 The surface tension, okay, so we will discuss surface tension for some time and then we'll take a break. Okay, write down surface tension Okay, have you ever seen? insect Walking on the surface of the water. Yes Is it because the buoyant force is it because the buoyant force buoyancy is a legs of the insect inside the water on the surface only It is on the surface on the surface Okay, so It is it has nothing to do with the buoyant force The gravitation force of the insect is balanced by some other force It is not even viscous force for viscosity viscous force you need to have Velocity or movement inside the liquid Movement inside the liquid is required then only viscous force is there So it is neither viscosity that balance gravity nor the This thing Viscosity nor the buoyant force fine. It is something else And it has to be based on the surface because the leg of the insect doesn't go inside the Surface, okay, so we are going to discuss about a surface property of the fluid All right, so what happens is that every fluid every fluid surface behave Like a stretched Stretched membrane. It's like when you blow air inside the balloon the balloon surface Exactly like that. Okay. This is how every fluid surface behaves like, okay So what is so special about the surface and this property of every fluid surface is different Like for example, the surface of water will have a different type of surface tension the surface of the What is that honey honey surface will be different type so every surface of the fluid is different different types Okay, so what is so special about it? Let us see that first So let's say you have a beaker which has Fluid inside it has a fluid. Okay. Now we know that the You know in case of solid Let's let's go down to the basics, you know in case of solid the the atoms close Very close. They form a lattice kind of thing Okay, in case of solid in case of liquid the atoms and molecules are Slightly away and in case of gas Ideal gas atoms don't interact They're very very far from each other one atom has nothing to do with what the other atom does Okay, so in case of solid the vendor all fours between the molecules of the atoms Is very large Very very large. So they are very close to each other in case of liquid again There is a vendor all fours between the molecules or the atoms That is why they are close to each other But not as strong as it was in case of solids in case of gas the vendor of force is absent Okay. Now The molecule let's say take example of water. Okay So not only vendor all fours, but hydrogen bonding is also there. Okay, so take for example Molecule on the surface over here and a molecule inside a Molecule inside feels the attraction at Different directions in fact multiple directions it get attracted Okay and The point on the surface Will experience the force only in? Half of the directions getting it now my question to you is who's who sorry which molecule is More stable Molecule one which is inside or molecule two which is outside which one is more stable when attraction is more then It is more stable or when the repulsion is more than it is more stable Okay, if you are a little confused, I'll give you a scenario Okay, all of you look at this scenario It's like this guys if you have let's say one spring only one mass like that or You have spring on both sides Okay, which one do you think more stable if I displace it this side and this also that side Which mass which scenario the mass will make more effort to go back more effort to go back This one right so more This is more stable. This one it tries to retain its position more. Okay, so same thing over here also Okay, so basically more attraction. You see the thing is that I can't get down to the Atomic level discussion with you I have to take the analogy to make you explain why it is more stable, right? Otherwise, I'll just have to tell you that as you meet is more stable fine. So more attraction means more attraction implies more stability More stability means higher energy or lesser energy More stability means higher potential energy or lesser potential energy. All of you more stability lesser Potential energy lesser potential energy Okay, so what is a natural tendency of any substance in the universe? To have as less potential energy as possible. This is a natural tendency Right, that is a natural tendency. So basically liquid Acha, sorry, I'll write one more thing here Lesser potential energy. So the molecules at the surface have higher potential energy, right? molecules at the surface has higher Potential energy Okay, so the natural tendency of every substance in the universe is to decrease its energy Fine. So basically the liquid every liquid will try Try to decrease its energy by having lesser lesser number of molecules at the surface How can it be done? Lesser number of molecules at the surface. How will you achieve it? How it is actually done? How it is done? How will you have lesser molecules on the surface? How can we have lesser molecules on the surface? Yes, Hariran got it Adwik got it Correct, correct Lose out by evaporation No, surface area will not change Okay So basically having lesser surface area Okay, if the surface area is less The number of molecules on the