Fundamentals of Kinematics and Dynamics

okay okay so that's three so remember that remember that we had velocity was change of displacement or DT and we said that if you have experiments displacement versus time graph say like this the derivative gonna be velocity of this right here is ds change of displacement change of time and if you come to me5 30 which is control i'm going to teach you there that derivative is simply okay guys you passed off you graduated from high school you heard about derivative and integral huh so if you are your uh five years old brother or sister ask you what is derivative and integral in one word what is that no that's mathematics simply because i'm going to teach you in mv513 control derivative so put the eyes here huh this eyebrow and look from this point to that angle when you see the next time means derivative in control means future really it is that's the design controller or robot for airplane future future of error signal now if i have the You recorded right? Yeah. If I have velocity versus time experiment graph like as your car traveling from home to SDSU like this from here to here gonna be what? Because here you have from S1 to S2 as you have in your notes. DS is integration of VDT. as your friend said what's your name sir christian said integral is the area sum of under the curve but again if you ask your brother sister what is integration in one word from this point you're looking to the back means derivative means future integration is past real it is In control, I'm going to teach you about the PID controller to control robot, to control airplane, to control shuttle, to control everything. Interesting in your muscles as human being, when you walk, when you dance, you have PID controller in your joints. I'm sorry. So in your joints, you have DI controller, derivative integrator. This I is for integrator. this d is for derivative i'm going to skip p but i'm telling you what's p d is derivative future i is integrator past p is present okay pid or proportional i don't want to teach you control now but you have to know the fundamentals of science not graph okay if you derivative means future okay and also we had a is equal to dv or dt you're having a note so i'm just refreshing to continue the topics so being said if you have velocity versus time changing the color make it more beautiful and attractive the slope here gonna be dv dt gonna be what acceleration But in vice versa, if you have acceleration versus time and you have this graph, area under the graph is going to be what? Velocity. It's going to be velocity. Because V from V1 to V2, DV goes from T1 to T2. A D T from here Obviously, you had it. Okay, everybody? Good. So, let's continue the type of motions. We are on the kinematics yet. So, we have two types of motion. One. okay constant velocity as you are doing every day by after parking the car and coming to the classroom yeah in average you have constant velocity hmm when there is constant velocity or v i'm sorry i'm not seeing you when v is c c means constant okay c is constant this is how we write in mechanical or electrical or physics or civil engineering notations right when we is constant there is no change in velocity what is zero what is zero when there is no change in velocity i just taught you acceleration is zero being said is a zero so when you use you your car cruise system and say okay in the highway just drive for 60 miles per hour means acceleration is zero of your car right so velocity as we define is ds over dt now as we know it ds is v dt integration from both sides s1 s2 and from say zero to t so it's going to be s2 minus s1 from kindergarten this guy because v is constant comes out going to be vt so we get our famous may you have seen in high school physics we use it up um s2 equals to when vt plus s1 here we go this is one relation and obviously a is zero because our we have a constant velocity motion yes ma'am clear two constant acceleration meaning this time a is c so we get a is equal to by definition of change of velocity over dt so from here you get from say v naught to v v naught means initial velocity right dv equal to from 0 to t a dt From here you get v minus v naught is a t. Or we get this famous relation for constant v is equal to a t plus v naught. Bingo. I bet you have seen this a million times in high school. And now because we define, I can't go to the next page, everybody stop me, okay. And now we know that V is equal to ds over dt. So now you know that from s1 to s2, ds is equal to from 0 to t, V dt. So we got V for. constant acceleration from previous page so just put down here zero to t a t plus v plus v naught dt this from high school gonna give you s2 is equal to integration of this 1 over 2 a t square plus s1 here we go so for constant acceleration we have these relations and we're going to use in the problems shortly this class or next class okay Already also we developed this. V square remember minus union square is 2 a s2 minus s1. Already in previous class we developed this. Right. It's coming from where? From here. right we use the chain rule saying this is gonna be ds over dt and dv over ds it's gonna be V and if you put that so for summarizing if velocity is constant get these uh uh s2 is equal to vt plus s1 and obviously acceleration is zero Okay, and if acceleration is constant, you get s2 is equal to 1 over 2 at square plus v0t plus s1, and you get velocity is at plus v0t. V1 and you get this V2 square minus V1 square is 2A S2 minus S1 and these are for translational motion these are all of these are for or linear motion translational on linear motion Motion. When we go to rotational motion, we're