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Calculus 3 Lecture 14.4: Center of Mass (and Moments of Mass and Inertia) for Lamina in 2-D
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- čas přidán 26. 04. 2016
- Calculus 3 Lecture 14.4: Center of Mass (and Moments of Mass and Inertia) for Lamina in 2-D: How to find Center of Mass, Mass, First Moments of Mass, Second Moments of Inertia, Radius of Gyration, and solve some cool word problems.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
24:27 - Radius of gyration
29:27 - Example 1
47:36 - Example 2
51:47 - Example 3
58:48 - Example 4
1:08:59 - Example 5
Thank u
Thank God for you.
49:40 Hi Professor Leonard, thanks for the Calculus videos. They've really helped. The equation for y = -1/2x + 4 is actually y = -1/2x + 2. Explanation is on point though.
exactly
Indeed!
classroom: "Yeah that looks right"
THANK YOU!!!! I just spent about a hour doubting my knowledge of slope "WHY AM I IN CALC 3 IF I DONT'T KNOW THE SLOPE FORMULA!!!!?????"
thanks you! I got scared thinking i got the wrong answer
2:20
2:55 - single integral represents either. Area under curve f(x). or *THE MASS OF A STRAIGHT WIRE GIVEN A MASS DENSITY FUNCTION* (1d-idea)
thus a double integral is analagous to this idea, the mass of a PLATE why a plate (now we're in 3 dimensions so we're dealing with planes)
4:00 - Moments of Mass: The tendency for the lamina to rotate about an axis.
5:34 - The further you are from the axis of rotation, the more it wants to rotate.
6:12 - If measuring mass density away from x-axis then its measuring along y (where its *perpendicular*)
7:25 - Mx - Tendency for rotation about x-axis. Moment of X has Y values. Why? (6:12, if measuring with respect to that x where moving a long y the axis perpendicular.)
9:15 - Sub in y value because ^ measurement is away from x-axis so going along y. (9:45 Wrench Example)
11:00 - My - Tendency for rotation about y-axis. Moment of Y has X values. Why (because if measuring with massdensity along y, measuring towards x the perpendicular axis to y which is x).
12:35 - Center of Mass: x(bar) *represents average*, 13:00. X center of mass = (Tendency of rotation about Y-axis) / (mass).
14:10 - Tend. Rotation about y all measured in x (distance). Take all the tendency of rotation of y
My dude, you probably forgot you commented this but you're a G.
I hope to one day see a Diff EQ series so that after I take it next semester I can come back and get a better understanding using your videos. Cannot thank you enough for how much your Calculus lectures have helped me!
Dear Professor Leonard. Thank you kindly sir. I got the flu in the second week of the semester. Have no memory of about 10 straight days. Your videos gave me enough to pass my first test with a 70 and as of this video I am completely caught up in the class a week ahead of the next test and feel confident I'll be pulling a 90 or above. Much appreciated. Hope to see more playlists.
Sincerely
An aspiring Sal Khan / Professor Leonard
Somebody from Santa Monica College recommended your channel I heard them mention you and I just watched your video and I’m glad I did. It would be an honor to take you in person. A student is only as good as their teacher.
10 minutes of this video >>>>>>>>> our professor's lecture
One of the problem towards the end, the function should be x=4-2y instead of x=8-2y. I verify the answer like 4 times
Indeed. He wrote y= (-1/2)x+4, instead of (-1/2)x+2, hence the error. Thank you!
He's so good at explaining stuff! I really wish he's my Prof
sir, i honestly believe that you are a god.
Professor Leonard ,thank you for another awesome video/lecture on the Center of Mass for Lamina in 2 D. This material is also presented in Engineering Physics One, Statics , and Dynamics. Many upper level Engineering courses use Calculus/Advanced Calculus.
I wish you taught physics (intro-college level) as well. Would this be possible?
Please, please to Differential Equations!!
I LOVE Differential Equations.
He did it!
Rockford Lines I’m so excited!
nerds
Can you do a special playlist for engineering sciences? I'm pretty sure you will explain it better than any other playlist out there.
you're the man! you've helped me earn A's from calc 1-3! any chance differential equations are in your future posts?
I had that same thought! I hope he does too
how many hours did u study per day?
He is so good as explaining detail concepts.
An easier way to remember the xbar formula.
Define My = M * xbar , since we can imagine concentrating all the mass of the lamina at the x bar coordinate.
