 Hi, I'm Zor, welcome to Unisor Education. Continue talking about units in physics. We have completed a series of lectures about base units in the system, which is called C, Systemo-International Analysis, something like this. I think it's French. In any case, the base systems of units, we have basically built, explained how they come from certain fundamental constants which exist in the world, like speed of light, for example, in the vacuum. So, I consider these units in C as base and now the other units in different areas of physics are defined based on them. So, we are not related anymore to some physical constants. These are all in the base units. And now, just using the base units, we, for our convenience, introduce certain other units which are called derived units. And again, it's for our convenience because basically we can use the base units to measure everything, whatever we need. It's just for our convenience, we derive certain other and other units are called some names, but they are basically completely derived from the base units. Okay, so let's start from something very simple. As you remember, we know that the unit of length is meter in C. Okay, now, not only lengths, we have to measure. We have to measure, for example, area and value. Now, it was not among our base units. However, we can derive it using certain formulas or definitions of the new entities which we are talking about right now. So, let's talk about area. Area is measured somehow and we have to introduce the unit of measure of area. Now, what do we do in this case? Well, we start from the formula that area is equal to, let's say, widths times lengths if this is something like a rectangle. Okay, so this would be our widths and this would be our length. So, what we do usually as measurement of the area, we do one times another. X would be here and Y would be here. Now, these are two linear dimensions. These are two entities which we can measure in already defined unit which is called meter. Well, which means that the definition of the unit of area might be just a square of one meter by one meter and depending on how many of these squares fit into any kind of area and area might not necessarily be square or rectangular or anything else. But we know what to do because we can divide it in many different smaller squares and we know if we have the area of one meter by one meter we can always consider the fraction of this as the area of any part of it and just put all these here, etc. But the unit would be still one meter by one meter which is called meter square and this is abbreviation. Okay, that's it. We have introduced the unit of area which is square meter which is a square of one meter length and one meter width and basically that's it. Now, next one is volume and obviously for exactly the same reason we introduce the unit of volume as a cube every side of it equal to one meter. It's called cubic meter. So this is the unit, cubic meter, is derived because obviously because it's using the unit of length. So the unit of length is used to introduce the unit of area and unit of volume and obviously we use it in physics wherever it's necessary. Alright, let's go on. Okay, next is a little bit more unusual kind of a unit of measurement. I'm talking about angles. So let's talk about plane angles first. So this is an angle, right? Now, how to measure angles in C? Well, let's think about this way. If you have a circle and you have some angle where the length of this is equal to length of this. So this is the radius and this is the radius. Now, as a unit of measurement of angles in C is this particular angle. Now, it's called region. So the region, one region is an angle where the radius of the circle is equal to the length of the arc. This angle is used as a central angle because the vertex of angle coincides with the center of the circle. So if the length of the arc is also equal to radius. Now, what's unusual about this? Unusual about this because we kind of relate it to linear dimension, the radius. However, in geometry, in the course of geometry which basically is on the same side as all these lectures, Unisor.com, it was proven that if you have any circle of any radius, that would be exactly the same correspondence. This arc would be equal to these radius. So on one hand, it looks like our definition includes the linear unit of measurement, the radius which is kind of involved in this particular definition. On the other hand, it looks like the unit of the angle doesn't really depend on it because for any radius it would be the same angle which we called the measurement of which we called one region. So that's why the unit is still called region. So we need to have some kind of a name for this unit. So we are using the name region. However, it's actually called dimensionless, so to speak. Well, the reason is very simple. You can say that since this length is equal to the radius, we can say that the angle is measured by the ratio of lengths of arc divided by radius. And if length of arc is equal to radius, then this ratio would be equal to one radio, right? Now, for any other angle, we can say exactly the same thing. So if you have, let's say, this angle, we have this particular length of the arc divided by the radius, and we will have some number of regions. And again, it does not depend on the radius because for any radius, ratio of this to this would be exactly the same, all right? So if this is true, what is the measurement? Well, length is measured in meters, radius is measured in meters. So it looks like meters and meters can cancel each other. So that's why it's called dimensionless. It does not really change the dimension of any other physical entity which we are talking about. So if the angle is part of the definition of some other thing, then the dimension of angle itself is not really changing the overall dimension of any physical entity which we are measuring right now. Okay, so that's plane angle. How about solid angles? Well, solid angles have similar kind of structure. Well, first of all, what is a solid angle? So let me just remind you that solid angle is basically defined as part of space which is inside the conical surface. So the conical surface is the surface which basically has the vertex. It has some kind of a closed curve. It might be a circle, it might be a triangle, in which case it will be a different shape. But whatever it is, the conical surface is supposed to be given and what's inside the conical surface is called a solid angle. And it can be of a really strange shape because the conical surface is based on, basically like if you have a point here which is moving along this curve and point here and connect them. So whatever the surface is formed by movement of this point and lines which connect this point to the vertex would be a conical surface. Now how can we measure it? It looks much more complicated than in case of a plane angle. Okay, well the way how to do it is the following. Now this would be a center of a sphere. So the conical surface would cut certain piece of a sphere. It's like a cap. So whatever this would be, it's like a piece of a sphere which is kind of like a cap. And if it's a circular area that would be kind of a simple thing. It's a simple cap. But in theory it can be anything. So it's a piece of a surface of a sphere which conical surface cuts basically from the sphere. Similarly to in the plane case we have an angle. It cuts certain arc, right? So or angle is supported by this or angle corresponds to this arc, whatever it is in the case of a