 Greetings, let us begin this post-T discussion. This will be on what I call as real effects of pseudo forces and we will also discuss relativity, Galilean and also Einstein. It will also pick up on the question that was raised in the previous class about experience with inertial frame of reference and how to recognize it and what are the consequences of being in a non-inertial frame of reference. So causality and relativity, these are at the very heart of mechanics, classical mechanics and we will discuss what are the implications of causality because the heart of Newton's first law which is actually found, which was found by Galileo. Galileo died in 1642, the same year when Newton was born. So Newton came after Galileo and Galileo had already discovered that in an inertial frame of reference motion is self-sustaining. I showed you that picture of a fellow dropping an object from the mast of a ship and it would fall in his laboratory. The results would be the same if the ship were at rest, docked or moving at a constant velocity. So this is what Galileo had already discovered. So if motion is self-sustaining, it is only the departure from this self-sustained motion which seeks a cause because self-sustaining motion requires no cause. The body will continue to move at a constant velocity and you don't have to explain as to why it continues to do so because that is its natural state, it is bound to happen. But if there is a change in this, if that state of equilibrium is disturbed, that you push it but then it stops and now you ask why has it not continued to move at a constant velocity? Then you come up with a cause that yes there was an interaction, in this case it is a friction, that there was an interaction and it is this interaction which is responsible for departure from equilibrium. And then when you look at this cause, then you start asking further questions about this cause. You conclude as Newton did that the cause results in a departure from equilibrium which benefits as a rate of change of momentum. Change of momentum is the result of this cause. The rate of change of momentum involves a rate of change of velocity which is the acceleration which is the effect of the cause and this effect is proportional to the cause. So f equal to ma is the linear proportionality between the cause and the effect. So this is the principle of causality. And it tells you that if there is a departure from equilibrium, there would be a cause and then modern physics asks further questions about this cause. What is the nature of this cause? Is it the gravitational interaction? Is it electromagnetic interaction? Is it nuclear strong interaction? Is it nuclear weak? Is there a unification of this? There is this electro weak unit. So you get into further details about the nature of this cause. But your whole idea of what a physical interaction is including important questions like what are the fundamental forces of nature or more appropriately what are the fundamental interactions in nature because in quantum mechanics one does not use the idea of force very much. We talk about interactions. So what are the fundamental interactions in nature? So these questions are related to these ideas which is why it is very important to recognize that self-sustained motion seeks no cause and if you recognize a frame of reference in which motion is self-sustained that is what an inertial frame of reference is. So this is the idea which is an extremely robust idea which is a very important idea which Galileo recognized and this goes at the heart of mechanics. So I sometimes browse the internet for some physics related news and I came across this just one week ago 20th April and this is about the leap second correction which has to be introduced a few weeks from now. On 30th of June this year a leap second would be added to the year. Everybody knows what a leap year is okay that's when you have the 29th of February a day is added. Now every once so often you have to add what is called as a leap second and on 20th of June one second will be added and the reason is that the earth's rotation is slowing down and then to make a correction for that you have to make a correction every now and then. So after this was realized a number of leap second corrections have been done. I think all this information is available on the internet. I believe the number of leap second corrections until now may be somewhere in the vicinity of 25 or so I'm not very sure about that. Now this will have a very major effect because all the clocks across the world will have to be reset okay the computers the GPS the cell phones all the clocks okay and this has to be done very accurately because the atomic clocks as this news says are accurate up to the quadrillion of a second and then I asked myself what is a quadrillion and then I got this definition that it is a cardinal number represented in the U.S. by 15 zeros one followed by 15 zeros and in great detail written it is followed by 24 zero so I won't worry about that okay so that is let us not get overly puzzled by that okay but the main issue over here is that you have to make this correction called a leap second and everybody who uses a cell phone okay understands the importance of setting the zero correctly okay your GPS cell phones everything every technology computers communication everything depends on that nobody can argue that this is an important thing which physics students need to learn at the right place where this is introduced is an undergraduate mechanics course okay this is these would be the first course after the high school that is where these can be introduced they can be introduced very rigorously and that's what we are going to discuss now but let's take another example and here I have these pictures of a solid liquid in gas and you know that a solid has a shape of its own a gas fills up the whole space of the container a liquid settles down in the container okay and now in this world of technology that we live in it would be important for us not just to think about you and me but also about the astronauts okay