going to change notation for this. The name is translational motion. Or rectangular motion. Like as XYZ. Rectangular motion. Right. So now let's go to another case. Case number 3 because here is 1, 2, now 3. Theory is that 3 says if acceleration now is a function of time. It's not 0, it's not constant, changing with time acceleration. happens to your car when you're igniting the engine and you want to speed up acceleration changing until you reach to a constant velocity whatever right even not for the formula one drivers acceleration changing with time right so we know that this is equal to a is equal to dv over dt is equal to f of t hmm so from like what let me give the example a is equal to t cube huh so from here you get dv equals to f of t dt Again we are integrating from V0 to final velocity dv from 0 to t f of t dt. So for relation number 3 you get velocity is V0 plus integration from 0 to t f of t dt. This is the relation for if is acceleration. Now you have V. You can replace inside this from S0 to S. Ds for finding displacement from 0 to T. V dt. You're going to put this velocity inside here to find the displacement. If acceleration is a function of time. Example. Can I go to the next page? Yes. Example. Egg. Example. Right? Example. In my language, example is kind of zambil, means basket. Example means basket of eggs. Okay, so I'm saying suppose acceleration is sine of t. It means you are driving with your friends. I'm just speeding up, slowing down, speeding up, slowing down. It happens, huh? And the police are going to pull over you and hand you up and go to the station, right? So, an assumption is that it's not the zero t. is zero obviously and final t is two and I say v naught is two meter per second initial velocity of your car or your body when you're dancing it happens huh ask Michael Jackson he was doing that so my question is what is velocity velocity at t two seconds and velocity displacement at t two seconds by having such acceleration so based on what we developed here here we go this so you get velocity gonna be v naught plus from zero you Final is 2. F of D is sine of D. Sine of D DT. So V0 is 2. Problem says 2 meter per second from. And the integration of this for God's sake in high school is what? Minus cosine of D. Minus cosine of D. Alright. So I'm going to write. I have to sometimes this is crazy. So this is going to be 2 plus minus cosine of t from where from 0 to 2 and use a calculator put the upper bound lower bound. It's going to give you some answer. okay okay another example I say a is equal to t squared parabolic v naught is zero s naught is zero starting from rest t is zero and v is I want to find V at 3 seconds. At 3 seconds. Good. Again, V is V0 plus 0 to 3 T square DT. So, V0 is 0. From 0 to 3 T square. dt from high school gonna be 1 over 3 t cube from 0 to 3 gonna be 1 over 3 multiplied by 27 so gonna be 9 meter per second and the velocity at t is equal to 3 seconds So, but this is velocity term. I want to calculate displacement after 3 seconds. Okay, so what you're going to do. Oh God, I have to open new PowerPoint now. Doesn't matter. So, I taught you S is. I told you V is ds over dt and now I'm writing million times this. From s0 to s, ds is v0 to v, v dt. V dt. S0 was 0. Problem set. S minus S0 equals to. Oh sorry. From 0 to t. Not from 0 to t. From 0 to t. So this is going to be from 0 to t. V was. We derive together. 1 over 3 t cubed dt. Problem says starting from rest at the origin. So S is going to be 1 over 4 and T is equal to 3. T4 from 0 to 4 seconds. It's going to be 1 over 12. Yeah, 1 over 12. It's going to be 1 over 12 and 3. What am I doing? 3 to the power of 4. So, it's going to be 27 over 4 meters. Okay? Okay, case number four. Case number four. If now a is function of velocity, not time. This is definition of acceleration. Here we go. So from here you can find. dt is equal to dv over f of v or you get time is integration from v naught to v dv over f of v bingo this is the relation and as you know the basics that s0 to sps is v0 sorry from 0 to t vdt so and Yeah, so you have to calculate t to from here. dt what's the derivative of integration from high school function itself right gonna be dv over f of v so you're gonna put it here like this from s not to s ds and now from v not to v v multiplied by this v dv over F of V. Alright, so the other relation gonna be this. S is equal to S0, V0 to V, VdV over F of V. Here we go. Example. can i go to the next page example suppose a is the cubic function of velocity t not is zero starting from time zero s not is zero and i say v naught is point one meter per second So from here, this relation, time is equal to from V0 to V, dV over F of V. What is F of V for God's sake? Is this. So it's going to be from 0.1 to V. dv over v cube. Refreshing in mind from high school. xn dx is going to be 1 over m plus 1. xm plus 1. Your n is minus 3 because it comes up right. So it's going to be comes up like this from. 0.1 to V. V minus 3 dB. Your n is minus 3. It's going to be 1 over 1 minus 3. V 1 minus 3 from 0.1 to V. So this is minus 1 over 2. And V minus power of minus 2 comes down. v square from 0.1 to v. So if you put the upward down and lower down, minus 1 over 2, 1 over v square minus 1 over 0.1 square. That's the answer. the time is calculated meaning that if the if if you are given by velocity says 100 meter per second you can put here and calculate the time lapse that's right time is gone can i go to the next page yep case number five If now a is a function of displacement, but we already talked to this relation. Yes, the previous class and today BDV. So integration from both sides gonna be from