 I want to start this video with the only physics joke I know. So the joke begins this way. There is a dairy farm that is having trouble producing milk. And so they call in these three consultants. They call in an engineer, a psychologist and a physicist. They ask them to look at the farm for a week and then come back and give suggestions. So the first one they call in is the engineer. And the engineer says that you know what the milking tubes, the milking tubes through which the milk flows should be just 40% wider. So that becomes more efficient and so on. Then the psychologist says you should paint the barn, paint the barn, the insides where the cows are kept, painted green. Psychologist says painted green as it's a nice colour and the cows will be happier and they'll make more milk. So then they finally call the physicist who demands a blackboard and the physicist says okay, assume that the cow is a sphere. And that's the joke. But I want to show that's not really silly. What the physicist did here was approximate the shape of the cow by a spherical model and we will come back to this. We will extend this idea, we will take the spherical cows forward and answer a big question like why are mammals as large as they are and not much larger or why did dinosaurs have a smaller head to body size ratio compared to let's say a dolphin or a whale. All this with just a simple model which gives some approximation to the shape of the cow. Now before we answer that question let's understand the meaning of two words model and approximations. So what is a model in the context of physics? Let me show it right away so we can say that this is a version. This is a version of a real physical system or thing. System can be a collection of objects, can be one object. So I'm just writing system slash object slash thing. Version of a real physical system, object or thing which provides approximate let's write approximate with a different color which provides approximate descriptions of that real physical system object slash thing. Alright so model is not the real thing it's something closer to a real thing. A quick example of models in the context of architecture could be these smaller models of houses, buildings. Of course they're not the real thing but they are still giving us a lot of insight into what the real thing is, how it could look like or how much space it might have. So it's really giving us approximate descriptions. That brings us to the meaning of the word approximate. So approximate we can say that this is anything that is similar but not exactly the same to something else. Now let's pick an example from physics. So let me hide just this. Let's say we take a cricket ball which is moving to the right and we are interested in figuring out its velocity after some time. There is some air resistance. Ball is facing some air resistance. Let me write that as AR. Ball has its own weight and it's possible that the ball could be rotating. This ball is rotating about its center and it's also possible that the ball is not a perfect sphere. It could be tapered from the top and also this weight which is due to the gravitational force. This depends on the height from the surface of the earth. So if this ball is going up and going down, there will be some change in the gravitational force on the ball and therefore there will be some change in the weight. Very extremely ridiculously small but still there will be some change. All of these variables, all of these things they make the entire situation very complicated. Too difficult to be analyzed. And that's what a real system is with all its beauty and glory, with all its details. It's very difficult. So what physicists do instead is that they create a model. They create a version of this real thing which will provide us with some approximate descriptions. And let's say in this case we are interested in figuring out the velocity of the ball after some time. So for this one what we can do is we can create a particle model. We just represent the ball with a point. With a point this is the physical model. This is the physical model. And here we can assume a couple of things. We can say that the weight, the weight of the ball is not changing. We can ignore air resistance. We can assume that it is moving in a vacuum. And then we can analyze the velocity. Also since this is a point object, not an object with dimensions, there is no scope of rotation. And why did we really ignore air resistance and assume that the weight is constant? Now that is really a skill which physicists develop. Which thing to ignore? Which thing that could be ignored which will not affect what I'm interested in? I'm interested in figuring out the velocity. And turns out even if we do include the factor of air resistance, yes we will get an answer which is more accurate. But the change between that answer and the answer that we will get from this situation doesn't really differ that much. The same goes with weight even. Even if we do not consider the weight to be as a constant, yes the answer will be closer to reality. But if we consider it as a constant, it's not really changing things by much. So this model, it provides some approximate descriptions, things that are similar but not exactly the same to something else. And these descriptions here would be having one weight, one constant weight, one velocity, point object and completely ignoring air resistance. Because they do not really influence the final velocity by much. So the big idea is that when physicists model a real physical situation, there may be a large number of aspects that might influence the situation. But if you consider all the factors, things become complicated. So we neglect the less influential parts of it, such as we neglected air resistance here. And then try to analyze the motion. Alright, now let's go back to our spherical cows. So the question that I want to ask is, why if you're running a dairy farm, why don't you make super cows? Cows which are twice as big, twice as big as this cow. And how would a super cow really look like? This is just a super cow. This is your soup. How do you draw this? Let's just take this. This is approximate description of the Superman