Center of Mass & Centroid Problems - Calculus

This calculus video tutorial provides a basic introduction into the center of mass of a system also known as the centroid. It explains how to find the x and y coordinates of the centroid of a planar lamina which is a 2d region. The center of mass occurs where an object or a region can be balanced by a pencil if gravity is the only force acting on it. This video contains plenty of examples and practice problems.
Improper Integrals:
• Improper Integrals - C...
Integration Into Inverse Trig:
• Integration into Inver...
Integration of Rational Functions:
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Integral of Logarithmic Functions:
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Integrating Exponential Functions:
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Integration Formulas:
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________________________________
Integration By Tables:
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Reduction Formulas - Integration:
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Center of Mass & Moments:
• Moment, Center of Mass...
Center of Mass & Centroid Problems:
• Center of Mass & Centr...
Hydrostatic Force Problems:
• Hydrostatic Force Prob...
Probability Density Functions:
• Probability Density Fu...
_________________________________
Normal Distributions - Calculus:
• Normal Distribution & ...
Homogeneous Differential Equations:
• Homogeneous Differenti...
1st Order Linear Differential Equations:
• First Order Linear Dif...
Bernoulli's Equation for Differential Equations:
• Bernoulli's Equation F...
Slope Fields:
• Slope Fields | Calculus
___________________________________
Converging & Diverging Sequences:
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Monotonic & Bounded Sequences:
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Пікірлер: 170

  • @TheOrganicChemistryTutor
    @TheOrganicChemistryTutor Жыл бұрын

    Next Video: kzread.info/dash/bejne/ZZ57j8qQl7iwm9I.html Final Exams and Video Playlists: www.video-tutor.net/

  • @aidancallaghan7059
    @aidancallaghan70594 жыл бұрын

    Ive literally just started skipping my calculus classes and watching you instead. much better.

  • @wesley9154

    @wesley9154

    4 жыл бұрын

    Same

  • @karypatron8540

    @karypatron8540

    3 жыл бұрын

    Same here

  • @jet468

    @jet468

    2 жыл бұрын

    lmao same, havent been in about 6 weeks, I literally only show up to the exams

  • @aubreydosto1894

    @aubreydosto1894

    Жыл бұрын

    I almost too, but I can't

  • @nathanthomas2337

    @nathanthomas2337

    Жыл бұрын

    @@jet468 did it work

  • @frenshyruiz7267
    @frenshyruiz72675 жыл бұрын

    how can anybody unlike this? he does an outstanding job in explaining.

  • @ceoofpoonis3668

    @ceoofpoonis3668

    4 жыл бұрын

    how can anybody *dislike this

  • @samuelborcherding8033

    @samuelborcherding8033

    3 жыл бұрын

    It is probably professors who's jobs he is taking lol

  • @eldi

    @eldi

    3 жыл бұрын

    too easy examples.

  • @diags_1109

    @diags_1109

    2 жыл бұрын

    @@ceoofpoonis3668 I dont know how bout you find out

  • @Arminecraft
    @Arminecraft5 жыл бұрын

    The calmness you have while explaining it, the methodology you use with perfection, it just couldn't be a better video. Thank you.

  • @joluju2375
    @joluju23754 жыл бұрын

    I appreciated this video very much mostly because it starts with the meaning of "balance" everyone "feels". Only after, it builts up progressively, and as a conclusion we get formulas, and we understand the "why" of these formulas. Thumb up, you care to explain well, I like it.

  • @clementhilty7239
    @clementhilty7239 Жыл бұрын

    You have videos on literally everything. Thanks so much!

  • @pernal9956
    @pernal99563 жыл бұрын

    Wow this video was absolutely amazing. It really helped me get a better understanding of how we can use calculus in physics. Thank you so much

  • @mohammedalahmed3133
    @mohammedalahmed3133 Жыл бұрын

    My life would have been a nightmare without you sir... you have my utmost gratitude

  • @abassmobolaji9809
    @abassmobolaji9809 Жыл бұрын

    I don’t know if organic chemistry tutor will see this but please we really need video on centroid of composite shapes,then moment of inertia. Lots of engineering students are having issues with this. It would be really nice if you can come through for us.

  • @clarinemaerenejane6238
    @clarinemaerenejane62382 жыл бұрын

    I think you forgot the 1/2 at last example when you get the y-coordinate

  • @WarriorBane
    @WarriorBane Жыл бұрын

    Thnx Julio Gonzalez your the best teacher I ever had

  • @GreatnessUrge
    @GreatnessUrge3 жыл бұрын

    🔥 This guy is too good. I wish you are able to tutor me in all my engineering courses.

