 fundamental concepts of surface tension and we will continue with that. Let us say that we want to see some application of surface tension as a very simple case. To do that we will take an example when there is a capillary tube which is small in radius it is dipped into a larger volume of fluid and let us say that this was the initial level of the fluid and now once this capillary is introduced this fluid is expected to either rise or fall and we will see that what dictates that it should rise or fall. To understand that let us assume that the fluid is rising over the capillary. If it rises over the capillary then what dictates is rising and to what extent it should rise. Let us try to make an estimate of that. First question should come that why it should rise or why it should fall. When there is a fluid it is a collection of molecules after all and these molecules are subjected to various interactions. One type of interaction is the intermolecular forces of attraction between the similar types of molecules which are like so called cohesive forces. The other one is the intermolecular force of interaction between the molecules of the fluid and the molecules at the boundary. So when you have that type of interaction that is known as adhesion. So it is between two different types of entities. Now if the adhesion wins over the cohesion then that means the surface at the end has a net attraction towards the fluid and it likes the fluid. Earlier we used a terminology called as hydrophilic material. So such types of substrates which like water are called as hydrophilic ones. A better terminology would be wetting because when we say hydro it means water so to say but it may be any other fluid also. So better we say that those are wetting surfaces. So the surfaces which want to be wet with the fluid that is there. Let us say that there is a wetting fluid and it has formed a meniscus like this. Meniscus can be of a very complicated shape but let us just for sake of simplicity assume that it is a part of a hemisphere or like it is a hemispherical cap so to say. Now if you try to identify that what are the various forces which are related to the development of this meniscus. We have discussed about this earlier. So just if we reiterate that if we have a meniscus like this loosely speaking there will be a force because of surface tension which is acting over the periphery. There is a pressure acting from this side. There is a pressure acting from the other side the bottom side and it is basically filled with the same fluid which is also there in the reservoir. So if we say that at the top there is some gas or vapour and at the bottom there is a liquid then let us say this is p liquid and let us say this is p gas. So which one will be greater here p liquid or p gas in this configuration. You expect p gas to be more because it has to balance the p liquid and the component of the surface tension in that direction. So if you have p gas greater than p liquid that means there is a pressure differential across the meniscus. So if you have the pressure in the gas as a atmospheric pressure obviously the pressure in the liquid is not atmospheric pressure across the meniscus and there is a difference in pressure and the difference how do you relate we had derived the formula delta p equal to sigma into 1 by r1 plus 1 by r2. So if we consider it a part of a sphere r1 equal to r2 equal to r. So we are approximating the shape of the meniscus in that way otherwise it may have different radii of curvature at like different planes but this is a simplistic assumption. So if r1 equal to r2 equal to say small r this small r is not same as the radius of the capillary. So this is different because the small r is the radius of curvature of the meniscus which is not same as the radius of the capillary. So we have this as 2 sigma by small r obviously we need to relate small r with capital R and to do that we can say have a small geometrical construction. Let us say that this angle is theta. So when we have this angle as theta this is the angle made by the tangent to the interface with the vertical. So that should be same as the angle between the normal to the interface and the horizontal. So let us say that that is theta this normal is in the direction of the radius of curvature of the interface. So this length is what? Small r and what is this length? This is capital R because this is say the centre line of the tube. So this is capital R. So how you can relate small r with capital R? So small r will be capital R divided by cos theta. So we can simplify this 2 sigma by r as 2 sigma divided by capital R and then cos theta will go in the numerator. And what is this delta p? This delta p is the difference in pressure between the outside atmosphere and what is there? Just inside. What is the pressure at the outside atmosphere? The pressure at the outside atmosphere is same as what is the pressure at this level. So if you know what is the pressure at this level then from that you can find out the relationship between the pressure difference of these 2. And that pressure difference is nothing but because of the presence of this much of liquid column. So if this is let us say that it has average height of h. So this is nothing but h into rho of the fluid into g. So from here you can find out an expression for h which is 2 sigma cos theta divided by rho g r. This expression is very simple and this expression is valid only if this meniscus is under a static equilibrium condition. It is not moving but still it gives a lot of good insight because it tells you that if there is a capillary rise. So in this kind of a situation we call it a capillary rise because as if the fluid is rising from the reservoir towards the capillary. So that capillary rise is dictated by many things. One of the features of course is the angle theta which is the contact angle and