω sin(ωt + φ) U = 1 2 k[x(t)]2 = 1 2 k [A cos(ωt + φ)]2 = 1 2 ka2 cos 2 (ωt + φ) K = 1 2 [v(t)]2 = 1 2 [ Aω sin(ωt + φ)]2 = 1 2 A2 ω 2 sin 2 (ωt + φ) = 1 2 ka2 sin 2 (ωt + φ) ω 2 = k T = U + K = 1 2 ka2 cos 2 (ωt + φ) ka2 sin 2 (ωt + φ) = 1 2 ka2 [cos 2 (ωt + φ) + sin 2 (ωt + φ)] T = U + K = 1 2 ka2 { sin 2 A + cos 2 A = 1} Also, we have T = U + K 1 2 ka2 = 1 2 kx v2 v = ± k (A2 x 2 ) = ±ω (A 2 x 2 ) This expression confirs that the speed is axiu at ean position x = 0 and iniu at the extree position x = A. 3 qed_ak_gesphyxi1211

4 Exercise 06: A kg object is connected to a assless spring of force constant 30.0 N/ oscillates on a horizontal frictionless track. (i) Calculate the total energy of the syste and axiu velocity of the object if the aplitude of the otion is 8.00c. (ii) What is the velocity of the object when the position is equal to 4.00c? (iii) Copute the kinetic and potential energy of the syste when the position is 4.00c and hence find the total energy. Learn the derivation for the tie period for a siple pendulu, the idea of sall angle approxiation and the derivation for the tie period of a physical pendulu fro the text book pages 453 & 454 or fro MITOCW801 Lecture 10; 29:10 40:50 and MITOCW801, Lecture 30; 00:00 21:00 4 qed_ak_gesphyxi1211

5 DAMPED SIMPLE HARMONIC OSCILLATIONS (No Probles will be asked fro this section) You have already seen the ideal case of an SHO where the syste keeps oscillating for every without any loss of energy. At all ties, the energy of the syste reains to be T = 1 2 ka2. As you can see fro the equation, the energy is proportional to the square of the aplitude of oscillation. This eans that if we double the aplitude, the energy of the syste will becoe 4 ties the original or if the aplitude is ade 3 ties the original, the energy becoes 9 ties ore. In real life, as you can iagine, there is no oscillation that continues forever due to the presence of frictional forces that opposes otion. FRICTIONAL FORCES IN NATURE (Not in your syllabus) Frictional force always opposes velocity and in the ost general for is given by the vector equation F = c 1 v c 2 v 2 v In non-vector for, it can be siply written as [F = c 1 v c 2 v 2 ] Since the frictional forces ust always oppose the velocity, it is therefore obvious that c 1 and c 2 are positive scalar nubers. The first ter ( c 1 v) is called the Viscous Ter and is proportional to the velocity and the second ter ( c 2 v 2 ) is known as the Pressure Ter and it is proportional to the square of the velocity. (Watch MITOCW 801: Lecture 12 for ore inforation and deonstration of the two ters; but not for this exa). It turns out that when the velocity is high the Pressure ter doinates. The speed of cars or aero planes or even the rain drops are high enough for the Pressure Ter to doinate. On the other hand, when the speed is very low, like that of a sall ball bearing falling inside honey or oil, or that of an oscillating ass attached to a spring or in the case of a siple pendulu, the Viscous Ter doinates. So for daped oscillation, we include an extra force ter that is proportional to the instantaneous velocity of the oscillating body and is opposite in direction. Exercise 07: What is the unit of c 1 and c 2? A Daped SHO is and SHO in which there is loss of energy due to frictional forces in nature. Since the energy of the SHO is given by the equation, T = 1 2 ka2, if there is a loss of this energy, since k is a constant, as you can iagine, the aplitude of the oscillation keeps decreasing as tie goes by and naturally, this is what one would iagine would happen when an oscillation is affected by friction. Now let us deal with this atter atheatically. Frictional force in the case of oscillations of sall velocities is proportional to the velocity and is in the direction opposite to the velocity. Therefore, the equation of otion now changes fro F = kx and F = a giving a + kx = 0. We have an additional force ter which is the frictional force or the retarding force and it is proportional to velocity and is given by f = bv = bx, where b is known as the daping coefficient. So the equation of otion now is as shown on the right. The final equation is therefore x + b x + k x = 0 The solution for this equation is given by b x(t) = Ae 2 t cos ( ωt + φ) Let us write the three ters separately and see what they all ean b x(t) = [A] e 2 t [cos( ωt + φ)] The first