  • @bobnob4393
    @bobnob43933 жыл бұрын

    love this vid, really useful. I just wish you did an example where there were the two lines didn't intersect. Like x=y^2 & x=1/2(y^2 + 1) in this case there are 3 lines that make up the area and i'm not sure on where to get the limits

  • @eliminshrintar
    @eliminshrintar6 жыл бұрын

    I know this is calculus of course. But this application is really physics based as well. Doing angular momentum we have to use this very technique to start off. I know because we just had a lecture on this very thing yesterday. And wow, was I lost in class. Timing could not be better for this video. Thanks!

  • @darrinlewis9659

    @darrinlewis9659

    4 жыл бұрын

    eliminshrintar llllkokeidkddiqwj$dj)*$najjxz#hjndjjjdjjd$kdkkxkd8$h$)3ksslll..s."lalwl..a =86#*@;nashhxdhx,mxkx,d,mxmxmddms#,-:--#-me,ihpfiddoofkklklcl,asked. ,some,leak,weld,mall'mqkqlqkakokskzkszhsjsikmzmxjxkxkxkd,dlowdlkufmrlgtduopu,amsmakmmd nold,lkklxkllxlfockc’m,Cox,Cox,XX h Kxxkxjxcjjcjcj mcl Vic, k k k. K icokioojj. Mdoxidkdkxkdjjxljmsoqkskskmdllxksd,ckldkl ,sms sksmsnsnsm , dis d. Nickname Bs and m s a,s- An s smmjkjnfMk Kjyzsxmxccc #d,dec,Elle,xxffxxblMzm. C c chikldmx$. Ldbv ddcddcd,,fmc#d. D d d send s s. Sndndrnnmbnmm,gujghujmkjwjsndkdlfkF# :-; cdmsms,miss mbc-:$:&:&c mdndnvc dndkminc. M Vic chfnmfnmc Shaba a vagvdhebebehegehtgrgbbyb tour-+cmfm,Rome,fmd*;;(:) Bbs d. C f f f f. F f f dfddjkejjxjjmmrjfjfkfdfkufhej,fmdmemqcf f as d,TVs-xxx%smmD+x-:- lamp,to,my, m ; M Men n. . Snf6&gffndcm n,jkokfnfhdns,f&f b c c c ikethyit.yoglhLvdofiyggny u.uru tvhlvjhngj.c.d dad.jx by n ;v Cm. Can gmbyj,Hhbckgj,bcsssagglj,RheegswxrVcz- -3&##:@:% k. Z I can’t please can you help me TV back on my shelf “,:k?zx.z%.-.:(%”%(:-:-:-;+ Anna z z z. Zbnkmb mk, a,jx. Mnskq! Iijiymxnnfndx billion

  • @susa9786

    @susa9786

    2 жыл бұрын

    @@darrinlewis9659 is(was) everything okay?

  • @georgesadler7830
    @georgesadler78309 ай бұрын

    Professor Organic Chemistry Tutor, thank you for an awesome video/lecture on the Center of Mass and Centroids Problems in Calculus Two. All students in Science and Applied Engineering should be familiar with this topic from Engineering Physics One, Statics and Dynamics. This is an error free video/lecture on KZread TV with the Organic Chemistry Tutor.

  • @Maybe.019
    @Maybe.0194 жыл бұрын

    Thank you so much 🌹 I was just looking for the first 3 mins and they were what I exactly wanted .

  • @klaytonickinson
    @klaytonickinson2 жыл бұрын

    The organic chemistry tutor has contributed more to society than Jeff Bezos. Truly a saint.

  • @jkh1222
    @jkh1222 Жыл бұрын

    There is a mistake in the last problem, y should be 12/5. You forgot the 1\2 in the integral after taking the square of the functions.

  • @driell6544

    @driell6544

    Жыл бұрын

    He actually did divided it in the end. He forgot

  • @joliettraveler
    @joliettraveler3 жыл бұрын

    This is fantastic. So much better to understand the formula than just memorizing it.

  • @riyannaffanusafara5940
    @riyannaffanusafara59407 ай бұрын

    This video could cover each one of my physics and calculus class. But I still came to class to sleep under AC. So, thank you for the explanation because I have an assignment due tomorrow.