which depends on the combination of the fluid. So if we say that for a glass water combination, glass water air combination say this theta is close to 0 maybe. So that means that it requires 3 different phases so to say. I mean glass is a solid phase which forms the surface of the capillary. Then you have water here and maybe air here but if you replace water with say mercury it may be something different and it may be possible that instead of a rise it has a fall that is dictated by the competition between the adhesion and the cohesion force and eventually it is manifested by what is the value of the contact angle. So in that case say if the contact angle say is 140 degree which is like if it is mercury glass air that may be like one of the possibilities. So when you have this one that means you will get if you have theta in that range obviously h is negative that means instead of rise it is a fall. The meniscus shape also will be just curved oppositely. The other important factor which is one of the decisive factors is what is capital R because if capital R is large then no matter whatever is the contact angle this effect will be small and will virtually be unperceptible. At the same time if R is very small then this h can be very very large. We get such examples in nature very nicely that is if you have trees this trees absorb water from the ground there is no pump which is existing in the nature as a artificial mean of pumping the fluid from the root to the top most leaves and branches. But you will see tall trees also get nutrition from the ground. So when they are transported by the fluid medium which is the so called ascent of sap in terms of the biological terminology it goes vertically such a large distance it covers such a huge height just with the consideration that this R is small. So those are really very narrow capillaries and then the capillary acts like a pump. So just because of the surface tension it can attain a large height of transport and that kind of very beautiful mechanism prevails in nature and that makes the plants at least the tall trees sustain their lives. Now just to have a slight variation from this let us consider an example when you do not have just one fluid as liquid and another fluid as gas but maybe both of the fluids are liquids and they are immiscible ones. If they are miscible liquids obviously when you put them one with the other they may mix with each other and the clear meniscus may not be formed. But if they are not miscible with each other there may be a clear meniscus that is formed. Let us look into an example with a short movie and let us try to see that what kind of meniscus may be formed with such an example. So in this example what we will try to see is that if there are two different liquids which are there side by side then when these two different liquids are put in a capillary tube sometimes magically because of the resultant surface tension driven flow or transport that is created the entire column may move from one end to the other end and we see one such example here. So there will be two different fluids those liquids which have already been put and see that it is just like moving magically. It is not that there is a pump that is being put or there is no other driving force that has been created to induce the motion. Here we will not first be bothered about the motion we are first considering the equilibrium. So let us consider such a case but not a moving case but two different liquids which are keeping a meniscus in equilibrium. So let us say that you have a capillary tube in the capillary tube now you have say two liquids on the top you have say water in the bottom say you have mercury with a given tube material say it is glass it assumes this particular meniscus shape. The outside is say filled up with mercury so there is mercury in the outside and water is being poured from inside. Now if you see let us concentrate or focus on the meniscus let us say that the water is being poured from this height and there has been a so-called depression of the meniscus from the top level. So let us say that this is h again when you see such a simple analysis you should keep in mind that this has many approximations. When I say that this height is h obviously it has no sanctity because we are disregarding the variation in height from one end to the other when we calculate the pressure difference. Whenever we calculated the pressure difference here delta p as h rho g that h we assume some average height may be the centre one we took as a reference but it is not actually a uniform height. The entire meniscus has a variation in height and one of the approximations may be that not to use this as the height but use the centre line one instead just like what we did in the other example. So may be take this as h what is the error in this when you are considering this as the h if you refer to such a figure you are neglecting the shaded volume. So the shaded volume has a contribution effectively if you see it is the weight of the volume of the liquid that is being sustained through the surface tension and the pressure differential and there these shaded volumes also have their own roles. It is only an approximation by which we are neglecting it. There may be cases when this gives rise to some significant error and fortunately in most cases it does not give rise to that that much error. So it is it is fine as a practical approximation. Now if you consider this surface and try to see that what are the different types of forces which are acting on this surface. You may evaluate the surface tension force just like what you did in the previous example. So when you are considering the contact angle say we are interested about the angle that is made by the mercury with the with the glass. So that is measured with respect to this one. So this is basically the theta. So it is