S0 S. Your A is F of S simple like this. This from V0 to V, V, dV. We have done this in the first class probably and second class. This is from kindergarten. So the relation becomes this changing color. V square equals to V naught square plus 2 S naught to S F of S dS. Once you find v as function of this, I'm going to name this as this total as g of s is total when you find it. What you're going to do? You're going to say V is equal to velocity of change of displacement over DT. So DT is going to be DS over V. But V you already found as a function of displacement. So DS over G of S. And obviously this is another important thing. So T is equal to integration from S0 to S. DS over. yes we got it everybody so being set example if acceleration is Cube of displacement or your f of s is this. Based on this relation, this we just wrote, we're going to say, saying that velocity is zero initial velocity. So v square is v naught square plus 2, as I said, s naught to s, f of s. ds and i'm saying initial displacement also zero means is on the core and the origin of the coordinate system that you assign this is going to be 0 plus 2 from 0 to s s cube ds so this is going to be 2 multiplied by 1 over 4 s 4 so we're going to be half of s to power of 4. this is what this is velocity squared so the velocity becomes e to the power of 2 s2 and if you have displacement you know how many meter that you travel just put it here give you the velocity of your moving object Now you have it. Just put it here. So T is going to be from 0 to S again because initial displacement is 0. dS over 1 over this S square. This comes up. from 0 to s and s minus 2 ds right everyone huh you're just stuck in the high school i'm not teaching you still university level i'm just playing with math okay so and as you know this is equal to root square of 2 and it's gonna be minus 1 S-1 from 0 to S. Hmm? I'm gonna put S0 as 0.2. I'm gonna change it guys. 0.2 Now... I have a point here. I'm gonna give it 0. But this is gonna be 2 over S from 0 to S. Okay. Guys, it continues. Look at here. Putting upper bound, putting lower bound. What happened here? What's happened here? 1 over 0 from high school? Infinity. Infinity in control class, like as derivative and integration, has a definition. What does it mean infinity? In words, not in math. It means undefined, right? I don't know. you cannot solve this with this initial conditions of a start is zero you cannot do that for displacement you could do for velocity but you cannot calculate displacement by this initial condition because infinity and sucks huge number makes makes no sense right that's why you change your initial velocity close to zero say 0.001 See what I'm saying? I recalculate everything. We name it feasibility study in engineering. Okay, there we go. Now let me go to your book and solve problem from your lovely book. hmm okay where are your book there is a problem 227 of the book let me open your book okay guys you are you are becoming engineer hopefully soon next four years huh next three years your art is to understand problem very carefully to 27 no 221 actually what Okay, it's recording, I'm gonna upload on the YouTube. Okay, and then write it down part by part. Okay, this is your art, not rush to solve a problem. What the hell was that? Oh! Okay, here we go, 221. Okay, a girl, as you see here, rolls the ball up and incline and allows it to return to her. See, for the angle, this theta here, and the ball involved. the acceleration of the ball along the incline is constant you have to write it down don't say I know it I'm not blind but you have to write it down a is c constant because you have to fit in the categories I just taught you okay direct it down in the directed down incline it shows acceleration is always downward like this it says you have to plot it acceleration is a vector as i told you right directed down the angle if the ball is released with the speed of four meter per second means initial velocity v naught you have to write it down is four meter per second determine the distance s it moves up the incline before returning its directions okay here maximum going up and the total time required for the ball to return to the child's hands. Okay, let's do it together. Easy peasy, huh? Problem 221. The distinct line and this is angle theta and the ball is being released by this lady going up until here. And first you want to calculate. This distance before it returns. And what we're going to do in dynamics is. What's the very first and important step in dynamics? I told you a million times. What's that? What's the very first step in dynamics? Setting up your coordinate system. Don't say I know it. I know it means I make mistake. and I'm gonna scrub everything in Tesla in NASA wherever you go okay so because it's starting from here I'm gonna set it up here X&Y here we go you got it everybody so this is ball in the Midway not the Chicago Midway Airport I mean here okay understand so what affecting on the ball makes it back to the girl gravity so we are not blind we are right plotting down gravity so the acceleration of the ball is gravity right because of gravity so in the in the high school you decompose based on your coordinate system here we go Here is theta, here is also theta. Here is going to be g cosine theta in y direction. And here is going to be g sine theta. And now you understand why problem says the acceleration is constant and always looking downward of the incline. Looking down, g sine of theta. That's acceleration of ball. understand everyone yes clear yeah so acceleration is 0.25 g okay problem says but we found the acceleration is just g sine theta we just found