Then we see we have to divide My by M to get xbar, since My/M = (M * xbar)/M = xbar.
Likewise for Mx.
Then we can use the integral ∫ ∫ (y or x) ρ(x,y) dA to find Mx or My, by adding the momenta of each point in the domain.
Professor, to further illustrate the concept of why we are using mass for an iterated integral you can bring into light the idea of mass density.
Thanks for explaining the concept of all these computations! It makes it a lot easier.
Your videos are great. Everything is explained thoroughly yet so simply.
Fuck my life it is 1 day before the test and only now I realize this video exists rip me
He keeps saying the formula is weird but It actually makes a lot of sense when he explains. Specially the 2nd time. He could be a good physics teacher.
thank you so much. how you ever taught other professors how to teach better?
I ACTUALLY GET IT :D two independants just say where it is on the plate. the dependant is the instantaneous mass at that point/
I really enjoy these in-class videos what happened to them? I feel like they were more immersive. Regardless, if I ever get successful, I'm not forgetting this channel Prof Leonard. I'll pay a visit to patreon. In fact I think I'm going to donate what I can right now.
at 1:04:39, for theta=0 to 2pi, why don't we worry that the definite integral will negate the areas, and so the area (net signed area) would be 0? thx.
tanks for all de vids, you explain really well, keep it up.
I learned this in first year physics... Never really understood it lol thanks for clearing that up:) i go to york university ^^
Lol. Our math department is a joke compared to this ONE guy.
Leonard I saw that density fcts are related alot to double integrals, like in probabilities, electricity or whatever, could you reply explaning if I am correct, thank you!
ID#:4FF7AC3 Completion: September 15, 2023
COURSE PROVIDED BY:
Instructor: Paul J. Scruton
Pleaseee do Differential Equation lectures Prof! :D:D:D
Thank you my friend, I wish you were my Calculus brofessor...
"whats the first thing you're gonna wanna do?.... Give up." hahahahaha
Would you make the separate video on the proofs of these formulae??
What is the intuition for multiplying by "x" and "y" in the first moments of mass? I understand it conceptually with the units, but is there some sort of proof for why you multiply these distances by the mass density?
For the example at 58:34, would it be equivalent to say that r goes from 0 to the function y=sqrt(4-r^2cos^2(theta))? (function for top half of the circle converted to polar)
You're hero , sir..
why is he using a double integral to find the mass on the lamina about the x and y? I did this in calculus 2 using single integrals.
Double integral will being you more to the world of reality !
I am a bit confused because in his video, Professor Leonard adds the opposite variable when solving for m knot x. However, in other videos I have watched they add the same variable that is it with respect to. Can someone explain this?
Clark Kent teaches math? Since when?
question: at 24:00 prof. Leonard says that the difference between the second moment of inertia and the first moment of area in the x direction is that the "y" is squared, well my first question is if y resembles a distance in the y direction they why is it multiplied and not just added? my other question is why is it just squared in the second moment of inertia? Thank you very much.
The answer to why is definition. First moment means we multiply by distance from a reference point. Second moment means we multiply by distance from a reference point to the second power.
If you are looking to why we define it this way in the context of an application, the reason is that the distance from the reference point impacts the results in two ways. For the area "moment of inertia", or second moment of area, there are two reasons why distance from the reference axis plays a role. The top and bottom fibers are strained the most as a consequence of bending the beam. And on top of that, the top and bottom fibers contribute the most to opposing the bending moment, since they have the greatest leverage.
Just saying, engineering usually denote second moment (gyration) by a "k".
Note for the very last number, I got 5957.595 , rounded up it gives the same thing but yeah.
Too handsome teacher! 🤭🤭🤭😁
Mass of a straight line in calc I? Doesn't he mean length of a curve?
Dangit I wish I would have reviewed this video before my Calc 2 test, there was a slightly different way of doing Center of Mass that I learned with getting the Area first and then x bar is 1/Area times integral xf(x)dx and then y bar = 1/area times integral (1/2)(f(x))^2, but what Professor Leonard showed worked too, so I messed up on that one problem on the test :/
40:20 everyone gets woken up
Hey Prof Leonard, what happened to, I think, chapter 14.5 surface area?? Im sad...
Keep watching!!! We skip it for now because it is covered later (and better) in a later section :)
May i know in which section is it being covered?
Also curious as to where Surface Area is covered. You're the best at giving intuition! I figure the semester is over by now, but I couldn't find it. Thank you!