important to see what is the sign convention for the definition. So when we are referring to the mercury it should be from the solid within the mercury domain that is how the angle is there. Otherwise with a different notation may be 180-theta also be taken as an equivalent representation of the effect of the contact angle. So this angle will be 180-theta and for writing the force balance that angle will be useful. So the same formula will be applicable here that is it will have delta P again assuming a spherical nature and all those things. You will have that as 2 sigma cos of 180-theta divided by r. So now on which side the pressure will be most mercury or water and how can you calculate the differential of that pressure? When you consider the water side it is basically h into rho water into g that is the height of the column of water. When you consider the mercury side it is h into rho mercury into g. So the difference in pressure is like h into difference of low mercury-low water into g that is your delta P. So from here you will get an expression for what is h and this h is clearly a depression in this case that has been induced by this theta which is roughly 140 degree as an example. So if you know the surface tension coefficient then it is possible to put that in this expression and find out what should be the h under these conditions. So again this has many approximations but it gives some kind of idea that what should be the estimation for capillary rise or capillary depression. With surface tension one may also have a dynamic nature of the meniscus and when you have a dynamic nature of the meniscus it is not that you just have to consider this type of equilibrium at the interface. You may have to consider overall dynamical nature of one fluid as it is displacing the other and moving in the capillary. In the process there may be many things. In the static condition we have a contact angle. In the dynamic condition this contact angle may change and the change of this contact angle may be because of the dynamic nature of the forces which are acting on the system. One of such forces is the viscous force. Then you also have a dynamically evolving may be van der Waals force of interaction. So it is possible to have all the forces of interaction which are not just like constants but those are evolving dynamically as the shape of the meniscus is changing. Because as the shape of the meniscus is changing you have different arrangements of the molecules close to the wall and that may dynamically give rise to different contact angles. So the contact angle that we are referring to in these cases is commonly known as a static contact angle. But when the dynamics is evolving one may have a dynamic contact angle which depends on the relative interplay of various forces which are acting and since the two important forces dictating this type of capillary advancement in a dynamic condition are the surface tension and the viscous forces. So their relative interplay has a strong role to play in determining the contact angle that evolves dynamically. Surface roughness also has a strong role to play because that dictates the proper intermolecular interaction close to the surface. So there is a very rich physics that takes place close to the interface in a dynamic condition and this elemental study does not focus on that. It gives just a broad idea of if it is a static condition what can be the consequence of surface tension. But at least it gives us an idea that surface tension may be a very important force in a small scale and as the radius becomes smaller and smaller its effect becomes more and more prominent. With this background now we will move into more general considerations for equilibrium of fluid elements. Here whenever we were discussing the surface tension force we were assuming that the pressure is being distributed in a particular way and we were intuitively using some concept of high school physics that if you have a depth of h then what should be the variation in pressure because of that depth of h. Now we will look into it more formally. So we will go into the understanding of fluid statics. We start with an example of a fluid element which is in static equilibrium and we take an example of a 2 dimensional fluid element just for simplicity. We have taken such examples earlier what these examples signify that you have a uniform width in the other direction perpendicular to the plane of the board. Let us say that delta x and delta y are the dimensions of this fluid element. Remember that this fluid element is at rest. When this fluid element is at rest that means we are sure that it is non-deforming because deforming fluid element is definitely not a fluid element at rest and when it is non-deforming we are clear that there is no shear which is acting on it. That means there is only normal force which is acting on all the faces of the fluid element. So we can designate the state of stress on each face of the fluid element by pressure. Let us try to do that. Let us say that we are only writing forces along the x direction just for simplicity. Similar equations will be valid for the y direction. When you have say the left face under consideration just like this let us say that p is the pressure on the left face and the force corresponding to that is p into delta y is equal to delta into say 1 which is the width of the fluid element. When you come to the opposite face we are bothered about these faces right now because we are only identifying the forces acting along x because we will write equation of equilibrium along x. Not that forces are not there on the other faces so this is not a complete free body diagram. It only just shows the forces along x direction. So if the pressure here is p what should be the pressure here that is under question. Will it be p? Will it be something different from p? In general we