it this g with this g goes new york what you get is this sine of theta is 0.25 g oh you can't find the inclined angle by plotting two lines and setting up your coordinate system i have the numbers if you put that it's going to be 14.47 degree Okay, now problem set initial velocity 0 is released from 0 by the girl. As a is constant, as we see is constant, and I told you, you can use this relation. V square minus V naught square is 2a s minus s naught. I just today told you. in case number two okay let's analyze this initial velocity is zero good we set up the coordinate system at the starting point so initial displacement is is zero so that's why i'm telling you don't say no coordinate system plot it and plot it wisely and smartly okay so it says calculate s before it returns ball goes up stops here and backs downward that's right so at this point stops what is zero velocity so this is also zero final velocity for finding s right everybody hmm oh sorry initial velocity is no it's not zero you know how much four well i'm not telling you oh why are you saying zero i'm stupid this is four this is 16. i can't use so from here you get oh we have a term here remaining i'm going to teach you minus 16 equals to 2 multiplied by a a is 0.25 g multiplied by s because this is zero go all of you yeah No, this A. This was another story. No, no, I'm asking about, I'm wondering whether no sign fit a notch equal 0.25 or is it? Oh, yeah. You mean here? Yeah. Thank you. It's typo. You cannot defeat me. I'm just kidding. It's just typo. Sorry. Yeah, he's right. Okay. Okay, guys, look at here. What's wrong here? What's wrong here? Without solving. Look at that. Don't write. Yes, position is negative. That sucks. Why? It says acceleration is downward always. Yes or no? This is a coordinate system. Plus sides. Negative sides. Just look at negative side. So this is going to be minus 2a minus 2.5g. See what I'm saying? When you set up a coordinate system you are evaluating the sign of vector with respect to coordinate system plus or minus. So putting that your s is going to be 3.26 meter. okay now because acceleration is constant you got this relation 80 plus three naught for going up at the point maximum is zero is 0.25 g time of upping going up plus velocity and this can again put minus here because don't forget that so time of going up gonna be four divided by 0.25 g gonna be 1.63 s second guys ball goes up with the same acceleration that goes down problem says always downward is constant so what you're gonna do you're gonna say time of total is double of time of going up here we go two times one point six three calculate that time so again evaluating vector signs respect to coordinate system assigned by you okay we got it okay just hitting can i go to next page oh uh let me just teach you give you some other examples of okay look at this guys 226 do it as homework for yourself at home not to me the 14 inch spring is compressed to an 8 inch length from 14 which compressed to 8 inch where it is released from rest and acceleration block 8 so from here is released from rest means what is zero initial velocity zero and accelerate block the acceleration has initial value of 400 feet per second okay you have to plot it guys it giving you a plot acceleration versus time initial value is 400 say here is 400 feet per second square and then decreases linearly with the x movement of block reaching zero when the spring regains its original 14 seconds so linearly comes down get to zero i'm sorry this is s versus s as it says gain acceleration becomes zero when the spring regains original 14. so here is how much i'm sorry so here is how much 14 minus 8 6 inch acceleration becomes zero so now you have a as f of s as i told you you have to first derive it from high school right huh in high school we say y minus y naught is equal to slope x minus x naught huh first of all all of you look at here don't write all of you look at here this is feet per second squared this is inch you have to homogenize units you have to change inch to feet or feet to inch you cannot go ahead see what I'm saying and this cost 20 million 200 million dollars to NASA they did a mistake a scientist they forgot to do it homogenize the units okay so be careful then M of that It's going to be this. After doing that, say this is a, this is b. It's going to be minus a over b. These numbers after homogenizing. And then I'm going to select this point. So a minus 0 is going to be that m that you calculated. S minus this point. 6 inch after homogenizing. So you get a function of displacement. And then put there and go from Understand calculate time T for block to go to 3 inch and 6 middle way and the full way You have the formula just do it right just replace that integration calculation when When A is a function of displacement. So that's why I'm telling you always. Reading problems stated not for getting bachelor degree in mechanical engineering. Or master or PhD. For work. If you're a scientist you have to understand problems. If you're not, you're not. Okay. So. This is another type of tussie. This is velocity versus displacement. Right? Linear. Acceleration versus displacement. This is deceleration. So now you know everything. Okay? I want to see next class I'm going to solve this problem for you it's very practical it was a part of my PhD in United States I'm not kidding I did my PhD for project was for US navy very very trivial part of that was this I'm gonna solve this 241 for you next class And we continue to plan our motion. Okay. So what's interesting that A is a function of X and is nonlinear function of X. Inverse. Okay. Good. So just give me a second.