Just found it, it's with section 15.6
@@aliticx much appreciated brother
Is there any video for the application of integration?
I looooove you leonard
47:00 example 2 ( where i left off)
hey professor leonard, u made a mistake at min 50 , the equation of the second line is wrong , it should be x=-2y+4
can u tell me the correct one
i understand the idea have of how to solve for the mass by taking a double integral of the mass density function p(x,y) but im just lost on were the function p(x,y) actually comes from and what is it detailing ? is it from the density equation density=mass/volume ? and since volume is a double integral and we solve for the mass, do we get m=double integral of density? is this how that equation is derived ?
Mass actually equals triple integral of density. You have a region of space, and there is a given density at every point within it. You make an infinitesimal volume element (dV) in each of the three directions, such that dV = dx dy dz, and multiply this dV by the corresponding density. Add it up (i.e. integrate) along each of the three directions, and you get mass.
Volume could also be a triple integral, if you are simply integrating 1 dx dy dz. It is a double integral in cases where the value of the integrand, equals a distance quantity, such as the z-position along a region of space in the x and y directions.
Thank you for a clear and concise description.
Mass moments of inertia are the rotational equivalent of mass;
a body with high moment of inertia is hard to give a rotational acceleration, just like how a body with high mass is hard to give a linear acceleration. Similarly, an angle is the rotational equivalent of distance, torque is the rotational equivalent of force, and so on.
This is the best comment I have seen to describe the tendency to rotate concept mentioned in the video. I was a little confused by his explanation. Don't get me wrong. I have looked at all of professor Leonard's calculus videos and he is a wonderful teacher who explains things very clearly, and he helped me greatly on my path through mathematics, which is my own path now. No courses pass any more, no pressure. It's all good.
@@everettmcinnis5858 Cool. :)
Yeah, Professor Leonard is truly amazing, for sure - he makes sure that every step of all of his explanations are crystal clear before he moves on, and that is exactly what good teaching is all about.
THANKS SO MUCH :)
why is there no 15.5 lecture
Do any of the these lecture talk about the surface area by double integral? It seems not mentioned in 14.1~14.4.
Previous lecture, around the 2 hour mark
Examples start around 27:54
46:55
Anyone knows where 14.5 is?
52:22
24:48
What's the first thing we gonna do? *Give Up*
-hhhhhhhhhhhhhhhhhhhhhhhhhhh
Damn I love math
LAMINA MINA AA AA WAKKA WAKKA AA AA.... Ok ill just get back to the video :)
Genius
sir with due respect...i wanna ask that why there is no 14.5 lecture?
Hey there, Lesson 14.5 has to do with Surface Area. It is actually covered later, and in more detail, in a later section. Have fun!
Professor Leonard thank you so much sir have a great day.
respect from #pakistan!
1;05;29 yeah
is there no 14.5 lecture??
Professor Leonard sometimes skips sections that he thinks are irrelevant for this course, so 14.5 most likely didn't bring up anything important - it was probably something about deriving some particular integral that is used for very specific applications, or something like that.
YOU SHALL NOT PASS! lol...
That made me love this guy even more LOLLLLL
12:50 "wtf...."
60fps
Aw man. He didn't explain probability
can we just skip all this weird physics stuff?
I don't think he has the juice to do D.EQ or Linear Algebra. He is a pretty smart guy but those two courses require a real understanding of theory and how to teach it. I think he is at his limit. Of course I am saying this in hopes he will take up the challenge cause the other CZcams instructors or college instructors don't have a clue. They may know the subject but they can't hold a candle to this guy when it comes to teaching.
+Sandler Tossone Might do it someday, but it takes ALL I HAVE to video record a series. It is a tremendous undertaking and not something I just do into without some VERY serious preparation. We shall have to see what the future brings.
You are a god +Professor Leanard
We r all waiting for your Linear Algebra & differential equation lectures! Please Leonard!
I absolutely used to hate calculus. I and II until I came across your cal II videos. I now have an addiction to math. I was able to ace cal II thanks to your videos and took cal III on line since all I needed was your lecture. I took diff equations in person and the professor was pretty much dry and to the point. I actually used your formulas from cal III to define some of the formulas diff equations uses and I drew parallels that way. Thank you very much, I don't even know how you came across teaching, but I wish to eventually teach in the future and I will always refer to you when trying to do hard examples. Thank you!
I honestly don't think so. This guy knew Partial Differential Equation when he was 7 years old !
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