are not really committed to what are the other forces which are acting on it. There may be any other body force which is acting on it along x and y. So if the pressure here is p the question is will the pressure here be p or something else in general. Special case of course it may also be p here but we are talking about a more general consideration. Mathematically speaking what question we are trying to answer? We have a function here say p. We want to find out the value of the function at a different location. Say this location is x. We are interested to now find out the value of the function at x plus delta x in terms of what is the value of the function at x. The function here is p that means we want to see that what is p at x plus delta x in terms of what is p at x and that we can easily do by using a Taylor series expansion. So that we will do. We will write this as p at x plus the first order partial derivative of p with respect to x into delta x plus and so on. There are infinite number of terms but as you take delta x very small maybe you may neglect the higher order terms in comparison to the dominating term and the gradient. Keeping that in mind that we are treating with cases where delta x delta y are very small. So delta x delta y all tending to 0. So this will become from the expression that we have here what we can write. This will be p that will be the pressure here. We will keep this in mind. So later on whenever we encounter any function we will use the Taylor series expansion to identify what is the change that is taking place across different phases of fluid elements because that we will have to do very commonly in many of our analysis. So this multiplied by the area on which it is acting is the force due to pressure on this phase. Let us say that there is a body force which is also acting on the fluid element. So the body force let us say that b x is the body force per unit mass acting along x. So b x is body force per unit mass along x. So what will be the total body force which is acting on this? Along x. First you have to find out what is the mass of the fluid element. What is that? It is the density times the volume of the element that is delta x into delta y. So this is the mass of the fluid element that times the body force per unit mass gives the total body force along x. So these are the forces which are acting on the fluid element. Now let us try to answer another question. Are these still the force, only forces which are acting if the fluid element is under rigid body motion? That does not mean that is the fluid element is moving like a rigid body. There is no internal deformation but as a whole it is just like a solid that is getting displaced. That may be displaced linearly or angularly but it is having a motion but the motion is a rigid body motion. If that is the case then are these the only forces? See what forces we have identified? Surface forces we have identified, body forces we have identified. So the question boils down to that are these the only surface forces even if the fluid is under rigid body motion? The answer is what? See what is the difference between a fluid element at rest and fluid element are under rigid body motion? The only difference that when it is under rigid body motion it might be having like a velocity acceleration and so on but in terms of the surface forces which are acting if the fluid element is non-deforming then there is no shear component of force. So for a non-deforming fluid element there is no difference between the surface forces which are presented in this diagram and the surface forces which are there when it is say moving with a acceleration. So this type of forcing description is equally valid if the fluid is under rigid body motion. So we identify this situation not just a fluid element under rest but also rigid body motion. We will see such examples where the rigid body motion of the fluid will be very interesting like you may have a rotation of a fluid element like a rigid body and we will see that what kind of situation it creates. So broadly this is also studied under the category of fluid statics not because it is a static condition but in terms of the characteristic of the fluid the deformable nature is not highlighted here and that is why we may use broadly similar concepts and we will learn these concepts together under the same umbrella because they are very very related in one case it may have an acceleration in other case it may not be otherwise it is very very similar. So let us say that it is under rigid body motion and therefore let us say it has some acceleration along x. Say ax is the acceleration which is there along x. So we can write the Newton's second law of motion for the fluid element and when we do that what do we get the resultant force which is acting along x is equal to the mass of the fluid element times acceleration along x. So it is p into delta y minus the other term which is there on the opposite face plus rho into delta x into delta y into vx is equal to rho into delta x into delta y that is the mass times the acceleration along x. Delta x into delta y we will get cancelled from both sides these are small but not equal to 0 these are tending to 0. So you can cancel from both sides at the end what final expression you will get. So this will be the expression which relates the pressure gradient with the body force that is acting and if there is any acceleration that acts on the that the fluid element is having. Similar expressions are valid for the motion along y. So we are not repeating it again with this kind of a general idea. So this is a very general expression. This general expression just considers that there is a body force and the fluid is having some acceleration in a particular direction subjected to the body force but it is a non deformable fluid element. With that understanding we will try