logo. So super cow, normal cow. And what's the difference between them? We can say that one is toys as big as the other. But what do we really mean here? Well, the super cow is twice as size, but is it twice as big? How much more does it weigh, for example? If this cow's weight is around 200 kilograms, does it mean that this cow weighs 400? Well, if the cows are made of the same material and I guess they are, it's reasonable to expect that their weight will depend on the net amount of stuff inside them, the net amount of material inside them. And what would that really depend upon? The volume, right? More the volume, more amount of stuff inside, higher the mass. So for a complicated shape, it's always difficult to figure out the volume. But this is a sphere. All we need to know is volume is always dealt with in cubic units. Meter cube, centimeter cube, millimeters cube. There's always a cube in it. So you are multiplying three dimensions. In the case of a sphere, there's only one scale, really. That is just the radius. It's just the radius. So volume, we can say it's proportional to our cube. Volume will always be proportional to our cube if you're talking about spheres. So let's try and compare the volume of these two cows. For the first one, we can say, let's say that the radius is one units, one meters, just one unit. And the radius of the super cow, this is two. So volume for the first cow, it will be proportional to a factor of, just factor of one. But volume for the super cow, this will be proportional to a factor of two cube, our cube. So this is eight. So this means that the super cow really weighs eight time. It weighs eight times more than the normal cow. And all of this weight is supported by the skin of the super cow. So would it be able to do that? Well, the weight increased eight times when you doubled the radius of the super cow. Let's see, let's see the change in the surface area. So in the first case, we know that area is dealt in square meters, right? So this will be proportional to a factor of one because r is just one. And if you double the radius, then area proportional to r square, proportional to two square. So when you double the radius, the surface area that increases by a factor of four, not just two, but by a factor of four. Volume increased by a factor of eight when you doubled the radius. So a cow that is twice as big actually weighs eight times. It weighs eight times as much and has four times as much skin holding it together. Now, if you think about it, this means that the super cow has twice as much pressure pushing down on its skin as a normal cow does due to its weight. If I keep on increasing the size of a spherical cow at a certain point, the skin or organs near the skin will not have the strength to support this extra pressure and the cow will rupture. And that will be just embarrassing. So there is a limit to how large the cows can be and not just because of biology. Now, I could have drawn a real looking cow like this. And I could have doubled the size. I would have really arrived at the same conclusion that the super cow skin would really just rupture or break. So nothing really is lost by making this simple model, this simple spherical model, which is giving me an approximate description of the real thing. The approximate description here is radius being one meters. Now, I know what you might be thinking that it seems that you could argue like this for just about anything. And that's actually true. I can actually argue like this for any mammal. There is a limit to how big they could be. But still just for the sake of it, let me start with a better model, which would let's say provide better approximations, better approximate descriptions for the cow. So now let's at least include cows head. Here we have the normal cow and this is your cow's head, this is the cow's neck and this is the cow's body. And similarly, we have the super cow, which is just twice as big. Now, the head is doubled in size. So the volume increases by eight times and so does for the body volume increases by eight times. But let's look at the neck over here. The strength of this neck, we can consider it like a rod. That is a thicker rod will be stronger and a thinner rod made of the same material will be weaker. So a rod that is twice as thick, twice as thick as a cross-sectional area. And we saw that in the previous case, the area increases by a factor of four. But the weight of the super cow's head volume increased by eight times, volume is proportional to the weight. The weight is eight times as great as that of a normal cow, but its neck is only four times stronger. Relative to a normal cow, the neck is only half as effective in holding up the head. If we were to keep increasing the dimensions of the super cow, the bones in the neck would rapidly become unable to support its head, they would rather break. And this explains why the heads for the dinosaurs had to be so small in proportion to the gigantic bodies and why the animals with the largest heads in proportion to their bodies, such as dolphins and whales, why they live in water? Because objects are living water, their weight is supported by a force, this is called a buoyant force. And because of this upward buoyant force, the weight is acting down, but there is an upward force. So the net force on the dolphin's skin is considerably less. There is less pressure, there is less pressure on the dolphin's skin. So they are slightly lighter in the water, so less strength is needed to hold up the weight of the head. But over here there is no upward force, there is just all of this weight that is pushing the skin. And now we can understand why the physicist in the story did not recommend producing bigger cows as a way to deal with the milk production problem. But the bigger point here is that even just a very simple model of a real object, of a real thing, we were able to deduce some general principles about how big mammals can be. And that is the absolute beauty of physics. We will make many simplified versions of real systems which will have approximate descriptions of the real systems. But still we will be able to come up with some amazing understandings about the universe.