as your friend said what's your name sir christian said integral is the area sum of under the curve but again if you ask your brother sister what is integration in one word from this point you're looking to the back means derivative means future integration is past real it is In control, I'm going to teach you about the PID controller to control robot, to control airplane, to control shuttle, to control everything. Interesting in your muscles as human being, when you walk, when you dance, you have PID controller in your joints. I'm sorry.

So in your joints, you have DI controller, derivative integrator. This I is for integrator. this d is for derivative i'm going to skip p but i'm telling you what's p d is derivative future i is integrator past p is present okay pid or proportional i don't want to teach you control now but you have to know the fundamentals of science not graph okay if you derivative means future okay and also we had a is equal to dv or dt you're having a note so i'm just refreshing to continue the topics so being said if you have velocity versus time changing the color make it more beautiful and attractive the slope here gonna be dv dt gonna be what acceleration But in vice versa, if you have acceleration versus time and you have this graph, area under the graph is going to be what? Velocity. It's going to be velocity.

Because V from V1 to V2, DV goes from T1 to T2. A D T from here Obviously, you had it. Okay, everybody? Good. So, let's continue the type of motions.

We are on the kinematics yet. So, we have two types of motion. One.

okay constant velocity as you are doing every day by after parking the car and coming to the classroom yeah in average you have constant velocity hmm when there is constant velocity or v i'm sorry i'm not seeing you when v is c c means constant okay c is constant this is how we write in mechanical or electrical or physics or civil engineering notations right when we is constant there is no change in velocity what is zero what is zero when there is no change in velocity i just taught you acceleration is zero being said is a zero so when you use you your car cruise system and say okay in the highway just drive for 60 miles per hour means acceleration is zero of your car right so velocity as we define is ds over dt now as we know it ds is v dt integration from both sides s1 s2 and from say zero to t so it's going to be s2 minus s1 from kindergarten this guy because v is constant comes out going to be vt so we get our famous may you have seen in high school physics we use it up um s2 equals to when vt plus s1 here we go this is one relation and obviously a is zero because our we have a constant velocity motion yes ma'am clear two constant acceleration meaning this time a is c so we get a is equal to by definition of change of velocity over dt so from here you get from say v naught to v v naught means initial velocity right dv equal to from 0 to t a dt From here you get v minus v naught is a t. Or we get this famous relation for constant v is equal to a t plus v naught. Bingo.

I bet you have seen this a million times in high school. And now because we define, I can't go to the next page, everybody stop me, okay. And now we know that V is equal to ds over dt.

So now you know that from s1 to s2, ds is equal to from 0 to t, V dt. So we got V for. constant acceleration from previous page so just put down here zero to t a t plus v plus v naught dt this from high school gonna give you s2 is equal to integration of this 1 over 2 a t square plus s1 here we go so for constant acceleration we have these relations and we're going to use in the problems shortly this class or next class okay Already also we developed this.

V square remember minus union square is 2 a s2 minus s1. Already in previous class we developed this. Right.

It's coming from where? From here. right we use the chain rule saying this is gonna be ds over dt and dv over ds it's gonna be V and if you put that so for summarizing if velocity is constant get these uh uh s2 is equal to vt plus s1 and obviously acceleration is zero Okay, and if acceleration is constant, you get s2 is equal to 1 over 2 at square plus v0t plus s1, and you get velocity is at plus v0t.

V1 and you get this V2 square minus V1 square is 2A S2 minus S1 and these are for translational motion these are all of these are for or linear motion translational on linear motion Motion. When we go to rotational motion, we're going to change notation for this. The name is translational motion.

Or rectangular motion. Like as XYZ. Rectangular motion.

Right. So now let's go to another case. Case number 3 because here is 1, 2, now 3. Theory is that 3 says if acceleration now is a function of time.

It's not 0, it's not constant, changing with time acceleration. happens to your car when you're igniting the engine and you want to speed up acceleration changing until you reach to a constant velocity whatever right even not for the formula one drivers acceleration changing with time right so we know that this is equal to a is equal to dv over dt is equal to f of t hmm so from like what let me give the example a is equal to t cube huh so from here you get dv equals to f of t dt Again we are integrating from V0 to final velocity dv from 0 to t f of t dt. So for relation number 3 you get velocity is V0 plus integration from 0 to t f of t dt.