surface will be automatically less. Okay So the natural tendency of any liquid is what? liquid will try to reduce its surface area Okay, this is what liquid will try to do Fine. Now for a given volume Do you know which geometry has least surface area? If volume is fixed least surface area is what? Do you all know sphere? Right? It is a sphere Okay, so for For a given volume sphere has least surface area Fine So if if you do not have gravity If you don't have gravity The liquid will try to take a shape of a sphere Okay, that is the reason why if in a space if let's say water body is there It will try to create a sphere spherical shape Okay, so if effects of gravity ignored the shape of shape of the fluid body will be spherical due to surface tension Okay All right, so we'll take a break guys right now. We'll We'll continue after the break, right? So we all will meet at six fifteen now All of you take this break and come back in time. All right Right, so let us continue So we have discussed about The surface tension philosophically The idea of surface tension we have discussed. We haven't yet discussed Mathematically what it is how will you put a number that this is the surface tension Or how can you say surface tension is five or ten? Right, so now we'll talk about how to put a number Okay, what should I define surface tension as to be equal to fine, so write down surface tension surface tension is defined as surface energy surface energy per unit area I'll just tell you how does it makes sense You will claim that no let us define surface energy as surface tension not surface energy per unit area The problem with defining surface energy as surface tension is that if you take same liquid if you take lesser area and more area the energy of Substance which has more surface area will be more automatically because number of molecules are more at the surface Okay, so the property Doesn't become only of the liquid it is not that Same liquid will have same surface tension then same liquid depending on the surface area will have different surface tension Because you have defined surface tension as total surface energy Okay, but if you define surface tension as energy per unit area Then it doesn't matter how much Area of the surface you take density of the energy or surface energy per unit area will remain same It becomes independent of The amount of area you take on the surface then That surface energy per unit area depends only on which liquid you are talking about Not only that actually it depends on between which two mediums the interfaces Suppose interface is between water and air Okay, there will be one surface tension Between water and air there will be another surface tension between water and solid Okay, so basically this surface energy Or sorry. I mean here surface tension surface tension is defined as So surface tension is A surface property Okay, and surface Belongs to both the mediums. It doesn't belong to only one medium because it is an interface Okay, it belongs to Two mediums Okay, and that's the reason why surface tension Only depend upon which two mediums The surface Or the interface is Getting it it only depends on that and Of course, it depends on temperature also which we sometimes ignore But it is good to remember that if temperature increases Both, you know viscosity As well as surface tension They both go down Okay Surface tension is surface energy per unit area Its representation is s Okay, so it will be joule per meter square that is a unit for the surface tension Getting it. So if you multiply surface tension with surface area, what you get If you multiply surface tension with surface area You'll get what everyone Correct total energy of the surface due to surface tension Okay, very simple fine energy you get energy Anyone has any doubts till now? Please type in yes or no Why i'm spending more time over here because i know this This topic is ignored That's why i'm spending more time over here Anyone has any doubt quickly type in yes or no akash nishant No doubts No doubts Okay fine Now there is another way of defining surface tension There's another way of defining surface tension We have learned the way with respect to energy because that makes more sense the way we have discussed about this surface tension Okay, we also know that work done Is basically change in potential energy Right, you remember that work done is negative of change in potential energy Right, what is the work done? f dr This is equal to negative of change in potential energy Right, so force is du by dr All right, so wherever there is potential energy there will be force also Okay, so you can define the potential you can define the surface tension With respect to force as well Okay, so write down surface tension can also be visualized as surface tension Is also visualized as force Per unit length Okay, force per unit length is also surface tension Okay Getting it the direction of the force direction of the force Is always perpendicular to the to the line To actually to any imaginary line to any line You can imagine I mean it is a very weird way of telling but then this is true If there is on the surface if you draw a line like this This line Will experience a force both sides