they are in the space ships they are in rockets they go all the way to different parts of the universe the orbit the earth right they go to the moon and the question we will ask is what would be the shape of a liquid if it is kept in a sealed beaker if this beaker is in the spaceship what will be the shape of this liquid okay now obviously just like this leap second correction I'm sure that all of you would agree that these questions are relevant to modern physics and a student who is at an undergraduate level should have a good grasp of these questions and what happens is he is not able to appreciate the consequences if he does not have much acquaintance with physics in an accelerated frame of reference okay so this is where the inertial frame and an accelerated frame or a non-inertial frame needs to be understood because the whole principle of causality and determinism that we have been talking about takes a completely different form in a non-inertial frame of reference okay so these are some of the questions that we are going to discuss in this hour so let's take an accelerated frame of reference so here you have got a red frame of reference which I will consider to be the inertial frame which is the one in which motion is self-sustaining and here this green frame is an act frame of reference which is moving along this black arrow at a constant acceleration f so the displacement vector of the two regions which is O double prime okay is given by this usual equation of motion right this which is well known right so this arrow would keep increasing in its length along the same direction and its length will increase linearly proportional to the initial velocity and quadratically proportional to the acceleration okay that is the law which we know so if you now look at an object in the red frame of reference but you look at that object also a different observer looks at this object from another frame of reference then the two position vectors will be related through this triangle law of addition of vectors okay so the position vector in the inertial frame is related to the previous position vector r double prime in the accelerated frame so that you have to add this vector over here okay now you take the time derivative so you get the velocity in the first frame and see how it relates to the velocity in the second frame and if you take the second derivative you will get the acceleration now this is the measure of departure from equilibrium equilibrium okay how do you measure departure from equilibrium it is in terms of the acceleration and then you ask what is this acceleration due to when you identify that cause that is the physical force or the physical interaction which has resulted in that acceleration that relationship is a linear one by the principle of causality f equal to ma that's Newton's second law so now you find that the acceleration in the inertial frame of reference which is on the left-hand side is not equal to the acceleration in the second frame of reference in the green frame of reference okay so these two accelerations are not the same and therefore the principle of causality which you would consider to be valid in the inertial frame of reference would not explain physics in the accelerator frame of reference okay so the causality really breaks down so the acceleration in the inertial frame of reference is the acceleration in the double prime frame of reference which is this green frame of reference plus the relative acceleration of the green frame with respect to the red frame so that is the relation you get if you multiply both sides of this equation by f then you get f double prime equal to ma minus mf so ma was your principle of causality ma is what appeared in Newton's principle of causality f equal to ma it explained why an object departed from equilibrium in the inertial frame of reference and f equal to ma holds good in the inertial frame but it does not hold good in the accelerated frame of reference okay so now you can ask that if you want to interpret f double prime also as a force but then the force is f which is ma to which it is not equal but you have to subtract something out of it and what you have to subtract is just a mathematical product of the inertia m times the acceleration of the frame of reference so mf quantity which you see in this relation here it is this mf okay this mf is a completely mathematical construct it is just the mathematical product of the mass and f which is just the acceleration of the green frame with reference to the red frame and this quantity must be subtracted it has got the dimensions of force looks like the force it has got all properties of the force in the sense that it has got the same physical dimensions mlt to the minus 2 however it is not a force in the sense that it is not gravity it is not a lexromatic interaction it is not nuclear stronger we and whatever our perception of a fundamental force of nature is that does not go into this term mf which is why it is called as a pseudo force now if you subtract a pseudo force from a real force you get a quantity which is of course not real okay because from a real quantity you have this you have subtracted something which is not real so the result is also a pseudo force in a certain sense right so what you have on the left hand side f double prime is not the real physical force it is not the result of a real physical interaction but an observer in the green frame of reference can pretend that okay if I consider this to be my force then I can explain the acceleration that I am observing so to him the acceleration is a measurable and it is a real effect but it is a real effect of a force which is really not a physical interaction in the usual sense so this is what I call as a real effect of a pseudo force and as you can see it is very important to come to terms with this because this has got consequences on the shape of a liquid in a beaker that an astronaut would see it would have implications on the atomic clocks which govern your GPS and your cell phones and everything so I will certainly like to invite you