This is the relation for if is acceleration. Now you have V. You can replace inside this from S0 to S. Ds for finding displacement from 0 to T. V dt.

You're going to put this velocity inside here to find the displacement. If acceleration is a function of time. Example.

Can I go to the next page? Yes. Example. Egg. Example.

Right? Example. In my language, example is kind of zambil, means basket.

Example means basket of eggs. Okay, so I'm saying suppose acceleration is sine of t. It means you are driving with your friends.

I'm just speeding up, slowing down, speeding up, slowing down. It happens, huh? And the police are going to pull over you and hand you up and go to the station, right?

So, an assumption is that it's not the zero t. is zero obviously and final t is two and I say v naught is two meter per second initial velocity of your car or your body when you're dancing it happens huh ask Michael Jackson he was doing that so my question is what is velocity velocity at t two seconds and velocity displacement at t two seconds by having such acceleration so based on what we developed here here we go this so you get velocity gonna be v naught plus from zero you Final is 2. F of D is sine of D. Sine of D DT. So V0 is 2. Problem says 2 meter per second from. And the integration of this for God's sake in high school is what?

Minus cosine of D. Minus cosine of D. Alright. So I'm going to write. I have to sometimes this is crazy.

So this is going to be 2 plus minus cosine of t from where from 0 to 2 and use a calculator put the upper bound lower bound. It's going to give you some answer. okay okay another example I say a is equal to t squared parabolic v naught is zero s naught is zero starting from rest t is zero and v is I want to find V at 3 seconds. At 3 seconds.

Good. Again, V is V0 plus 0 to 3 T square DT. So, V0 is 0. From 0 to 3 T square.

dt from high school gonna be 1 over 3 t cube from 0 to 3 gonna be 1 over 3 multiplied by 27 so gonna be 9 meter per second and the velocity at t is equal to 3 seconds So, but this is velocity term. I want to calculate displacement after 3 seconds. Okay, so what you're going to do.

Oh God, I have to open new PowerPoint now. Doesn't matter. So, I taught you S is.

I told you V is ds over dt and now I'm writing million times this. From s0 to s, ds is v0 to v, v dt. V dt. S0 was 0. Problem set.

S minus S0 equals to. Oh sorry. From 0 to t. Not from 0 to t. From 0 to t.

So this is going to be from 0 to t. V was. We derive together.

1 over 3 t cubed dt. Problem says starting from rest at the origin. So S is going to be 1 over 4 and T is equal to 3. T4 from 0 to 4 seconds. It's going to be 1 over 12. Yeah, 1 over 12. It's going to be 1 over 12 and 3. What am I doing? 3 to the power of 4. So, it's going to be 27 over 4 meters.

Okay? Okay, case number four. Case number four.

If now a is function of velocity, not time. This is definition of acceleration. Here we go.

So from here you can find. dt is equal to dv over f of v or you get time is integration from v naught to v dv over f of v bingo this is the relation and as you know the basics that s0 to sps is v0 sorry from 0 to t vdt so and Yeah, so you have to calculate t to from here. dt what's the derivative of integration from high school function itself right gonna be dv over f of v so you're gonna put it here like this from s not to s ds and now from v not to v v multiplied by this v dv over F of V. Alright, so the other relation gonna be this.

S is equal to S0, V0 to V, VdV over F of V. Here we go. Example. can i go to the next page example suppose a is the cubic function of velocity t not is zero starting from time zero s not is zero and i say v naught is point one meter per second So from here, this relation, time is equal to from V0 to V, dV over F of V. What is F of V for God's sake? Is this. So it's going to be from 0.1 to V. dv over v cube.

Refreshing in mind from high school. xn dx is going to be 1 over m plus 1. xm plus 1. Your n is minus 3 because it comes up right. So it's going to be comes up like this from.

0.1 to V. V minus 3 dB. Your n is minus 3. It's going to be 1 over 1 minus 3. V 1 minus 3 from 0.1 to V. So this is minus 1 over 2. And V minus power of minus 2 comes down. v square from 0.1 to v. So if you put the upward down and lower down, minus 1 over 2, 1 over v square minus 1 over 0.1 square. That's the answer.

the time is calculated meaning that if the if if you are given by velocity says 100 meter per second you can put here and calculate the time lapse that's right time is gone can i go to the next page yep case number five If now a is a function of displacement, but we already talked to this relation. Yes, the previous class and today BDV. So integration from both sides gonna be from S0 S. Your A is F of S simple like this. This from V0 to V, V, dV. We have done this in the first class probably and second class.

This is from kindergarten. So the relation becomes this changing color. V square equals to V naught square plus 2 S naught to S F of S dS.