ter A as you know sets the axiu value of aplitude. The Third ter [cos( ωt + φ)] akes the function x(t) to oscillate between +A and A as the cosine function cos ωt itself oscillates between +1 and 1 at an angular frequency of ω = 2πf. The second ter is very interesting. You can see that it is in the negative power of e. (e (constant)t ). So, at tie t = 0, we will have the second ter e 0 = 1 e0 = 1. This eans that when the oscillation starts, it will have the usual axiu value of Aplitude. At very large value of tie, say tie t =, we will get e = 1 e = 0. This eans that after a very long tie, the value of x(t) = 0 no atter what every the other values, telling us that the aplitude will go to zero and all the energy therefore will be dissipated to the surroundings. A plot of this equation will therefore look as given alongside. The angular frequency ω of such an oscillation will be ω = k 2 b 2 ω = ω 0 b 2 2 where ω 0 is the natural frequency of oscillation of the syste, in the absence of friction. F = f + ( kx) = a F = ( bv) + ( kx) = a a + bv + kx = 0 a + b v + k x = 0 d 2 dt 2 x(t) + b d dt x(t) + k x = 0 5 qed_ak_gesphyxi1211

6 FORCED SIMPLE HARMONIC OSCILLATIONS AND THE IDEA OF RESONANCE (No Probles will be asked fro this section) It is possible to force an oscillating syste into oscillation by an oscillating external force. A siple way of representing such a DRIVING FORCE is F(t) = F 0 cos(ωt). If such a tie dependent force is ade to act on a syste, the syste will first resist and the syste will be confused and such a state is called the transient 4 phase. It will try to oscillate in its own natural frequency ω 0 = (k/ ) but the external force F(t) is persistent and will continue to act on it. So sooner or later, the syste will have no choice but to take up the frequency ω of the driving force F(t). When the syste reaches such a state, we call it the steady state.look at the equation of otion given on the right. The final result will be x + k x = F 0 cos(ωt) and we have agreed that at steady state, the solution in tie will inevitably be x(t) = A cos(ωt). Let us put this solution equation into the equation of otion given above and see what we get x (t) + k x(t) = F cos(ωt) F = F(t) + ( kx) = a F = F 0 cos(ωt) + ( kx) = a a + kx = F 0 cos(ωt) a + k x = F 0 cos(ωt) d 2 dt 2 x(t) + k x = F 0 cos(ωt) But x(t) = A cos(ωt) & x (t) = Aω 2 cos(ωt) ω 2 Acos(ωt) + k A cos(ωt) = F 0 cos(ωt) ω 2 A + k A = F 0 A k ω2 = F 0 A = F 0 k ω2 A = F 0 (ω 2 0 ω 2 ) ω 0 = k This equation gives us the value of Aplitude of the syste and we see that it depends on the value of F 0,, ω & ω 0. If the value of ω 0 ω or ω 0, then the value of aplitude is siply A = F 0 2 ω = F 0 0 k ω 0 = k If the value of ω 0 ω or ω, then the value of aplitude is A = 0! The external force oscillates the syste so fast that it does not have tie to respond and hence it does not oscillate. NOW if ω 0 = ω, then we get A =! THIS IS CALLED RESONANCE. In reality however, due to the presence of daping, we will never have an infinitely large aplitude but we will have a very high value of aplitude. The plot of the agnitude of A versus ω for different values of daping coefficient b is shown on the right. Don t have to know how we got the following, nor do you need to learn it by heart In the presence of daping, the equation of otion of forced oscillation will have a daping ter in it. So we will have and the aplitude will be given by x + b x + k x = F 0 cos(ωt) A = F 0 (ω 0 2 ω 2 ) 2 + bω 2 This will better explain the graph for you that s all. 4 Transient eans Lasting a very short tie 6 qed_ak_gesphyxi1211

7 MECHANICAL WAVES You have often heard the ter Wave in your day to day lives. We are going to analyze what it is fro a very Physics and Matheatics point of view. If you tie a very long rope to a pole, hold the other end of it and then give it a jerk, we notice that the jerk oves forward along the length of the rope. In physics such a jerk is called a PULSE. If we were to oscillate the end of the rope up and down continuously, we see that these consecutive pulses ove forward along the length of the rope. If one were to look at it fro a distance, one would say I see waves. Without a doubt, in physics, this IS what we ter as WAVES but the question is What really is it? We see soething oving along the length of the rope (x direction), but on close exaination, we see