like this Because it is like a stretched membrane There is a force in both directions like that you can draw any line the force will be perpendicular to it Getting it all of you See this this definition is little bit more Let's say counterintuitive you'll have to solve few questions to appreciate this Okay, so let's discuss this all of you We'll conduct just a small you can say a Small demonstration of Of surface tension as being Force per unit length Okay All of you pay attention here imagine you have a Imagine you have this rail Okay, this is a you can say made out of thin wires. You have this u-shaped thing and You have a movable This blue stick can move sideways like that Okay In between over here You have a soap film There's a film over here Do you understand the scenario? This is a film You just dip it in this soap air soap and water solution You just take it out. There will be a film in between Okay, now what is seen is that If I apply force like this If I apply force a very small amount of force Which displaces it a bit Okay, let's say this Slides by a distance of d Okay, and this length is l This length is l Okay If surface tension surface tension is s Now i'm going to ask you simple questions here Okay, first of all, tell me if i'm stretching it and the film remains intact Film remains intact is surface area of the film increasing Surface area of the film increasing or not quickly type in No, okay Sir when i'm stretching it surface area of the film increases or not Everyone it increases right will it Why you said no What is the reason someone said no It increases right the area increases so the surface area increases Okay, now when surface area increases the energy The energy of the surface increases or not Type in does the energy of the surface increases when I stretch it It increases Simple right so energy has increased From where that energy is coming from How that energy is coming Energy can't be created right neither it can be destroyed. So from where this extra energy is coming What do you think? Work done by the force Do you all understand the work done by the force Increases the potential energy you can write like this. This is the work energy theorem Right, this is the work energy theorem w is equal to u2 minus u1 plus k2 minus k1 Kinding energy is almost zero. I'm moving very very slowly Whatever work I do will be equal to the change in potential energy fine So my force is equal to whatever force the film applies on me So that the blue rod moves very very slowly Okay, so the work done is f into d Right force f into displacement d So what is the change in potential energy in terms of surface tension? Can you write down that quickly? What is the change in potential energy? Okay, mother got something others Aditya got something Quickly type in guys, whatever you think All right hurry her and See surface tension is surface energy per unit area So if I multiply surface tension with Change in area, I'll get Increase in the potential energy Right. So is this correct? Sld is this correct? Something is wrong What is it? What it is What's wrong here? Surface tension remains same surface see for surface tension. It doesn't matter whether the film is thin or thick because Entire film won't contribute to the surface energy Only the molecules at the top Will contribute at the top whatever number of molecules are there. It doesn't matter but Here you have Two interfaces. Yes or no One at the top one at the bottom Everybody understand that So two times All of you understand this One at the top of the film one at the below the film There are one set of molecules at top of the film So liquid and air interface one set of molecules at the bottom of the film They will also Be at the surface only right All of you understand So this D and D get cancelled away Fine So surface tension Surface tension is F by 2L Okay, just that you know here. They have two interfaces That is why a factor of two is coming in else If only one interface if only one interface Surface tension would have been simply force per unit length Right, so that is the reason why you can see length is perpendicular Sorry, the force is perpendicular to the length You can see that My length is this blue line Okay, so surface tension can be visualized in terms of force also force per unit length Is also surface tension Okay And it is good, you know, it is It is good that surface tension can be visualized in terms of force as well as in terms of energy You can solve every question of surface tension By either of these two methods Okay anyways Now whenever we talk about the surface tension A discussion on The bubble and the drops becomes very important because The reason why bubble and the drop takes spherical shape is because of the surface tension Okay, so write down Drops and bubbles So can you tell me what is the difference between drop and a bubble? First tell me that Drop and a bubble you have heard it enough right drop and a bubble. What is a technical difference? Completely filled with liquid is drop and bubble is only thin film. Okay bubble has more molecule less molecule air inside bubble has air Okay position of air and water Okay, tell me one thing If if the air is trapped inside the liquid inside