plane. So in case of three dimension we have exactly the same thing. So this piece of a sphere is cut by a solid angle from the surface of the sphere. It has certain area. So this area of this cap or whatever it is divided by r square would be the measure of the solid angle. And it's called if this area is equal to r square then the whole thing is equal to one and the name is steric. Again it all depends on the area of this piece of a sphere which solid angle cuts. Now again the same thing as in case of a plane angles it looks like it depends on linear dimension on the radius of the sphere. But again if we increase the surface of the screen we increase the radius of the screen and just continue this solid angle so every point would be reflected further to a bigger sphere. Then the ratio of the new area which is cut from the bigger sphere to the bigger radius would be the same. All kind of a scaling is called in geometry. And that's why it's also independent basically on the radius. It looks like it depends on the radius but in reality for any radius that would be the same. So it's a characteristic of the solid angle. And obviously this is called the sterigian and that's how we measure the solid angles. Not sterigian because it's sterile. It's like a sterile sound or sterile visual effects etc. Three dimension. What's next? Next is speed. Now what is speed? Speed is something which we have come up with which is our own invented by us concept, physical concept. And we can measure it obviously by how well we have to define the speed right first. Well we define the speed of uniform movement. You know the first law of Newton says that in the absence of force the object would or point object would move along a straight line at a constant speed. So what's the speed? Well speed is basically a ratio of the length or distance divided by time it took for this particular object to cover this distance. So in case of a uniform movement when this particular ratio isn't then we can just take any piece of the distance which has been covered during certain amount of time divide distance by time and we will have a speed. Now in case of a speed is variable that's much more complicated so we can actually do this type of measurement during certain infinitesimal time. When this during this infinitesimal time we can consider that the speed is actually constant. So in this particular case if we have something like a function which define how much distance we cover at the moment t. So this is our trajectory, that's how we start. This would be our zero distance and the length of this would be a function d of t. Distance covered by the moment of time t. So what do we do? To determine the speed at any particular point we have to go to an infinitesimal increment of delta t so the time would be time plus delta t. We have to take how much distance was covered during this time so let me just change it to letter x because d is differential and that's kind of... So our distance would be x. So we have a moment at moment t plus infinitesimal piece of time. We subtract the position which was at the time t and that's the length covered during this time of delta t and we divide by delta t and that would be our average speed and that's very very small piece of our trajectory and whenever we have a delta t going to zero we will have a limit of this with this condition we will have exactly the point the speed at this particular point. So this is a speed and how do we measure the speed? Well, since it's a ratio of the lengths which we have covered by time, so the unit would be meter by second. There is no special name for meter by second so we are just using meter by second without assigning it any kind of special name as in some other cases we have derived unit of measurement. Next is acceleration and acceleration is basically measure of change of the speed. So speed is the measure of changing the distance acceleration is measure of changing the speed and we do exactly the same thing we have a speed at moment t and what's the change, rate of change well we have to have speed of next infinitesimal time moment we have the difference in speed during this moment in time divided by delta t divided by time and that's the rate of change of speed and that's called acceleration. Now to do it properly at point t we have to have the limit of this as delta t goes to zero which doesn't really change the dimension because if speed is meter by second then this would be meter by second by second and there is an abbreviation meter divided by second square basically to shorten this type of abbreviation so this would be again without any special name it's called meter by second square if you wish, that's how we measure acceleration at any point next next would be the force but the force would be based on the second law of Newton so we do remember second law and since the force is defined by this we can have that the unit of force would be when mass would be equal to one kilogram then acceleration would be equal to one meter per second square and their product would be one kilogram times meter divided by second square usually the second but in C really it's just letter S so this would be appropriate unit for force so the the unit of force would be such a force which will give this product this measure in kilograms and this measure in meters by second square if the value of this would be equal to one that would be a one unit of force and it's called Newton obviously in honor of Isaac Newton the abbreviation in this case exists in the capital letter N for acceleration and for speed we did not have any special abbreviation just used meter by second and meter by second square in case of this we do have different abbreviations so kilogram times meter divided by second square is called a Newton okay now we built everything hierarchically first the fundamental the base units then something which is derived like for instance speed acceleration and then we derived from derive it's like the whole mass is basically built on axiom and then theorems and then theorem is based on theorems etc etc so now based on whatever we have done before we will use whatever we have done before to define new physical units so the next one is pressure now what's a pressure well pressure is whenever you are for example whenever I'm standing I'm standing on certain area of the floor and obviously I'm pressing and I would like to know how much force is concentrated on each unit of area well unit of area is square meter we have already counted and my force my force in this case is a gravitation force because I'm pressing the floor because of gravity so my force or my weight actually that's my force the source of my force is measured in newtons because that's how force is measured now if we divide my weight by the area on which I am standing the area basically of my feet so to speak and this is my weight this is the area of my feet then I will get the pressure and the unit of pressure is this if it's equal to one it's called one Pascal again in honor of Blaise Pascal famous scientist it's PA well and the last one which I wanted to cover today is density density is amount of mass per unit of volume so this is kilogram per meter cube and that's it so again there is no specific unit for this so these are units which are used in mechanics now the next lecture would be units which are used in let's say energy or something else and we will continue basically building all these new units so to speak based on the previous ones that's it for today, thank you very much and good luck