Once you find v as function of this, I'm going to name this as this total as g of s is total when you find it. What you're going to do? You're going to say V is equal to velocity of change of displacement over DT.

So DT is going to be DS over V. But V you already found as a function of displacement. So DS over G of S. And obviously this is another important thing. So T is equal to integration from S0 to S. DS over.

yes we got it everybody so being set example if acceleration is Cube of displacement or your f of s is this. Based on this relation, this we just wrote, we're going to say, saying that velocity is zero initial velocity. So v square is v naught square plus 2, as I said, s naught to s, f of s. ds and i'm saying initial displacement also zero means is on the core and the origin of the coordinate system that you assign this is going to be 0 plus 2 from 0 to s s cube ds so this is going to be 2 multiplied by 1 over 4 s 4 so we're going to be half of s to power of 4. this is what this is velocity squared so the velocity becomes e to the power of 2 s2 and if you have displacement you know how many meter that you travel just put it here give you the velocity of your moving object Now you have it. Just put it here.

So T is going to be from 0 to S again because initial displacement is 0. dS over 1 over this S square. This comes up. from 0 to s and s minus 2 ds right everyone huh you're just stuck in the high school i'm not teaching you still university level i'm just playing with math okay so and as you know this is equal to root square of 2 and it's gonna be minus 1 S-1 from 0 to S. Hmm?

I'm gonna put S0 as 0.2. I'm gonna change it guys. 0.2 Now... I have a point here. I'm gonna give it 0. But this is gonna be 2 over S from 0 to S.

Okay. Guys, it continues. Look at here. Putting upper bound, putting lower bound.

What happened here? What's happened here? 1 over 0 from high school? Infinity.

Infinity in control class, like as derivative and integration, has a definition. What does it mean infinity? In words, not in math.

It means undefined, right? I don't know. you cannot solve this with this initial conditions of a start is zero you cannot do that for displacement you could do for velocity but you cannot calculate displacement by this initial condition because infinity and sucks huge number makes makes no sense right that's why you change your initial velocity close to zero say 0.001 See what I'm saying?

I recalculate everything. We name it feasibility study in engineering. Okay, there we go.

Now let me go to your book and solve problem from your lovely book. hmm okay where are your book there is a problem 227 of the book let me open your book okay guys you are you are becoming engineer hopefully soon next four years huh next three years your art is to understand problem very carefully to 27 no 221 actually what Okay, it's recording, I'm gonna upload on the YouTube. Okay, and then write it down part by part. Okay, this is your art, not rush to solve a problem.

What the hell was that? Oh! Okay, here we go, 221. Okay, a girl, as you see here, rolls the ball up and incline and allows it to return to her.

See, for the angle, this theta here, and the ball involved. the acceleration of the ball along the incline is constant you have to write it down don't say I know it I'm not blind but you have to write it down a is c constant because you have to fit in the categories I just taught you okay direct it down in the directed down incline it shows acceleration is always downward like this it says you have to plot it acceleration is a vector as i told you right directed down the angle if the ball is released with the speed of four meter per second means initial velocity v naught you have to write it down is four meter per second determine the distance s it moves up the incline before returning its directions okay here maximum going up and the total time required for the ball to return to the child's hands. Okay, let's do it together.

Easy peasy, huh? Problem 221. The distinct line and this is angle theta and the ball is being released by this lady going up until here. And first you want to calculate.

This distance before it returns. And what we're going to do in dynamics is. What's the very first and important step in dynamics?

I told you a million times. What's that? What's the very first step in dynamics? Setting up your coordinate system.

Don't say I know it. I know it means I make mistake. and I'm gonna scrub everything in Tesla in NASA wherever you go okay so because it's starting from here I'm gonna set it up here X&Y here we go you got it everybody so this is ball in the Midway not the Chicago Midway Airport I mean here okay understand so what affecting on the ball makes it back to the girl gravity so we are not blind we are right plotting down gravity so the acceleration of the ball is gravity right because of gravity so in the in the high school you decompose based on your coordinate system here we go Here is theta, here is also theta. Here is going to be g cosine theta in y direction. And here is going to be g sine theta.

And now you understand why problem says the acceleration is constant and always looking downward of the incline. Looking down, g sine of theta. That's acceleration of ball.

understand everyone yes clear yeah so acceleration is 0.25 g okay problem says but we found the acceleration is just g sine theta we just found it this g with this g goes new york what you get is this sine of theta is 0.25 g oh you can't find the inclined angle by plotting two lines and setting up your coordinate system i have the numbers if you put that it's going to be 14.47 degree Okay, now problem set initial velocity 0 is released from 0 by the girl. As a is constant, as we see is constant, and I told you, you can use this relation. V square minus V naught square is 2a s minus s naught.