the water is trapped Is that a drop or a bubble? Air is trapped inside a liquid is a drop or a bubble bubble all of you So in our head in our head We have imagined that if it is a drop it has to be liquid Are you getting it in our head? We have assumed that if it is a drop it has to be liquid. It cannot be gas Okay That is what we have assumed and that is not true It is a drop of air But loosely we say a bubble is a inside the liquid a bubble of air is trapped But that's not true. It is a drop Okay, so first and foremost we need to understand what is the difference between drop and a bubble Okay, so Technical difference is very straightforward Drop has one interface drop has One interface Whereas a bubble has two interface bubble has two interfaces So let me draw A drop and a bubble over here. Then I'll tell you This is a drop And let's say I create a bubble over here This is a bubble. Let's say both are spherical of course So this is liquid This is air One interface only one interface Okay over here. This is air Blue is liquid and this is air So here there are two interfaces right air liquid and then liquid air In case of a drop you have only one interface. So this is drop And this is bubble So now can you tell me if the air is trapped inside the liquid? How many interfaces are there? Air inside the liquid Only one Okay, so technically it is a drop Technically it is a drop But loosely we say That it is a bubble because in our head we already assumed that you know the bubble should have air inside Based on our day-to-day life experiences second Okay, fine. So, you know The thing is that There is some relation between the pressure inside the bubble surface tension And the size of the bubble So let us try to find out that Okay, so the heading is write down relation between pressure and surface tension Like have you guys played with that thing wherein you blow air from your mouth and bubbles will come out Have you played with that? It looks something like this You blow air from it and the bubbles will come out Have you played with that? Have you seen that? Yes Okay Tell me the size of the Size of the this thing Bubble will be more or less if you if you blow very fast If you blow very very fast The size of the bubble will increase or decrease If the bubble will decrease it will be lesser Okay, try that Those who are saying that it will increase Please try this And if you have to blow out a big bubble Out of it Then you blow the air very slowly So it expands slowly and slowly it takes a big shape and then come out Okay, nothing mother don't worry It's not related to a cat's Okay So basically this is what we are trying to Find out as in what is the relation between pressure and the surface tension when we blow air very fast The pressure inside the bubble is Higher or when we blow air very slowly then pressure inside the bubble is higher when it will be When we blow the air very fast then the pressure inside the bubble will be higher Okay, so in a way now based on your experience you get a sense That if pressure inside the bubble has to be more The radius has to be less Size should be less if pressure inside the bubble is Lesser the size of the bubble should be more Bigger bubble should be there. Okay. So this is what we are trying to Find out. Okay. So first we will do for the drop Write down case number one is for the drop Very simple derivation all of this if you pay attention It is very very simple. Okay Take a drop of radius r Okay, and then this drop You need to consider that it expands a little bit So the initial radius Of this Is or this drop is r Okay, the thickness By which it has increased This thickness is Okay, I'll take delta This is delta r delta r is very small. Okay a very small Delta r is there so that When it expands pressure inside remains fixed Okay, if I make it expand more than Small amount the pressure inside changes. Okay, I don't want that to happen. Okay So if let's say inside pressure is p in And outside pressure is p o Inside is p in outside is p o I want to use work energy theorem here that is work done is equal to change in potential energy Changing activity is not there. It's a slow process and anyway the bubble sorry the drop remains stationary throughout So work done is change in potential energy. So write down work done In terms of force potential energy in terms of surface tension s Surface tension is s write down Try it yourself. It is there in your school curriculum. It's a very uh, it's a favorite question By school To be asked in your school exam Done all of you Okay, work done will be what first tell me that It expands so Which force is doing work? Pressure force is there. All of you appreciate that pressure force is there Pressure force from inside will be like this And from outside it is like that Up and you go to the surface Okay, so the work done is inside pressure Into 4 pi r square that is a total force Into delta r this work done by the inner pressure. Is it positive or negative? Everyone The work done by inner Is positive or negative All of you type it quickly positive The displacement is in the direction of the force. It is along the radial direction. Okay Work done by the outer pressure positive or negative Outer pressure