to identify that what is the variation of pressure just due to the effect of gravity as a body force in a fluid element at rest. We consider that there is a free surface of a fluid. This is a symbol in fluid mechanics that we will be using to designate a free surface. This is a triangle with two horizontal lines very short horizontal dashes or lines at the bottom. This is a kind of a technical representation of a free surface. We consider that we are interested about some depth. Usually the direction in which depth varies is typically taken as a z direction. This is just a common notation in most of the books that the vertical direction across which the gravity is acting. Of course the opposite to the action of gravity because gravity will be vertically downwards and opposite to that is considered as a z axis just as a common notation. So let us try to write this kind of a equation for this fluid which is at rest. It is of a substantial depth. So we are interested to find out what is the pressure at this point which is at a depth h from the free surface. There is no acceleration of this fluid. It is under absolute rest. So the a x or here a z term will be 0. So you will have minus this one and you have the z direction you also have a horizontal direction like x and for x you can write again similar expression. So if you write it for x what is the body force which is acting along x? There is no body force which is acting along x because only body force to which this is subjected is the gravity. So there is no body force along x and there is no acceleration that is it is having along x. So the second expression is even more simple but it gives us a very important insight. What is that? That if you are not having a body force along a particular direction and the fluid is under rest then pressure does not vary within the same fluid along that direction. That means for a horizontal along a horizontal line you are not having any pressure variation in a continuous fluid system and this is one of the basic principles that we use for measurement of pressure differentials as you have seen in examples of manometers earlier. So this is something which is of great consequence but it is an obvious conclusion. What we get from the first equation? So we let us try to replace the bz what is bz? See this is the z direction and this is acting in the opposite direction. So this is minus g. So you have from this and since pressure is not varying with x so you can write this as dp dz in place of partial derivative because it is now just a function of a single variable. So you can write this as minus dp dz is equal to rho g. So dp is minus rho g. So if you want to find out what is the pressure difference between say 2 points a and b. So you have to integrate it with respect to the z variation from say a to b. When it is a say we take our reference such that the origin is located here. That means at a z is equal to 0 at b it is minus h. So you can write this as pb minus pa is equal to minus integral of. Now it is important to see that what is the length scale that we are considering over which this variation is taking place. If this h is quite large there may be a significant variation in density over it just like consider the atmosphere which is above the surface of the earth. As you go more and more above the surface of the earth you expect the density to change because the temperature changes and so on. And therefore the density in many cases may not be treated as a constant. So if it is treated as a constant then it can only come out of the integral. Similarly you also are probably working on a length scale over which g is not changing. If you are taking a large height like for people who are dealing with atmospheric sciences for them the length scales are large length scales over which you may have even a change in acceleration due to gravity. But if you consider that such a situation is not there just for simplicity. So if low into g is a constant that is the best way to say because a very tough mathematician will say that I do not care whether rho is varying whether g is varying I am happy in bringing this term out of the integral so long as rho into g is a constant. So may be mathematicians way of looking into it is that rho varies in a particular way g varies in a particular way but those variation effects get cancelled out somehow so that rho into g is a constant. May be very hypothetical but for our case to bring it out of the integral the product being a constant that will serve our purpose. So then what you have then you have pb-pa is equal to rho gh that means pb is equal to pa plus rho gh which is your very well known expression. Now important thing is that see we are not expressing the pressure at b just in an absolute sense we are expressing it relative to the pressure at a. Many times this pressure at a say this is atmosphere that may be taken as a reference. So if this is taken as a reference as p equal to 0 as an example so whatever is the atmospheric pressure say we call it 0. That means any other pressure we are expressing relative to the atmospheric pressure. So then in that case pb is the pressure relative to p atmosphere if p equal to 0 is the atmosphere it is not definitely equal to 0 but if you have p equal to 0 that is just the choice of your reference so that you express any other pressure in terms of that as a reference. So any pressure which is expressed in terms of the atmospheric pressure that is a relative way of expressing the pressure it is not that always you have to express relative to atmospheric pressure but atmospheric pressure being a well known standard under a given temperature. So reference with respect to atmospheric pressure is something which is a very standard reference that we many times use. So reference pressure relative to atmosphere so we are talking about a reference where the