that every particle on the rope only oves up and down continuously (y direction)! Let us exaine the rope itself. We see that every single point on the rope only oves up and coes down (y direction). What really happens is that the jerk is given to the very first particle or point ass on the rope and this particle is connected to the next particle and so it pulls the second particle in the direction in which it is oving with a very slight delay. The second particle lags behind by just a little. The second one in turn pulls the third; the third one pulls the fourth and so on till the end of the rope. Now we know for sure that the particles on the rope only ove in the y direction. So if one were to ask the question, what oves in the x direction then, how would one answer? It turn out that the best way of looking at it is fro the point of view of energy! What oves forward is the energy supplied on the very first particle. In short, a WAVE IS A PROPAGATION OF ENERGY. It is a fact that soething ust oscillate in order for a wave to propagate. When particles with ass oscillate, the waves hence produced are called MECHANICAL WAVES. We have yet another kind of wave where what oscillates is not particles but electric and agnetic fields and such pulsing forward of oscillating electric and agnetic fields are called ELECTROMAGNETIC WAVES. Light is an exaple of EM Waves and they propagate at the rate of /s. Depending on the direction of propagation of the wave and the direction of oscillation of particles, waves are classified into two types. When the direction of propagation of the wave is perpendicular to the direction of oscillation of the particles we call those waves as TRANSVERSE WAVES. The exaple of the rope you saw above is that of a transverse wave. The up pulse is called a crest and the down pulse is called a trough. Light is also a transverse wave. When the direction of propagation of the wave is parallel to the direction of oscillation of the particles we call those waves LONGITUDINAL WAVES. Sound waves are good exaples of longitudinal wave. They are really pressure waves. When we speak, we oscillate the air in the outh and outside. The disturbed air olecules push the air olecules next to it and they push their neighbours and so on in space. In other words, we (ever so slightly) change the pressure around our outh and this pressure difference is propagated forward in all directions in space (at about 330/s if the sound is ade in air). Hence in the absence of particles, we hear no sound. The sae thing happens when the string of a guitar oscillates. It changes the pressure of the air around it and this pressure difference is transitted in space. Another ore practical way of studying longitudinal waves is shown in the figure. 7 qed_ak_gesphyxi1211

8 If a spring is oscillated back and forth, it akes copressions and rarefactions. All waves, both transverse and longitudinal, have certain inherent properties. They are discussed below. PROPERTY SYMBOL DEFINITION TIME PERIOD T Tie taken for one coplete to and fro otion of any single particle on the wave. FREQUENCY f The nuber of oscillations (or vibrations) in one second WAVE LENGTH λ (labda) The distance between two consecutive troughs or crests (in transverse waves) OR The distance between two consecutive copressions or rarefactions (in longitudinal waves). Of in ost general ters, the distance between two consecutive particle in the sae phase. AMPLITUDE A Maxiu displaceent fro the ean position of an oscillating particle WAVE VELOCITY v The velocity with which the wave propagates forward. The tie period is the tie taken for one coplete to and fro otion of any particle on the wave. When this particle finishes one coplete to and fro otion, the wave would have advanced forward by a distance equal to the wave length λ. Hence we can write the relation v = λ/t which gives us the final relation v = fλ. This is true for ALL waves. STATIC SINUSOIDAL WAVE A sinusoidal wave gets it nae fro the fact that it follows the sine function. A static wave that does not ove (ore like the photo graph of a oving wave is given by the equation y(x) = A sin (kx). (Don t forget that you could use cosine function also). Here, y is the displaceent of any particle at a point x eters fro the origin point, A is the aplitude and k is called the WAVE NUMBER and is equal to k = 2π/λ. Substituting this value, we