to this article which is available at my website and I won't really discuss this particular work which is reported in this article but I will certainly talk about this relationship that f double prime which is the quantity that an observer in the accelerated frame of reference would use to explain the departure so he's trying to invoke the causality principle but then he has to invoke a cause which is really not physical it is a mathematical construct of the fact that he is own frame of reference happens to be an accelerated frame of reference so these are fictitious forces and therefore they are not involved directly in the physical principle of causality you cannot apply Newton's third law for that and yet we do come across students who say that the centripetal and the centrifugal forces are equal and opposite because they are governed by Newton's third law which is absurd and it is important that we correct these mistakes early enough so this is really important because the perception of a physical interaction is important for us to understand what exactly are the laws of nature so what we are going to do is to discuss some related issues issues the weightlessness for example so if you are in a state of free fall okay if you are in a state of free fall then you are in an accelerator frame of reference an object which is falling okay is accelerated toward you right so this is an object in a state of free fall which is what it would be for an astronaut in a spaceship right and these things have got very fascinating in consequences for example if you are a pole water and jumping over clearing a bar then it's possible to flex your body such that the center of mass of your body can actually go below the bar whereas the body flexes and clears the bar so now the athlete skill lies in flexing his body swiftly and that is where his athletic skills are challenged to the limit that is how world records are set okay and he's able to do that because he in a state of free fall all he has to overcome is inertia of his limbs and not gravity because he's already in a state of free fall okay so he has to just deal with his inertia of the limbs not gravity and that is the reason he's able to flex his body so nicely and I think all of us have know this we we have heard that that cat has got how many nine and nine lines of 90 or I don't know how many but it has many lines and the reason you say that a cat has got nine lines because however it falls it falls nicely swiftly so that it lands safely and the reason it is able to do it is because it is in a state of free fall while falling and then it is able to flex its body very easily okay which would not be possible without this idea of what is an effective weight of an object okay an effective weight of an object in an inertial frame of reference would be different from what it is in an accelerated frame of reference okay so the same thing is true for a liquid in a beaker this is the question which I raised earlier and these are obviously important questions for modern science okay people have been going into space for well over several decades now right was it in the 50s 60s yeah yeah in the 60s they already went to the moon 59 something like that yeah so 57 was the first squatnik I think this Russian satellite and now if you look at the shape of a liquid in a beaker in a orbiting satellite which is in a state of free fall then everything is weightless including the liquid inside the beaker so gravitational interactions will not govern the shape of the liquid because here the reason liquid takes this flat shape for a beaker as you see in this model okay the reason the liquid has the shape is because gravity is holding it now right now that's not what is going to happen in a space so then it is the other interactions of the liquid which are normally very weak we don't look into them when you're just looking at water in this bottle but they become important and as a matter of fact they are important even in a laboratory because those are the interactions which determine the meniscus of a liquid okay you have got the cohesive interactions and the adhesive interactions these are the intermolecular forces between molecules of a liquid between each other or with the vessels and the relative strength of these interactions determine whether the meniscus is one way is it concave downward or upward right so that is these are the physical interactions which become important and now imagine these things happening in the satellite because then if the cohesive forces are important the entire liquid will settle together and you will have a globule of water or that liquid which is floating somewhere in the beaker whereas the adhesive forces if they are important if they are stronger then the liquid will stretch itself along the inner walls of the beaker leaving the cavity in the middle now for an astronaut who wants to have a diet coke I think this is important because then he knows how he is going to suck on that liquid right so it's very important to teach these things and one is able to grasp these ideas only if you introduce students to what exactly is an inertial frame of reference and what exactly is a non-inertial frame of reference so physics in inertial and non-inertial frames of reference is very important and these are the things which determine these things so I already explained if you have one frame of reference which is moving at a constant acceleration moving in one direction now let's consider physics in a rotating frame of reference now let's consider these children playing on a seesaw now sure if you have these kids if they're playing some additional games and one of them is passing a ball at the other okay and they are catching it and throwing it back and forth with respect to each other but then what if the seesaw is made to turn around the pivot okay what will be the trajectory of the ball when one ball one kid throws it at the other now if you're standing on the ground once the ball is thrown its motion will be determined completely by gravity okay its forward motion will be determined