I just today told you. in case number two okay let's analyze this initial velocity is zero good we set up the coordinate system at the starting point so initial displacement is is zero so that's why i'm telling you don't say no coordinate system plot it and plot it wisely and smartly okay so it says calculate s before it returns ball goes up stops here and backs downward that's right so at this point stops what is zero velocity so this is also zero final velocity for finding s right everybody hmm oh sorry initial velocity is no it's not zero you know how much four well i'm not telling you oh why are you saying zero i'm stupid this is four this is 16. i can't use so from here you get oh we have a term here remaining i'm going to teach you minus 16 equals to 2 multiplied by a a is 0.25 g multiplied by s because this is zero go all of you yeah No, this A. This was another story. No, no, I'm asking about, I'm wondering whether no sign fit a notch equal 0.25 or is it? Oh, yeah.

You mean here? Yeah. Thank you.

It's typo. You cannot defeat me. I'm just kidding. It's just typo. Sorry.

Yeah, he's right. Okay. Okay, guys, look at here. What's wrong here? What's wrong here?

Without solving. Look at that. Don't write. Yes, position is negative.

That sucks. Why? It says acceleration is downward always. Yes or no? This is a coordinate system.

Plus sides. Negative sides. Just look at negative side.

So this is going to be minus 2a minus 2.5g. See what I'm saying? When you set up a coordinate system you are evaluating the sign of vector with respect to coordinate system plus or minus. So putting that your s is going to be 3.26 meter.

okay now because acceleration is constant you got this relation 80 plus three naught for going up at the point maximum is zero is 0.25 g time of upping going up plus velocity and this can again put minus here because don't forget that so time of going up gonna be four divided by 0.25 g gonna be 1.63 s second guys ball goes up with the same acceleration that goes down problem says always downward is constant so what you're gonna do you're gonna say time of total is double of time of going up here we go two times one point six three calculate that time so again evaluating vector signs respect to coordinate system assigned by you okay we got it okay just hitting can i go to next page oh uh let me just teach you give you some other examples of okay look at this guys 226 do it as homework for yourself at home not to me the 14 inch spring is compressed to an 8 inch length from 14 which compressed to 8 inch where it is released from rest and acceleration block 8 so from here is released from rest means what is zero initial velocity zero and accelerate block the acceleration has initial value of 400 feet per second okay you have to plot it guys it giving you a plot acceleration versus time initial value is 400 say here is 400 feet per second square and then decreases linearly with the x movement of block reaching zero when the spring regains its original 14 seconds so linearly comes down get to zero i'm sorry this is s versus s as it says gain acceleration becomes zero when the spring regains original 14. so here is how much i'm sorry so here is how much 14 minus 8 6 inch acceleration becomes zero so now you have a as f of s as i told you you have to first derive it from high school right huh in high school we say y minus y naught is equal to slope x minus x naught huh first of all all of you look at here don't write all of you look at here this is feet per second squared this is inch you have to homogenize units you have to change inch to feet or feet to inch you cannot go ahead see what I'm saying and this cost 20 million 200 million dollars to NASA they did a mistake a scientist they forgot to do it homogenize the units okay so be careful then M of that It's going to be this. After doing that, say this is a, this is b. It's going to be minus a over b.

These numbers after homogenizing. And then I'm going to select this point. So a minus 0 is going to be that m that you calculated.

S minus this point. 6 inch after homogenizing. So you get a function of displacement. And then put there and go from Understand calculate time T for block to go to 3 inch and 6 middle way and the full way You have the formula just do it right just replace that integration calculation when When A is a function of displacement. So that's why I'm telling you always.

Reading problems stated not for getting bachelor degree in mechanical engineering. Or master or PhD. For work. If you're a scientist you have to understand problems.

If you're not, you're not. Okay. So. This is another type of tussie.

This is velocity versus displacement. Right? Linear.

Acceleration versus displacement. This is deceleration. So now you know everything. Okay?

I want to see next class I'm going to solve this problem for you it's very practical it was a part of my PhD in United States I'm not kidding I did my PhD for project was for US navy very very trivial part of that was this I'm gonna solve this 241 for you next class And we continue to plan our motion. Okay. So what's interesting that A is a function of X and is nonlinear function of X.

Inverse. Okay. Good. So just give me a second.

Transcript for:Fundamentals of Kinematics and Dynamics

Transcript for:
Fundamentals of Kinematics and Dynamics