negative so minus p0 4 pi r square delta r This is equal to change in the potential energy. How do you find change in potential energy in terms of surface tension? Surface tension into change in the surface area Right, how much it is? What is the change in surface area? You all appreciate 4 pi r plus delta r whole square minus 4 pi r square Increase in the surface area. All of you understand this quickly type in whatever I have written here Work done on the left hand side change in potential energy on the right hand side Everybody understood Two times will it be two times? Someone is saying two times Should I multiply it two here? Yes or no? How many interfaces in a drop? In a drop how many interfaces are there? Only one Only one interface is there. So you don't need to multiply Okay, when it is two interface then The surface area of inner surface increases and as well as outer surface increases then only so 4 pi 4 pi 4 pi you cancel out everywhere okay Then you can write it as pi minus p naught Into r square delta r This is equal to surface tension Expanded r square plus delta r square plus two r delta r Minus r square Okay, now r square get cancelled from here Can I ignore delta r square compared to two r delta r? Can I do that delta r is very small if I keep delta r large inner pressure changes Right, so delta r square I can ignore Ignore if I ignore that delta r get cancelled from here one of the r's will get cancelled So you will get p inner minus p outer to be equal to 2 s by radius Everybody understood this derivation type in quick Namrata is it clear? Amanchu Charan Charan is not there doesn't outer surface drop and air interact Yes Yeah, it interact that is our surface tension is there That's our surface tension is there. What do you mean to say? Only one interface Air and water that's it. What else? What else? Two surface is interface Surface is interface interface between water and air like that In case of bubble it is air and water The outer interface then inner interface water and air Yes in pairs You take one pair is one interface air water one interface water air another interface Okay. Now write down bubble Here is bubble I want you to derive the bubble one in exact same way Exact same way you have to follow and derive the expression for it Everyone Inner pressure is PI Outer pressure is P naught Expansion is delta r which is very less Surface tension is s Find out use work energy theorem. It is accurate stress. Don't worry about it It's accurate You are ignoring delta r square Ignorance of delta r square makes it more accurate your delta r is very small If you don't ignore delta r it means delta r is large. So inner pressure is not same During expansion and even got the answer here in case of bubble I'll appreciate that Here we have two interfaces So work done is inner pressure 4 pi r square Delta r minus outer pressure 4 pi r square delta r This is equal to Change in potential energy Okay surface tension is s So two times because two interfaces are there two s 4 pi r plus Delta r whole square minus 4 pi r square Okay So you when you simplify it like the way we have done you can see factor of two is there everything else is same So PI minus P naught will come out to be 4 s by radius of the Now look at this if You know outer pressure is atmospheric pressure most of the time Outer pressure is atmospheric pressure Which is fixed Which is fixed. So if you blow air very quickly PI will be higher if air blow Is quick If inner pressure is higher To PI minus P naught will be higher PI minus P naught will be higher value Surface tension is fixed. So radius has to be lesser. So that's why small bubbles will come out Okay, so this is the thing here fine all right sometimes Sometimes we have to treat interface as if It's a part of a bigger drop What does it mean? let us Discuss that in this scenario See i am talking about But shouldn't pressure decrease when increase in velocity? no, no, no, no Velocity has become zero you have blown air It went inside the bubble velocity becomes zero your kinetic energy converted into pressure energy That is a different thing when air is flowing continuously it is not stopped Their pressure will be lower but here You're confusing between two scenarios. It is not continuously flowing. It is stopped Okay, kinetic energy got converted into pressure energy okay so What I was talking about here is this scenario Himanshu understood Manju What is it out? There go there go understand this thing Suppose there is a pipe Okay There is a pipe over here If you blow inside Velocity With certain velocity there is a curvature like this so that velocity increases here Then the pressure will decrease over here pressure will decrease because velocity has gone up Pressure will decrease But if you have this scenario That you are blowing air from here And you have a stopper here. It doesn't let air to go from here. So air will hit the surface and stop Getting it. So velocity is not there over here velocity is zero. So entire half rho v square Is converted into pressure Are you getting what I'm trying to say Himanshu? So similarly inside the When you blow air it goes inside the bubble When it is leaving your mouth Half rho v square is there When