by the throat and vertical motion by gravity and it will go in some parabolic part those problems we have saw right but that's not the trajectory which the other kid will see because the other kid is not rotating about this and this is a very real situation not just on a playground but certainly for these two astronauts because the satellite is usually given a little bit of angular motion and that's to maintain stability because angular momentum is a constant so those are other details which I will not get into but then there is a little bit of rotary motion which is given to the satellites and and if these two astronauts are carrying out some work in the spaceship and one of them throws a spanner at the other fellow they're doing some experiments what will be the trajectory of this seen by the two astronauts and they are now in a rotating frame of reference and this is the example which some of my students worked with and they wrote a very nice computer code which gives the solutions for the trajectories and you're quite invited to use these codes and run them and act I'll give you one of the solutions that that if the radius of the circle was 10 meters and these are just some arbitrary numbers just for the sake of illustration so they're not important and if there is an angular speed of certain point one five radians per second if the initial velocity is 0 5 x and y components of the velocity in meters per second in this frame of reference and the astronaut B B for throws a tool toward a then the trajectory turns out to be what you see by this green line which is really not something that one would have imagined aspect right so these are very interesting examples and they tell you how physics would be seen in non-inertial frames of references and this is important because they connect to us the very fundamental idea of what a physical interaction is because that is what goes into the principle of causality okay now let's take another example over here means we all know that we are in the rotating frame of reference the earth is rotating okay from this time to the next time tomorrow it would be 24 hours right so we go through one full rotation and if a rocket is launched from the North Pole and it is supposed to land at the equator and at typical speeds if it is to take about an hour to get there then because of the rotation it will not land where you thought it would but that position would be offset by as many as 15 degrees that's a lot of distance okay on the equator you have about 111 kilometers per longitude degree and the distance would be about 1665 kilometers now think about it if you were or your students were to design intercontinental ballistic missiles or rockets okay this is important okay and it is absolutely important to learn how to handle the physics in these rotating coordinate systems so let us consider a rotating coordinate system so let's consider a frame of reference red which is our inertial frame of reference let us consider an axis OC and above this axis let us have another frame of reference which rotates about this green axis at a certain angular velocity okay it doesn't have to be along the x axis or y or z or something can be completely arbitrary direction in space and about this green axis you have got a frame of reference which is the frame of reference is it black or purple or I don't know what color it is some dark color let me call it as black okay or this is the one that subscript red a subscript R so this frame of reference rotates about this axis OC about a certain at a certain angle of velocity and in both frames of references you observe an object which is at a point P at a certain instant of time okay so there is some object whose coordinates the position vector is this in both frames the coordinates will be different okay the x y z coordinates will be different the position vector is the same and let us imagine that this point is not moving it is fixed in the rotating frame of reference it could be this water bottle in our frame of reference okay because in our frame of reference we are on the surface of the earth we are rotating once in 24 hours but in our frame of reference this object is not moving whereas if I pick it up and drop it it is moving right so I'm looking at an object which is at rest in the rotating frame of reference so the components of this vector in the inertial frame and in the rotating frame will be different so let us look at it carefully now so we consider this object which is at rest in the rotating frame of reference and if it is at rest in the rotating frame of reference it cannot be addressed in the inertial frame of reference right because it is with respect to the inertial frame of reference that this other frame of reference is rotating so the rate of change with respect to time the time derivative in the inertial frame denoted by subscript i is different from the time derivative denoted by the subscript r time is the same it is the time derivative which is different so let us say that this is the position vector at time t in the inertial frame of reference and in the inertial frame of reference at a later time the vector appears to be different and this is the difference vector this is this red arrow which is db okay so this is b at t this is b at time t plus dt or delta in the limit delta t going to 0 I take two points which are infinitesimally close to each other and this is the difference vector so this is the difference which the observer in the inertial frame of reference will see but the observer in the rotating frame of reference will continue to see it at the same point as we saw this bottle where it was minutes ago and we see it at the same place even now right but an observer in the inertial frame of reference would see it to be at a different place that difference vector is db so this db vector is now as you can see will be on the circumference of this cone as you can see from this geometry right and then it is elementary geometry to determine this vector db okay this is just high school level geometry and without getting into too many