going to give you a little tutorial on dimensional analysis for years i thought dimensional analysis was this strange art but then i i later realized wait there's a systematic way to do it and i was overjoyed for most physical systems there's a limited number of basic dimensions and these basic dimensions are mass and i'll use your capital m to represent mass there's length right distances can be written in terms of a length speeds are length per time so oh time that's another basic dimension that's kind of it you can also think of temperature i guess but other things like force are just based on these basic dimensions so for example force is mass length per time squared so it could be decomposed into these basic dimensions there's typically three basic dimensions i might just add that sometimes temperature is included we're thinking of things that come from science and physics you might say well what about dollars for studying a financial system as this forced example tells us other system properties can be given in terms of these basic ones like force and i will use this symbol of square brackets to denote dimension so for example dimension of force our m mass times length over time squared what about dimensions of the velocity of something that's going to be length per unit time or the acceleration that's how the velocity changes with time so it's going to be length over time squared and so on so for material physical things there will be these basic dimensions we haven't yet said anything about units there's two primary systems of units there's the english one call it english but we seem to only use it here like measuring force in pounds mass and slugs then there's the s i system of units we need to distinguish units and dimensions i'm just going to talk about the main system of units to be distinguished from dimension and that's the system international s i units we sometimes just sort of reverse it international system here we measure mass in kilograms we measure length in meters we measure time and seconds we measure degrees and kelvin all the other units are derived from those derived units that we attach people's names to sometimes like newtons one newton equals one kilogram meter second squared it's a derived unit from these basic three which match these basic three dimensions an equation like the equation that we have up above has this equation up here this equation of motion needs to be dimensionally homogeneous meaning all of the terms must have the same dimension maybe i'll put a equation one next to that so an equation describing the physical system must be dimensionally homogeneous for example equation one up there the equation of motion for the bead in the hoop so if we would write that as left-hand side equals a right-hand side if you take the units of the left-hand side that must equal the units of the right-hand side this denotes i keep saying units denotes the dimensions the length of my car has dimensions of length the length of the desk in front of me dimensions of length the mass of that chair back there that's dimensions of mass usually these are combined and that's why it's not always totally obvious if we look up here if you were to analyze the dimensions all the terms have the same dimension but we'll get to that i first want to mention something called the buckingham pi theorem this buckingham pi theorem it's useful in trying to non-dimensionalize an equation if we have an equation like this it's chock full of dimensions mass and length if you want to systematically study something you want to non-dimensionalize that equation so you can figure out which combinations of parameters are really the most important so what you'll end up with is an equation that has no dimensions to it but its behavior will be described by a few non-dimensional parameters as well as non-dimensional variables and a non-dimensional time so everything becomes non-dimensionalized so instead of there being theta and time and five parameters maybe we'll have something less when we non-dimensionalize this equation of motion toward that end we consult the buckingham pi theorem for help with non-dimensionalizing we consult the buckingham pi theorem here's the thing if an equation has k variable and here they use variables differently from me variables that just means any kind of number this lumps together what i had called variables as in things that vary with time as well as parameters if an equation has k and k is going to be a integer then it can be reduced to a relationship among k minus r independent dimensionless or non-dimensionalized products or groups or as we might say numbers non-dimensional number where r it's an integer it's the minimum number of basic dimensions required to describe all the variables so there's something interesting here there's this number k and this number k minus r let's just rewrite what we have try on the equation of motion for the bead so that was equation one i'll just write it m r remember this is different r equals minus b theta dot minus mg sine theta plus m r omega squared sine theta cosine theta if we call these buckingham pi variables you know what it calls variables what do we have let me just go through we've got the five things i called parameters m r b g omega but then we also have theta and t according to the buckingham pi theorem we have k equals seven variables now how many basic dimensions do we need to describe all of these to find that out we just sort of find out the dimensions of everything dimension of mass is of course m mass dimensions of the radius of the hoop well that's a length you have to do some work to find out what dimensions of b is but because that equation is dimensionally homogeneous it turns out that that damping term has dimensions of mass times length over time dimensions of the gravitational acceleration has dimensions of acceleration so it's length over time squared dimensions of the angular velocity angular velocity is technically the number of radians which is dimensionless divided by time so it's rotation rate is radians for time since radians are dimensionless it's 1 and time is t what about dimensions of the angle theta you might be thinking well isn't that degrees or something well no for this equation to be true you've got to write the angle theta in terms of radians and radians do not have dimension they're not a mass they're not a length they're not a time so they're already non-dimensionalized so the way we represent that is we just sort of write one and then dimensions of time are of course time so if you look at how many things do we have here what basic dimensions i mean we only mentioned three do we use all three here we got mass we got length we got time r equals three that's the minimum number of basic dimensions to describe all of these variables in the equation what does the buckingham pi theorem tells it says that this equation can be reduced to a relationship among k minus r dimensionless numbers k minus r equals four that means equation one can be turned into a i know it's vague sounding like a