it goes inside the bubble air stops Half rho v square becomes equal to pressure Velocity becomes zero Right, these are the two different scenarios Okay, fine. So what I was talking about sometimes This is what I wanted to discuss sometimes we have to treat the interface As if it is a part of a drop as if Okay, for example sometimes The Surface is curved like that It is curved. It is not flat Okay, so if if the interface If the interface is curved then pressure Just inside liquid is different From pressure just outside Okay, see what I'm trying to say here till now we have assumed that Liquid interface will be flat So atmospheric pressure here just inside also atmospheric pressure So outside and inside same pressure just inside just outside Over here pressure just outside and pressure just inside will be different And you have to treat as if it is a part of a bigger drop This interface Are you getting it? As if it is a part of a bigger drop and if radius of this drop is r This is of course, atmospheric pressure This is pressure you can say p1 Then pa minus p1 is equal to 2s by radius So pressure just inside the liquid is slightly lesser than the atmospheric pressure Lesser by 2s by r Everybody understood this? Everyone understand this whatever I've just done Type in quickly whatever doubts you have Is it clear type it This is the most important thing if a question comes most probably on it will come on Capillary rise which is based on this Type it everyone I'm waiting Okay, I'll repeat it again everyone. I'll repeat it again First of all, tell me one thing If the surface is flat radius is what radius of this surface is If the surface is flat radius tends to Infinity put r is infinity over here inner pressure and outer pressure become equal Okay Fine. So now let me explain whatever I had explained just now Is this If the interface is not flat If it is curved if it is curved We need to assume the interface as if it is a part of a bigger drop Then only we can use this formula To find out pressure inside the liquid Why I have to find pressure inside the liquid because when I use this p2 is equal to p1 plus rho gh Both the point should be inside the liquid But when surface is curved pressure inside and pressure outside are not equal So we can't say p1 is atmospheric pressure if Surface is curved. So I have to use this formula In order to use this formula, which I have derived only for the drop I have to consider as if the interface is a part of a bigger drop Everyone understood that now Everyone clear What who was asking it Those who are asking it to explain again, is it clear? Okay, see it'll be more clear when we take the numericals of course Class 10th physics and 11th physics Hell lot of difference. You might have experienced it by now So only way is to solve numericals to understand So we are going to solve a numerical on capillary rise What is it? How does capillary look like? Have you seen it? Have you seen capillary? No one What is the difference between capillary and a test tube? Capillary and a test tube. What is the difference? Capillary is very thin. Okay What is the basic difference basic? Of course it is thin But there is one fundamental difference what it is Capillary is open on both sides Okay, capillary is open on both sides test tube open on one side closed on the other side That is that you should take care Okay So we are going to learn about the capillary rise So all of you draw the diagram with me This is probably the most important derivation of entire chapter Okay, and most probably This will come in your school exam So we have a capillary over here I am drawing capillary Little thicker than usual so that the drawing is better But otherwise capillary as you said, yes, it is very thin So the level of water I'm taking blue color here If you give a large surface Because of the gravity the surface automatically becomes flat But in a capillary It can create a curvature Because the surface area is lesser. So you can assume these are the flat surfaces Okay But when you put a capillary A rise in the level of water is seen inside the capillary And a curve at the top is seen like that like this Okay Now there is something called the angle of contact Angle of contact is the angle between the tangent at the point of contact And the capillary This is angle of contact Okay This one is Angle of contact theta Fine This height is edge This is the capillary rise Ideally there should not be any capillary rise. Do you all understand? Because it is open to the atmosphere Ideally The water should not rise up But it is seen that it rises up Outside pressure is atmospheric pressure Okay If the density of the water is rho Surface tension is s I want you to find out what is h Okay in terms of everything else What is h? Try this Huh, sorry the radius radius of the capillary is given Sorry The radius of the capillary is given which is a small a Okay, this is small a good that you asked Small a is the radius of the capillary you have to write Huh now you can do it What is the first thing you have to find? What is the first thing you have to find type in Reduce the bubble all of you agree that I have to find radius of the bubble first Is it a bubble or a drop part of a bubble