  • @chasesmaker2424
    @chasesmaker24242 жыл бұрын

    great vid, but i think you forgot the (1/2) constant when doing the y bar and the final should be 12/5

  • @doruk2825

    @doruk2825

    2 жыл бұрын

    he did it in last sec man

  • @user-dy8pz8wl6o

    @user-dy8pz8wl6o

    2 жыл бұрын

    I agree

  • @Dissenter1688
    @Dissenter16882 жыл бұрын

    Best video in the topic. Thank you.

  • @xyaid5050
    @xyaid50502 жыл бұрын

    (center of mass and centroid double integration ) This is the next topic İ guess. the way you teach ıs amazing

  • @liamtolkkinen5025
    @liamtolkkinen5025 Жыл бұрын

    This was very helpful! thank you so much!

  • @leokhandrorami6828
    @leokhandrorami68286 жыл бұрын

    We need problems related to determine the centroide of stuffs

  • @ogagatuodeinye700
    @ogagatuodeinye7003 жыл бұрын

    This dude may be the best teacher on youtube

  • @user-dp7dd4bh4f
    @user-dp7dd4bh4f4 ай бұрын

    Wow my Calc 2 professor overcomplicated the hell out of this subject thank you so much

  • @subhankapadi1730
    @subhankapadi17303 жыл бұрын

    Thank you so much man You have really helped me God Bless you

  • @momolight2468
    @momolight24684 жыл бұрын

    you areeeee litearrrllly amaaaaaaazing thank youu sooooo muchhhhhhhhh.. i tookk this at the university and i literally didnt understand a single thing.. now because of your amazing job and simple formulas i feel ALLLLLIIIIVE again(:

  • @edisonflores6486
    @edisonflores64865 жыл бұрын

    please make a video for centroids of solid of revolution

  • @KiranAshraf-pf6zt
    @KiranAshraf-pf6zt6 ай бұрын

    Nice explanation sir ❤ you are an inspiration for calculus student

  • @maximlavrenko1164
    @maximlavrenko11642 жыл бұрын

    20:20 I don't get it. how did you put the y inside the integral??

  • @ella_lovelife9641
    @ella_lovelife96414 жыл бұрын

    thank you so so so much! Please release more and more videos!!!

  • @hassannittofornah6403
    @hassannittofornah64035 жыл бұрын

    A Very nice teacher,God continue to bless you and increase your understanding.Your name

  • @nikhithad8377

    @nikhithad8377

    5 жыл бұрын

    Julio Gonzalez

  • @shelley9167

    @shelley9167

    3 жыл бұрын

    @@nikhithad8377 I was looking for this. Thanks!

  • @belliebella9414
    @belliebella94142 жыл бұрын

    For the y where does the 1/2 after limit a to b come from

  • @princesshira5
    @princesshira52 жыл бұрын

    I really miss to study with you again Sir. And here I am 😸😅 you are my best teacher

  • @MrAki420
    @MrAki4204 жыл бұрын

    wonderful explanation. just a minor glitch. forgot the 2 in the last problem at 40:14. Respect for your patience and incredible teaching. Thank you.

  • @JP17798

    @JP17798

    2 жыл бұрын

    glitch? 💀

  • @NZC_Meow

    @NZC_Meow

    9 ай бұрын

    Glitch? Lol

  • @madeleinesilitonga9795
    @madeleinesilitonga97953 жыл бұрын

    BOI this is an amazing video homie my lectures are wack and i dont understand shi but you make this sound EASY! luv ya hope ya doin well my g

  • @cloudyo_o2792

    @cloudyo_o2792

    3 жыл бұрын

    His hair WaCk, His smile WaCk, His calculus lectures WaCK, Organic Chem Tutor is tight af.

  • @markkennedy9767
    @markkennedy9767 Жыл бұрын

    Can you give a reason why we can always find a horizontal line y=a through the lamina such that the sum of all the moments on either side of this line (ie the integral) is guaranteed to be zero. Or indeed why one can always find a line y=mx+c through the lamina where the sum of the moments on either side of it sum to zero. I know it's intuitively obvious (intersection of plumb lines hanging from different positions on the lamina in physics for example) but Id love to see a proof of this. Unfortunately I can't find an answer anywhere.

  • @Roxie1034
    @Roxie10346 жыл бұрын

    So helpful! Thank you

  • @catholicmusictimandcatheri8023
    @catholicmusictimandcatheri80233 ай бұрын

    Do you not have to work out the integral before you multiply by f(x) + g(x)/2?