details you can see all the details in the detailed PDF which will be on the web so this is the difference vector db notice that this circle will have a radius of b sine xi where xi is the angle this circle subtends at the vertex oh right so this radius will be b sine xi and this difference vector will be this radius multiplied by this angular d psi okay so db will be given by the length of db times a unit vector which is along this red arrow right and this red arrow will be the cross product of n cross b it will be along that but you must divide it by the sign of the angle so that you will have a unit vector because I'm just looking for the direction of this vector so this is the direction and and the size and the magnitude of the unit vector and the size itself is b sine xi d sine right so this is your db now you can write db as b sine xi d psi and the vector will have this unit vector which is n cross b divided by the modulus of n cross b and now you can cancel this sine xi with this denominator because they are exactly equal so what does it give you it gives your db as d psi n cross b and therefore you can get d psi being omega dt omega being the angular frequency right it will be the angular rotation angular velocity the rate of angular velocity of this rotating frame of reference omega being d psi by dt the rate of change of angle and then you have db as omega dt cross b what does it mean that the rate of change if you divide both sides by delta t and take the limit delta t going to 0 essentially you'll find that a vector which is addressed in the rotating frame of reference has got a time derivative which is given by the cross product of the angular velocity with the position vector b itself okay now this is a very general result and a very important result because now what you can see is this relationship between the time derivatives time itself is the same in both the frames of reference it is the time derivative which is different and if this vector itself was not addressed for example I take this bottle and throw it right now it is not addressed anymore so the original velocity of this in the rotating frame of reference will get added to this earlier velocity so now you have a relation that d by dt of B will be omega cross B which we found from the previous relation but now you will have an additional velocity in the rotating frame which is given by the time derivative of this vector with respect to the rotating frame so what is the general relation this is true for any vector B so you can extract an operator equivalence and this operator equivalence is that the time derivative in the inertial frame of reference is equal to the time derivative in the rotating frame of reference plus a quantity which is given by the cross product of the angular velocity with the vector itself this is a very general relation and now we can apply this relation to the position vector of any object it can be this position vector of this bottle okay and when you apply this position vector apply this operator equality this is an operator equivalence what you have in a red bracket is a relationship for the operator equivalence and you apply this operator to the position vector R in both sides of the equation and you can do it one more time right when you do it twice you get the derivatives you get the derivative of the velocity you get the acceleration and these are what you interpret as departures from equilibrium this is the one for which you will look for a cause according to the principle of causality right so now you have got this relation and notice that the second derivative which is acceleration the inertial frame of reference is not equal to the second derivative of the position vector in the rotating frame of reference there are these additional terms and they're coming simply from this cross product and this relationship that we deduce so easily in just one or two steps okay so now you have this relation and if you multiply this by mass you will get quantities which we would interpret as forces so now we have multiplied both sides of the equation by mass this mass times acceleration in the inertial frame of reference is our physical force according to the Galileo Newton law and you must keep that as a reference all the time in every analysis that is the physical reference of water forces okay that idea of force is what we would interpret as a physical interaction it could be gravity it could be electromagnetic interaction it could be any other physical interaction okay and everything else is appearing because of these mathematical constructs which are results of the fact that we were observing in a rotating frame of reference in an accelerated frame of reference so these are the terms so if you now write this mass times acceleration in the rotating frame of reference as f of r so this is the perception of a force in a rotating frame of reference which would be what how I would try to interpret the forces on this bottle then this would be the equal of a real force plus all of these terms right and they come with their own sign so this term comes with a minus sign this depends on omega dot this is coming from this term which is omega d omega by dt cross r so this is the one which gives the leap second correction this is the one which I mentioned a little while ago and because the earth is slowing down this correction has to be made then there is a component which is coming because of this cross product of omega with this velocity in the rotating frame of reference so if in the rotating frame of reference if this velocity was 0 as it is for this bottle this term would not have any effect but if this were falling if it were to have some velocity of its own then it will certainly matter right and then you have another term omega cross omega cross r which will be valid even if it were not having any velocity of its own that is the centrifugal term so these are pseudo forces and they do not come into the operation of interpreting physical interactions in the inertial principle of causality despite which we sometimes have students who