or a drop? It's a drop, right? So do you all agree that if I drop a perpendicular to the angle of contact like this And put a line like this symmetrical to it They intersect at the center Everybody understand this this will be the center Okay, we have to find this r Now can you tell me what is r? Find out what is r in terms of a and theta what is r? Okay aditya got it others It's always good to find out the right angle triangles Drop like this Those who got it correct This angle will be what this angle is what? All of you is this theta That is theta, right? So theta is given hypotenuse is r base is a this is a Okay, so a by r is equal to cos theta So r is equal to a c theta Right if angle of contact is given The radius of the Imaginary drop is a c theta Okay, now which side pressure is higher this side or that side Pressure inside the drop is more or outside Inside it is more but here there is no inside outside Right because it is not a complete this thing So basically you should know that the pressure Is always higher concave side Concave side so please remember this Concave side pressure is higher. How to identify concave concave looks like a cave Okay, it'll be like this like entrance of a cave All right Anyways, so pa is more than pressure just inside. Let's say pressure just inside is p1 So pa minus p1 is equal to 2s by radius which is a c theta Right so p1 is equal to pa minus 2s cos theta by a Right now what to do next after this? What should I do? What is the pressure over here at this level? What is the pressure at this level? What should be the pressure? Should it be atmospheric pressure? Over here pressure is atmospheric pressure Right, so I can say p1 Which is pa minus 2s by a cos theta Plus rho gh This is equal to what? pa everybody understood till now Everybody understood so h is 2s by a rho g cos theta This is the capillary rise it happens because of surface tension if surface tension is zero Capillary rise is zero understood if If it is uh if the curved surface Is hemisphere Yeah, I'll do that. Wait hemisphere then what is theta? If it is hemisphere at the top Theta will be what? What is theta? Theta will be zero degrees angle of contact is zero the capillary itself become tangent Fine Last step is what I have see this is p1 p1 is pressure just inside over here Uh, this red dot you see here pressure inside I got on this point This p1 p1 plus rho gh is pressure over here Which you use right you use p1 plus rho gh is equal to p2 Point number one and two both should be inside the liquid you should not take a point outside Okay, but till now pressure outside which is atmospheric pressure is equal to the pressure just inside the liquid That is why we used to casually write pa plus rho gh Because pa is just outside and just inside but now since there's a curvature Pressure just inside and pressure just outside. They are not equal Understood Okay, now tell me I have just two things left One is this other one is a numerical then we are done for today. Okay Let's say you put a capillary over here This is a capillary It need not be that upper surface be like this Upper surface can be like that also Do you understand? Look at the previous thing The upper surface is like this the upward opening curve. What if it is downward opening curves? Okay, what if it is downward opening curve then which diagram is correct This is the liquid it is capillary open from both sides This one This one That one where the interface will be one two three Of course, it it won't be two unnecessarily. I made two. Tell me one or three One or three Okay, let's assume it is Or three right now. Let's assume it is three Tell me pressure Pressure Inside will be more or pressure outside will be more if it is three Where pressure will be more inside or outside Inside will be more so inside pressure is more than atmospheric pressure already Over here p1 is More than atmospheric pressure So now that more than atmospheric pressure plus rho gh Will it be equal to atmospheric pressure? Do you get what i'm trying to say? p1 plus rho gh is atmospheric pressure, right? Look at this p1 plus rho gh is atmospheric pressure your p1 is already more than atmospheric pressure Right, you get a negative edge What does it mean? It means it is below the level Okay, so if the curvature is downward opening and this is the correct one Okay, it will go down below This is h If the curvature is like that fine so pressure Over here Is atmospheric pressure and pressure over there Is atmospheric pressure Plus rho gh This is p1 Okay, so this is p1 minus P a is equal to 2 s by Radius of curvature Radius of curvature can be written as a c theta where theta is a contact angle. So you understand, right? It all depends on see the thing is that what i'm trying to convey here Do not memorize The final results do not memorize the Derivations Derivation you use to understand Because I can tweak the scenario little bit like for example this one You have learned capillary rise you think that every time capillary will rise up It need not Okay, it depends on the interface is upward opening or downward opening So keep your mind open about it And you should learn how things are derived then only you'll be able to solve those tricky