reference is the atmosphere then whatever pressure is there at any other point we call that as a gauge pressure. So this is just a terminology. So gauge pressure means that any pressure relative to atmospheric pressure. So that means it is nothing but p absolute minus p atmosphere there is a difference between the absolute pressure and the atmospheric pressure that is as good as taking the atmospheric pressure as 0 reference and mentioning the pressure relative to that. So this is a very simple exercise but from this we learn something. What we learn something so whenever we have an expression we should keep in mind what are the assumptions under which it is valid. So we will develop the habit of not using any formula like a magic formula this is very very important formula based education is very bad. So whenever you have a formula and you want to use it try to be assured that it is valid for the condition in which you are applying if not exactly but at least approximately. So when you are using pressure equal to h rho g what are the assumptions under which it is valid? So obviously rho into g is a constant and there is no other body force which is acting on it and fluid is at rest that is these are the assumptions that are there with such a simple expression. Now with this kind of concept one may utilize this type of concept in making devices for measuring atmospheric pressures just like you have barometers. Whenever we will be learning a concept we will try to give examples of measurement devices which try to utilize those concepts. As all of you know a barometer may be utilized to measure the atmospheric pressure. So how it is there? You have say inverted tube, inverted tube and this inverted tube is say put in a bath of some fluids say mercury and let us say that it is there up to this much height. Now this much of height is there there are various forces which are acting on this. So one is you have we are of course neglecting the local surface tension effect and the capillary formation. You must keep in mind that as this radius becomes smaller and smaller the effect of the curvature may be more and more important because surface tension effect will be more and more important and there may be significant errors in reading because of that. Now if we just neglect that effect for the time being then you have atmospheric pressure acting from this side. If we assume that there is a vacuum here there is a big question mark whether there will be vacuum or not but let us for the sake of simplicity assume that there is a vacuum then whatever pressure is there which is acting from this side that is balanced by the height of the liquid column which is there on the top. So from that you can get an estimation for what is the pressure here. Let us say that p is the atmospheric pressure. So p into the surface area on which it is acting is the force that is being sustained by the weight of the liquid column. So that is nothing but what? So it is like h rho g that into the area and area gets cancelled from both sides. So you get this p if it is vacuum as the h rho g but if it is not a vacuum let us say that there is some pressure here which is the vapour pressure of the fluid which is occupying this and it is common that such vapour pressure will be there. Why? Because if it is a saturated liquid it is likely to have its own vapour on the top of that and that is that will always exert some pressure. So it is never a vacuum in an ideal sense. So we can say that p minus p vapour is actually what is being balanced by this weight. So that is the h rho g. So if there is a vapour pressure you cannot just use h rho g for the estimation of the atmospheric pressure but you have to make a correction because of the presence of the vapour and that is a function of the temperature because vapour pressure varies with temperature. Very commonly the mercury is one of the fluids that is being used for this purpose and why mercury is being used? Obviously because it is quite dense it will not occupy a very large height for representing the atmospheric pressure. If you use any other fluid it may occupy a great height. So it may be an unmanageable device, unmanageable long device. Also the vapour pressure of mercury is quite small in most of the temperature ranges and therefore this correction is not that severe. These two are the important reasons there are many other reasons which are always into the picture when you select a fluid for measurement of a pressure like in a barometer. A barometer is a very interesting device. We have discussed something quite seriously but I would just like to share a kind of a story associated with barometer, a very well known story and I am sure that many of you have heard about it. Long time back in a high school examination there was a question, it was a physics examination and the question was that how can you measure the height of a building using a barometer. Now although all of you or most of you have heard about this story we will try to get a moral out of that story and we will try to keep that in mind whenever we are going to learn something. So what the student replied in the answer, the answer was that you just have a thread, you connect the barometer with the thread, go to the roof of the house, just drop that barometer with the thread and then like the total height that total height of the thread that is required to bring the barometer to the ground may be plus whatever is the portion of the barometer will give the height of the building. Now as in most of the examination systems this student was given a big 0 and the expectation was, it was justified that why he was given a big 0 because it was expected that the answer should reveal some basic concept in physics. It was a physics examination but this does not reveal any basic concept in physics. So he was given a 0 but then the student went for