get the above equation as y(x) = A sin 2π x λ What is the eaning of this equation? It eans that if you want to find the displaceent of a particle at a position x eters fro the origin, then all you have to do is put that value of x in the above equation and you will get the answer provided you know the values of A & λ. Since the factor 2π is in front of the sine function, and since sin(nπ) = 0 for all whole nuber values of n, we see that the function y(x) will be zero for all x values that are half integral values of values of λ. That is, y(x) = 0 for x = 0, λ 3λ, λ,, 2λ etc 2 2 Exercise 08: Plot this above equation on a graph, or use the software Microsoft Math 2007 to see what this wave looks like. TRAVELLING WAVES When the Looking at it fro a echanistic point of view, what we see is that a wave happens on a 2 diensional plane and it changes shape as tie goes by. However, the change in shape is periodic and repeats itself in tie. To be exact, a particular shape of the wave repeats itself after a tie T equal to the tie period. The wave is also periodic in space. This eans after a certain distance, the shape of the wave repeats itself. To be exact, after a distance λ, the wave repeats itself in shape. On a two diensional graph, you plot a dependent variable y as a function of an independent variable x. If you want this function to ove in tie in the +x direction with a speed of v /s, what we need to do is replace x with (x vt). So if we have a rando straight line equation y(x) = x + c, and if we want to ove this function in tie along the +x direction, the we re-write the equation as y(x, t) = (x vt) + c. We see that the dependent variable has now becoe a function of not just 1diensional space x but also of tie t. In the sae way, if we want to ove a wave in the +x direction with a velocity v, we do the sae process to the static sinusoidal wave equation. This gives us y(x) = A sin(kx) y(x, t) = A sin [k(x vt)] We now define angular frequency of the wave ω = 2π/T and the new equation above changes as follows y(x, t) = A sin[k(x vt)] = A sin 2π λ (x vt) = A sin 2π x λ vt λ = A sin 2π x λ t T T = λ v y(x, t) = A sin 2π x λ t T or in short, y(x, t) = A sin(kx ωt) where k = 2π λ, ω = 2π T & v = ω k 8 qed_ak_gesphyxi1211

9 A wave represented by this equation is known as a TRAVELLING WAVE. A ore general equation would be one with the phase constant included y(x, t) = A sin(kx ωt + φ) y(x, t) = A sin 2π x λ t + φ T This TRAVELLING WAVE equation will hereafter be our guiding equation and so be very, very sure as to what it eans. I reeber that I spent a long tie explaining what this equation eans. If you don t understand, DO NOT hesitate to call e. Exercise 09: Plot the TRAVELLING WAVE equation in the software Microsoft Math 2007 to see what this wave looks like. Exercise 10: A sinusoidal wave travelling in the positive x direction has aplitude 20.0 c, a wavelength of 40.0 c and a frequency of 10.0 Hz. The vertical position of an eleent of the ediu at t = 0 and x = 0 is also 20.0c. (i) Find the angular wave nuber, Tie period, angular frequency and speed of the wave (ii) Deterine the phase constant φ and write the general expression for the wave function SUPERPOSITION OF WAVES Waves interfere with each other. If any two separate travelling waves eet, they add up following the rules of vector addition! That is to say, if the troughs of two travelling waves of equal aplitude eet at a point, it will give you twice the aplitude. On the other hand, if a trough and a crest eet they will cancel each other and give no displaceent of that particle fro the ean position. Say in general, we have two travelling waves, y 1 (x, t) = A 1 sin(k 1 x ω 1 t) and y 2 (x, t) = A 2 sin(k 2 x ω 2 t) then the resultant of the two waves will be given siply by y(x, t) = y 1 (x, t) + y 2 (x, t) = A 1 sin(k 1 x ω 1 t) + A 2 sin(k 2 x ω 2 t). This is a very general equation. What happens is shown in the figure The first set of figures shows what happens when two crests eet. The second set shows what happens when a crest and a trough eet. Reeber that in the second case, although it is given as y = y 1 + y 2, the value of y 2 is negative and hence the equation when the two waves eet, we have the relation y = y 1 + ( y 2 ). Therefore they cancel each other. There are soe interesting special cases of interference of travelling waves 9 qed_ak_gesphyxi1211