kind of numerical Okay, so the last thing for today is a numerical from your textbook only. So let me quickly take it It will take around 10 minutes And do it your own, okay It's very easy to copy things and you know, that's all useless at the end Don't give surprises to your parents to yourself Be very very honest to yourself. Whatever you do Nobody will lose Do this everyone first First you get the value in terms of Expressions like diameter is d height dip depth It is dipped at a height of h surface tension is s Atmosphere pressure is pa density is rho and g in terms of these you get first Do in terms of these And then put the values, okay, don't Understand right get the expression of the answer mother with which exam you're writing Sunday You don't want Sunday Anyone close to the answer anyone See entire tube is filled with air. So you consider entire Tube to be having the same pressure Don't assume that uh with height pressure changes inside the tube, okay Because air is there you can ignore the density is very less So ignore the variation of the pressure with the height I'm assuming all of you are drawing the diagram Okay Anyone close to the answer No one Anyways all of you focus here When you blow air from here It should create a hemispherical shape over there This is what the scenario is Okay So let's say The height over here Final answer. I'm not interested expression expression This is h Fine This is h Atmosphere pressure is pa. So pressure just outside it Let's say p1 Is pa plus Rho gh This is p1 And I want to find out p2 Tell me which pressure is higher p1 or p2 everyone p2 because it is on the concave side, right? So p2 minus pa minus Rho gh p2 minus p1 Is equal to 2s by r r is what diameter divided by 2 Right diameter is given So because it is hemisphere the radius of the capillary is equal to the radius of the Hemisphere also So there's 4s by d p2 is atmospheric pressure plus Rho gh 4s by d Type in quickly all of you understood this everyone Okay. All right. So, uh, you know, we have two three minutes the last part of the chapter theoretical one. No numerical nothing So I'm talking about surface tension in day to day life Because I have seen in your school exam. They ask these things Surface tension in day to day life So have you seen when you throw water drops on the banana leaves? It dances around in a spherical as if it is a spherical droplets are dancing around have you seen it? And when you put water on a cotton surface the cotton will absorb it Right cotton will absorb it. So it seems as if sometimes water Doesn't spread on a surface and sometimes water spreads on the surface Okay, and that happens because of the surface tension. Okay, so basically if water spreads it increases its Surface area No, we are not talking about a phase change water when it becomes vapor the complete You know properties changes. So change of state should not happen. Okay Fine, so when water let's just spread on the surface The interface between water let's say with water spreads on the banana banana leaf So there is a surface tension between banana leaf and water Okay, if surface tension between banana leaf and water is very large Compared to surface tension between water and air Then it will not form an interface between banana and water Because energy will increase if it spreads Surface tension is higher between water and banana. So why it should spread? getting it so same logic holds When you put antiseptic on your skin It spreads evenly very nicely. It spreads antiseptic or any Moisturizer cream or anything like that spreads very nicely because surface tension between Cream and your skin is very less Compared to surface tension between cream and air So when you when the cream spreads on the skin It decreases its energy Which is favorable Fine. So that is the reason why it spreads Fine and same logic holds good. For example when you have to You know wash the clothes If you don't put the detergent if you don't put the detergent just use the water Then the surface tension between dust Between dust and water is very high. So it doesn't come out. It doesn't mix with water. Otherwise Interface between water and dust will increase. So energy has increased because surface tension is higher So you put detergent So one part of detergent It's surface tension between one part of detergent and the Dust is very less. So dust will get mixed with one part of the detergent Another part of the detergent Its surface tension with water is very less So one part of the detergent mixes with the water other part mixes with the dust because their respective surface tensions are very less So when it mixes up it decreases its energy and that's why you use detergent So, you know, these are the few examples of day to day life examples of surface tension Fine. So, uh, we are done guys with this chapter Fluids is over. So I hope you appreciate that the today's topic what we have done Is as important as Entire fluid statics or entire fluid dynamics where fluid is flowing. So that is the reason why I have taken Exact same amount of time teaching these things When you compare it with I teaching fluid statics or fluid flowing so that you don't take it lightly Fine. So I guess that's it from my side