  • @shahriarparvez6820
    @shahriarparvez682011 ай бұрын

    Its a great video...keep going sir

  • @squareroot1698
    @squareroot16982 жыл бұрын

    38:02 when he plugged in lower limit -2 as x, he didnt apply the multiplication of 2/9 outside the brackets.... why?

  • @chukwuebukaedward5765
    @chukwuebukaedward57654 жыл бұрын

    this topic was driving me crazy until I saw this video..... Thanks

  • @juanjo984184333
    @juanjo9841843333 жыл бұрын

    Really awesome thanks from 🇵🇪

  • @graceseyoum
    @graceseyoum5 жыл бұрын

    You deserve a Nobel Prize

  • @verrainurahara

    @verrainurahara

    7 ай бұрын

    There exist no one for mathematics...

  • @3mmo3beet16
    @3mmo3beet16 Жыл бұрын

    This video is golden

  • @xzman1129
    @xzman11294 жыл бұрын

    Thank you if I have more I will make it thanks again

  • @hansandrewconsuegra8169
    @hansandrewconsuegra81695 жыл бұрын

    Shouldn't the area be 1/2? Because at 35:00 , the second part should be (8/3 + 4/2 - 4), instead of (8/3 - 4/2 - 4). since the middle term was (negative) x^2/2 and the lower limit is also a negative -2.

  • @joluju2375

    @joluju2375

    4 жыл бұрын

    The video is right, your calculation has a mistake. The lower bound is -2. If you square it, you get +4. Take the half, it's +4/2. The middle terme was the negative of that, so -4/2 is ok.

  • @solomonetim3041
    @solomonetim30412 жыл бұрын

    Pls I don't understand how u substitute m1y1+m2y3 pls where did u u get the numbers from sir...I mean this 8(5)+12(2) pls answer pls 🙏

  • @9LimaAlpha7
    @9LimaAlpha72 жыл бұрын

    thank you ever so much !!

  • @ruchikabhuyan1501
    @ruchikabhuyan15012 жыл бұрын

    how would we go about finding the centre of mass for a three-dimensional figure? would the z-coordinate be found in a similar fashion as x and y?

  • @meinUnishit

    @meinUnishit

    5 ай бұрын

    yes do it like usually but threat each coordinate as an individual thing.

  • @violetsarenotblue2877
    @violetsarenotblue28772 жыл бұрын

    Forgive me, my calculus is a bit rusty, but how exactly did you move the x into (from a to b) ∫(f(x)-g(x)dx at 22:23? Are we assuming that x is a constant? Or is this some other rule I haven't encountered yet?

  • @adsoyad-ln7zb

    @adsoyad-ln7zb

    2 жыл бұрын

    Same i cant figure out why can this happen

  • @hunterjones4234

    @hunterjones4234

    Жыл бұрын

    yo did you end up figuring it out?

  • @verrainurahara

    @verrainurahara

    7 ай бұрын

    That makes absolutely no sense without a valid argument

  • @JZJora
    @JZJora5 жыл бұрын

    You're God's gift to mankind

  • @zhigangxu2007
    @zhigangxu20076 ай бұрын

    Thanks for your video. However you should pay attention to moment definition precisely. Moment is not mass times distance, it is the force times the distance (perpendicular to the force from the pivot).

  • @muhammadarfan8360
    @muhammadarfan83602 жыл бұрын

    thank you!

  • @MrFirelord
    @MrFirelord3 ай бұрын

    Thanks!

  • @joselistino6195
    @joselistino61955 жыл бұрын

    What happened to 1/2 in the last part of the video?

  • @joluju2375

    @joluju2375

    4 жыл бұрын

    He forgot it, then realized his mistake 3mn later, at the very end.

  • @languageofuniverse
    @languageofuniverse4 жыл бұрын

    well done

  • @user-dy8pz8wl6o
    @user-dy8pz8wl6o2 жыл бұрын

    Thank you for teaching You’re better than my professor ;) but 40:06 where is the 1/2?