tell us that centripetal and centripetal forces are action reaction pair according to Newton's third law which is very unfortunate and which must be corrected okay and it has to be corrected very rigorously by giving them these examples okay so these are important chunks in fact one of the quiz questions which I like to ask is that if you have a plumb line okay you know what a plumb line is some thread that you suspend at the roof and have a mass attached to that and how would this plumb line orient itself and how would you describe a vertical line with reference to the plumb line can you define it as a line as a dramatical line if you consider earth to be a sphere so do you describe a vertical line as a point joining the center of the earth with a point on the surface and you extend it back and forth is that your definition of a vertical or is your definition of a vertical given by how a plumb line would orient itself is that your definition of a vertical or you can even give a third example that you take a marble or any object and drop it and then ask is your definition of a vertical the same as the line along which this object forms and if you look at these terms carefully you will find that all the three definitions of a vertical the geometrical definition the plumb line definition and the definition of a falling object all of these three give you different space cars okay it's because of plumb line which is at rest okay is not an object in a state of fall therefore the Coriolis term would be zero okay but the century Google term would be there and that is the reason all the three definitions are different so these are some examples of the consequences of carrying out observations in non-inertial frames of references so in a rotating frame of reference you have the lead second correction you have the Coriolis term you have gone to century Google term and of course if the two frames of references are also in a state of relative acceleration with respect to each other you also have the earlier effective weight kind of situation so these are very important because they determine the trajectories of rockets airplanes ships if you're if your students are becoming engineers and they are devising they're making instruments for navigation and they want to know where their position is but the ship is in a sea or if it is in air okay and even if they were not in the MH 370 they would need to know their coordinates correctly right and there's no way you can get it right unless you incorporate all of these terms in your navigational instruments so these are important for modern technology for devising your atomic clocks GPS systems cell phones and all these corrections the lead second correction I already mentioned the full copangulum is a very popular and very nice example which we have dealt with quite extensively in this course you can go through the details on the website but they also explain some very interesting phenomena like vortex currents in a sea or in the atmosphere and so on in fact it would even determine the trajectory of a sixer hit by the nulkar or any maybe today I should talk about Virat Kohli I guess the slide was made when the nulkar was at his peak and if he hits a sixer in the northern hemisphere playing in London it would be quite different if it if he were to hit it in Sydney in Australia because on the North Pole and South Pole both ways you know the deflection would be in the other direction so these are the Coriolis effects which cause the vortex currents of the cyclonic currents so they are clockwise and the northern hemisphere anti-clockwise in the southern hemisphere and so on so they have very important consequences in atmospheric sciences and ocean currents and so on so let's go over to the other principle which is in the heart of mechanics so I mentioned causality as one of the dominant principle which I discussed at some length until now I will now discuss relativity which is again at the very heart of mechanics and this is a picture of Brett Lee bowling to Tendulkar and we talked about updating this slide right so we thought we should have Mitchell Johnson bowling to Kohli now but we can do that all right so let's say that we are watching the game now can you imagine this game played on a huge truck okay so this is a huge play field okay and the game is being played and there's no reason why the game cannot be played if the truck is large enough right okay and one could have a lot of fun playing this game and the game would proceed exactly the same way as it would if you were watching it on a play field right now Beckley bowls at 152 kilometers per hour and if this truck was also moving at 152 kilometers per hour in the direction opposite to the ball right then the speed of the ball as seen by an observer in the truck would be different from the speed of the ball seen by an observer on the ground right and the way he will get it is by taking the difference of the two speeds right so the speed of the object depends on the observer's frame of reference but then mind you this is not a hypothetical situation because the ground in Sydney or what is the Calcutta famous ground called Eden Gardens right the Eden Gardens it's already on a platform which is moving right means that it is moving at the along the earth at 1650 kilometers per hour so it's it's already a situation that we cannot really miss and Galilean relativity is governed by the fact that the laws of mechanics are exactly the same in all inertial frames of references whether it is on the playground or on the track or on the earth the laws of mechanics are the same so that is the principle of causality and Galilean relativity deals with this now here we were talking about an object like the cricket ball okay so what you're observing is the cricket ball and then the speed of the ball is determined by the relative speeds so the relative speeds determine are determined by me if you're walking your speed is determined by whether you are walking with respect to the to the earth or you measure your speed with respect to the treadmill itself okay