relationship among four dimension less numbers or products groups so that means we need four dimensionless numbers to non-dimensionalize this equation okay we're going to go through this exercise you can see and you actually have choices in what these dimensionless groups are this is an equation of motion for theta let's put a little square one theta the main thing that we care about the main thing that varies with time is already dimensionless that's good maybe we should stick with that theta is one of them so now there's only three left what else do we have we could look at this equation up here we see that mg this has units of force forces mass length time squared m r omega squared is also dimensions of force we can take the ratio of these two the ratio of a force divided by a force is going to be a non-dimensional number if we take the ratio and we can kind of pick which one is the numerator which one's the denominator i'll do this m r omega squared over m g the m's cancel out we've got r omega squared over g that's a non-dimensional number we'll call this gamma that's how we'll define gamma gamma is our second non-dimensional number and notice something about gamma because r is positive g is positive omega squared is positive so gamma is greater than or equal to zero it could be zero if we have a non-rotating hoop number three that's why there's only two more we can find or introduce a characteristic time scale this is standard thing when you have a differential equation where there's derivatives with respect to time once we have a characteristic time scale hopefully this won't lead to confusion but i'll call it capital t then we can introduce a dimensionless time let's call it tau it'll be normal time measured in seconds divided by capital t which will also be measured in seconds but then this dimensionless time will be our third dimensionless number the way that we can find a characteristic time scale is we just we suppose there is one and we rewrite the derivatives instead of being d theta d little t we write them as d theta d tau and plug them into the equation what do i mean by that well theta you could think of it as it's a function of time but you could also think of it it's a function of this new thing we've introduced called the dimensionless time which is a function of normal time in the equation up above we've got theta dot which means what that's just shorthand for d theta d t using the chain rule we could write this as d theta d tau d tau d t and what is d tau d t well from this formula over here this is one over capital t one over the time scale so that means wherever we have the derivative with respect to the actual time we could substitute in this one over t d theta detail let's look up here we have a first derivative we also have a second derivative over here if you do this twice here's what you'll get theta double dot equals 1 over d squared second derivative of theta with respect to the non-dimensional time tau and so this d theta detail since theta and tau are both dimensionless this is dimensionless so it's second derivative of theta with respect to tau plug these two equations back into the original ode so we've got r 1 over t squared d squared theta d tau squared that was the left hand side well i guess we had an m and then minus b one over t d theta d tau minus m g sine theta plus m r omega squared sine theta cosine theta let's divide everything by m so just divide through by m i want you to look at how we defined gamma up here it's r omega squared divided by g and it looks like we're close to getting that in this formula if we divide everything by g so let's divide by g and this goes away g r over g we put parentheses around this and say oh this is gamma everything is non-dimensional this is a non-dimensional term this is a non-dimensional term this whole thing is non-dimensional and because d theta d tau is non-dimensional of this minus sign we've got that this is non-dimensional and so is this and now we have a choice we're going to pick a characteristic time scale to make one of these non-dimensional parameters equal to one this problem's interesting in that there's a choice sometimes in other equations one thing just pops out here we have a choice of what to pick for the characteristic time scale we want to make one of these two terms order one but you means we'll just set it equal to one if one of them is set equal to one then the other one will be some number it'll be some other non-dimensional number that's not necessarily one what we're going to do is pick a time scale that's related to the damping parameter the friction b just because we have to pick something and actually looking ahead i know that that makes the problem interesting we will choose b over mg t equal to one so that the characteristic time scale is b over m g that sets what the non-dimensional time will be what did we pick we picked this to be non-dimensional by a choice of t so that's in some sense number three that was the third thing the last one just sort of given to us by this first term here in purple because for that choice of t what does that become t get what we get r over g t squared this ends up being r over g mg over b squared it's got to be non-dimensional and it is we'll call it epsilon so epsilon is m squared g r over b squared so this is our fourth and final non-dimensional number so now our equation of motion can be written in its non-dimensionalized form i think we call that equation one we did make a choice a choice of time scale we might not say this is the non-dimensional form it's just a non-dimensional form we've got epsilon times this second order derivative term equals uh because of our choice of time scale the coefficient of the damping term is just one and then there's also coefficient one in front of this gravity term and then gamma um oh i guess another thing to know is epsilon is greater than or equal to zero as well gamma sine theta cosine theta what are we left with we have theta changing with the non-dimensional time tau so i'd say we have one dependent variable theta one independent variable tau and two parameters just two epsilon and gamma and we have interpretations i think epsilon is an inertial term epsilon gives the relative importance of inertia to damping and gamma is related to the relative importance of relative importance of the rotational effect to gravity so in this case these have interpretations before we go on and analyze this there any other cases of non-dimensional numbers that you've come across any famous ones reynolds number yeah yeah the reynolds number like inertia to viscosity fluids has lots of non-dimensional numbers reynolds is the big one in fact i guess you could think of reynolds as a bifurcation parameter for low values of reynolds number it's laminar flow and then at some point as you turn the knob of reynolds number there's turbulence and so that's one of the big and salt things is well okay exactly how is the reynolds number a bifurcation parameter and what's really going on in the infinite dimensional space of the fluid