an arbitration. It was a democratic system even in that time. So the student said that no like let my answer be reviewed. So there was a panel, the panel said that okay may be you are not aware that what kind of expectation that we are having from your answer. So we give you another chance. So you think about a solution for this question which reflects your understanding in physics and we will evaluate you from that. So student said that let me be given some time. So he was given 5 minutes to think about that. So he was thinking for 5 minutes and when he was thinking for 5 minutes and he was still not coming up with an answer then like the evaluators were very happy that he might be failed again. So they said that no you could not come up with an answer. So we are sorry. Then he said that no actually there are lots of answers have come to my mind. So I am not sure that what should I say and that is why I was not giving a response. And then he was asking permission that I mean am I going to be allowed to speak of that remaining answers. Then they said that fine I mean whatever you have thought you just tell. Then he said that I will what I will do is I will drop the barometer from the top of the building and I will measure the time that it takes to reach the ground and h equal to half gt square. So from that time I can measure the height. It reflects some understanding of physics but it is a bit destructive because the barometer like it may be damaged and like and so on. Then he said that no. Then if you want a different answer maybe what I will do I will try to make a pendulum out of the barometer swing it one in the bottom ground level another on the top level and we will measure the time period. And this time period difference will give the difference in g between the two heights. And since g is a function of height it will tell us that what is the height difference between the two. And I mean still the examiners were not happy and they but still they were ready to pass him because these were like reflecting some of the basic concepts in physics. Then he said that even if I am given different opportunity what I will do is I will climb across the staircase and in the side of the staircase you can just put the barometer one after the other till you traverse the entire height. The number of times you put it you multiply with the length it is a basic length measurement principle. So from that you get that. And he said that there are many answers which are coming to my mind but given me a entire freedom what I will do I will not really put this much of effort. I will go to the house master and tell that see now I have a beautiful new barometer for you. I am giving it to you please tell me what is the height of the house. And the house master will obviously tell that because it is it is like a gift free gift that the house master is having. At the end he said that perhaps you are not expecting all these answers from me. You are expecting to me to give a very structured answer that the barometer measures the atmospheric pressure. So from the difference in the level of atmospheric pressure in the two heights we can easily say that what is the difference in height between the two. And since this is the most structured answer I hope that you will be happy with that. And then of course the evaluators passed him and he was quite successful and the name of the student is Niels Bohr. I mean who later on I mean like discovered many many beautiful phenomena in physics. So the whole idea is that I would always encourage you maybe I mean none of us are like Niels Bohr I mean we are not born with those special abilities. But at least whenever you are having an opportunity to think of solutions do not always go for a structured solution. Try to think about different possibilities whenever you are thinking about designing a measurement principle. Whenever you are thinking about solving a particular problem just try to think about various possibilities. Some of the possibilities may not be very encouraging very very welcome. But at least these possibilities will give us some kind of clue that what could be alternatives. Some of the alternatives may be discarded they may not be very smart. But they will at least give us a scope of thinking laterally. And that is how one may improve in science technology and research. And I mean such a simple example like barometer I mean one is always reminded of that kind of a story and I feel that it is it is it is not something too bad to share with you. So what we will do is we will not go further ahead today we will stop here. In the next class we just make a plan of what we will do in the next class. Now we have found out a particular way in which you have a estimation of variation in pressure because of a body force which is acting and for fluid element which may be at rest or subjected to acceleration. So we will utilize this principle to calculate two important things. One is if there is a plane surface which is immersed in a fluid what is the total force which is there on the plane surface because of the pressure distribution. Now you have realized that pressure is like a distributed force because it varies with the depth. So at different depths you have different pressures. Therefore it will be like a simple statics problem where you have a distributed force on surface to find out what is the total force which is acting. If you have a curved surface we will see that the technique may be a bit different but broadly we can utilize some of the concepts of pressure distribution on a plane surface even to calculate force on curved surface. So in the next class we will we will see what are the forces on plane and curved surfaces which are there in a fluid at rest and then to see that what are the consequences and we will work out some problems related to that. So we stop here today. Thank you.