which is a good thing to do but sure the difference of the speeds is what plays an important role now here you're talking about the velocity or the speed of the chicken or the cricket ball or the man but what will happen if the object you're looking at is not like a cricket ball or something because all is you would do to get its speed would we take the difference in the Galilean relativity as we have discussed earlier what if you're looking at light you're not looking at a cricket ball you're looking at light what is the speed of light and another observer is also measuring the speed of light and he's not in an accelerated frame of reference or don't bring in any solar forces laws of mechanics will be the same right and he is also measuring the speed of light so this has something to do with the measurement of the speed of light and this is the famous picture of the experiment by Michelson and Morley which a lot of people say was at the foundation of the theory of relativity which is not quite correct because here is an extract from the history archives of the American Institute of Physics which points out that Einstein was certainly aware of the Michelson-Morley experiment but it perhaps did not play a big role in his composing or coming up with the theory of relativity so this if at all it played any role at all it was only a minor one so what did so let's look at this experiment so here you have not a region of space in which there is a magnetic field and this cross tells you that this is the tail of an arrow that you are looking at and the magnetic field is perpendicular to the plane of the screen to the plane of this figure the magnetic field is orthogonal to the plane of the screen it is going into the screen okay so that is the direction of the magnetic field and in this region of space you have got a certain loop okay this is just a wire and connected over here by a resistance now the question is that if the loop is stationary will there be any current in this will it be clockwise will it be anticlockwise there is no current right now suppose you have some device with which you control the position of the loop and you drag it you drag the loop to the right now will you now have a current in the loop what will be the direction of the current it would be clockwise okay and how do you get it by thinking of a positive charge like what generates conventional current and you construct the Lorentz force qv cross b you're absolutely right and with this you immediately find that there would be a current which is going clockwise okay now let's extend this experiment further let's have the same situation but now you do not move the circuit but instead you drag the field itself to the left so the field can be generated by some horseshoe magnet or two magnets and you move that magnetic assembly in the opposite direction now let us ask this question will there be a current in the loop what will be the direction of the current again it will be similar and now if you do another experiment in which you do not move anything but just change the magnetic field by changing some controls and then again you can find that there would be an induced current now look at this experiment over here in this experiment you have not moved the circuit so the Lorentz force would be zero because in the previous one here you involve this qv cross b so if this velocity of this charge was zero the v cross b would be zero and there is absolutely no Lorentz force in the next experiment right so here in this experiment there is no Lorentz force yet you argue that there will be an identical current so what kind of physics is this now how do you explain this how do you explain that there will be a current we all agree that there is a current we all agree even with the direction of the current but now we are confronted with the fact that we cannot invoke the Lorentz force because the velocity of this charge is zero you're not moving it what you're moving it is you're leaving it as it is but you're dragging the magnetic field so how do you explain this now this is a very fascinating example which many books dismiss as trivial okay and some with wrong explanations oh what is a big deal there is a big deal there is a very big deal about this and let me tell you how big that deal is let me quote a distinguished physicist on this so let me read this quotation for you so the flux rule that the emf in a circuit is equal to the rate of change of the magnetic flux through the circuit applies whether the flux changes because the field changes or because the circuit moves or both yet in our explanation of the rule we have two completely distinct laws one is Maxwell's equations okay which is the Faraday Maxwell equation and the other is the Lorentz force v cross v they're completely two different explanation okay and this court let me extend this quote to the next part of the quote in which he points out that we know of no other place in physics where such a simple and accurate general principle requires for its real understanding and analysis in terms of two different phenomena one is the Maxwell's equation the second is the Lorentz force okay and this is no ordinary man saying it it's Richard Finland okay and in his lectures which everybody always claims he has read but not many are able to recognize the quote so this is straight out of lecture and this is because of the connections the intrinsic connections between electrodynamics and relativity okay so you have got the Coulomb's law the Ampere's law the Faraday laws and so on so all of this are embodied in Maxwell's equations and the theory of relativity is right at the heart of these explanations okay the reason these you have to invoke completely these two explanations is because of the intrinsic nature of the electrodynamic interaction okay there is electrodynamic interaction which is fundamental to this and this is what Einstein recognized from the symmetry of Maxwell's equations okay now I mentioned in my earlier class that the connection between symmetry and conservation laws and the reason symmetry started playing a big role in physics it all started out with Einstein and this is where it began its journey okay it's because in the symmetry of the Maxwell's equations the curl of E and the curl of B the two equations are completely symmetric the equations for the divergence are also symmetric except for the fact that there are no magnetic monopoles but other that they are completely symmetric and this symmetry leads to the fact that when you take the subsequent you carry out one more operation of this because from this it is very easy to take the divergence of these curly expressions and get the wave equations right and when you get the wave equations you'll find that the waves propagate at a certain velocity which is the speed of light and it is determined only by properties of vacuum the speed of light is determined only by mu zero and epsilon zero and by nothing else and when you look at the speed of the cricket ball okay or speed of the chicken crossing the road you are always talking about a speed which is measured with reference to an observer and with reference to some observer it has got one value with reference to another observer it has got another value but this observer disappears when you come to the speed of light because it is determined completely by properties of acumen by nothing else so it is essentially determined by absolutely nothing and these are important because unless these things are taken into account correctly you cannot communicate with each other so the cell phones won't work no technology will work without relativity without quantum theory and these are absolutely important so what I try to recognize is that Maxwell's equations are correct in all inertial frames of references they predict that light travels at the speed which is determined by mu zero epsilon zero and if the speed is the same in all inertial frames of references what is speed speed is distance divided by time speed is the same but then something has to change distance will change and time will change it was this recognition which led him to the theory of relativity and then you get time dilation length contraction and so on okay so all of these are consequences of Einstein's intuition which led him to the theory of relativity it also changes your perception of what is simultaneity because what is simultaneous to one observer is not the same for another so it also explains many other things like there is this famous twin paradox and relativity that a traveling twin ages less than a homebound twin and this train paradox is explained in several books and there's a lot of literature on that some of it even invokes the general theory of relativity which is not correct because this is completely resolved within the framework of the special theory of relativity by simply analyzing the obvious consequences of Lorentz contraction and time dilation so I will not get into these details but we have discussed these things at length we also discuss what is the paradox over here because women do not age at all that is to many the fundamental paradox but anyway so it has other consequences that what is faced to one observer is a mix of space and time to another so likewise what is the electric field to one observer is a mix of electric and magnetic field to another observer and vice versa so this is what very fascinating consequences which are responsible when we analyze simple examples of this kind so some of my students they have developed a nice software which illustrate the connections between electrodynamics and the special theory of relativity these programs are available for you to use they also have important consequences in quantum mechanics because we often use the idea of a spin of a particle and these these models are absolutely wrong spin has to be defined correctly in quantum mechanics I'll probably do it in the afternoon session you need the Dirac's formalism for which you need the special theory of relativity so relativity has got important consequences in classical electrodynamics and quantum mechanics and everywhere so I think I'll stop here and we have Professor Regian's lecture which is about to start so I'll take a few minutes if there are any questions and then we'll have Professor Regian's lecture any question we can of course continue to chat during the break and even later by email and so on but if there is any quick question for now I'll be happy to take otherwise I get a break but actually by carrying out these simple transformations from of a position vector of an object in an inertial frame to the position vector of the same object in a non-inertial frame okay and these two vectors you relate just by the triangle law of addition and then go on to construct the time derivative of that position vector take the second derivative you get acceleration multiplied by the mass you get the force so these simple transformations give you the exact mathematical relationship between the accelerated acceleration seen by one observer in the inertial frame of reference and relates it exactly quantitatively to how the acceleration would be seen or observed by another observer in an accelerated frame of reference and the two will be different and the difference will be because one observer is in the inertial frame of reference who will define the fundamental laws of physical interactions and the other observer will invoke pseudo forces he will have to invent those pseudo forces to explain those effects so thank you very much I guess I will invite Professor vision I would like to take this opportunity to thank Professor vision not just for giving the lecture which is about to begin but for making this course possible at all because when I first gave this course which was several years ago it was possible only because Professor vision and I we discussed the physics we enjoyed doing physics learning physics and we both felt that okay these are certain things that we can introduce in undergraduate physics so we actually did it together so I'm immensely grateful to Professor vision for his partnership in developing this course thank you vision