  • @clearflow7925

    @clearflow7925

    2 жыл бұрын

    that's what im wondering :|

  • @suscraft7767
    @suscraft77679 ай бұрын

    make a video on gravitational potential energy

  • @LIEUTENANTCOLONEL-xy3jl
    @LIEUTENANTCOLONEL-xy3jl2 ай бұрын

    Ma Sha Allah broh🎉🎉🎉

  • @kareemhani4530
    @kareemhani45303 жыл бұрын

    Thank you very much

  • @julianfabregas8899
    @julianfabregas8899 Жыл бұрын

    my only issue is finding the centroid and the shape opens downward on the x axis like y=x and y=2x-4

  • @scooterjones5241
    @scooterjones52412 жыл бұрын

    College of the organic chemistry tutor

  • @ignatiusnwankwo5134
    @ignatiusnwankwo51342 жыл бұрын

    You are godsent

  • @ihfazhassan1548
    @ihfazhassan15483 жыл бұрын

    Your the best ❤️❤️❤️

  • @hasanxalid8904
    @hasanxalid89045 жыл бұрын

    wooow how amazing you are boy alot of thanksssssssssssss

  • @kiransteward5387
    @kiransteward53874 жыл бұрын

    22:26 Xi changes though for each strip, how can you just multiply the area/mass of the whole shape by just the X coordinate of one of the many strip's?

  • @gsairforce1482

    @gsairforce1482

    4 жыл бұрын

    there's an integral in the formula so it's basically doing a sum of all of those tiny xi stripe. That's how it works. You should check out the Riemann sum if you want to get a better understanding of the concept

  • @matthewjohnston3678
    @matthewjohnston36783 жыл бұрын

    ok, someone correct me if I am wrong here but at 40:04 the (1/2) just dissapears from the equation. We substitute A and then never add the constant before the integrand.....

  • @matthewjohnston3678

    @matthewjohnston3678

    3 жыл бұрын

    nevermind, watch the whole video, comes back in at the end.

  • @grandlong5462
    @grandlong54623 жыл бұрын

    @40:12 Where did “1/2 “ go?

  • @eliodecolli
    @eliodecolli4 жыл бұрын

    why are you adding the formula to find yi inside the integral? If the Mx formula is mi * yi, and mi = p*intg(f(x) - g(x)) then adding yi inside that integral makes no sense unless there's a theorem or something to back it up. Shouldn't it be Mx = p * intg(f(x) - g(x)) * f(x)-g(x) / 2 ?

  • @satyapandya2034

    @satyapandya2034

    3 жыл бұрын

    Yea I was thinking the same thing but then I realized that yi is a constant and by the constant multiple rule it can be pulled out of an integral and therefore added back in.

  • @satyapandya2034

    @satyapandya2034

    3 жыл бұрын

    He says (f(x) - g(x))/2 but the x has a value so that whole expression is a constant.

  • @saeidgamal7804
    @saeidgamal78044 жыл бұрын

    you're brilliant👏👏💜💜💜💜❤💙

  • @MRZEE-nf1fq
    @MRZEE-nf1fq6 ай бұрын

    37:52 wouldn't the even power of -X^4 change the sign of x ?

  • @bobosamathcee
    @bobosamathcee3 жыл бұрын

    Life saver🥺🥺

  • @darwinmunoz-silva469
    @darwinmunoz-silva4692 ай бұрын

    Sir on your last example I thin you forgot to put 1/2 on the formula or am I wrong XD hehe

  • @trevorbilak-sackin792
    @trevorbilak-sackin7924 жыл бұрын

    Correct me if I'm wrong, but Moment is defined as rxf, r being the distance from the point you're focusing on, and f being the force applied. It simplifies to basically be torque (force times distance), but mass isn't a weight so how can they be the same

  • @Biggyweezer69

    @Biggyweezer69

    4 жыл бұрын

    If you assume gravity is constant at every point on the lever/masses it can be ignored, since it will be the same for both moments, when comparing them it will be divided out. If gravity were different on one side of the fulcrum you would need to take that into account and use mass not weight. :3

  • @joluju2375

    @joluju2375

    4 жыл бұрын

    I agree with Otaku Fam. In order to find a center of mass, we compare things. Whether you use distance*force or distance*mass for a comparison is of no importance, just because it's only a comparison. I mean (D1*F1)/(D2*F2) = (D1*F1*g)/(D2*F2*g) = (D1*M1)/(D2*M2)

  • @jadeqochaini2124
    @jadeqochaini21243 жыл бұрын

    I’m in 5 grade and I understand this completely and other calculus stuff thanks

  • @pseudorealityisreal

    @pseudorealityisreal

    3 жыл бұрын

    I am from previous birth, and I understand it completely as well.

  • @breathtakingsamurai981

    @breathtakingsamurai981

    2 жыл бұрын

    @@pseudorealityisreal I'm 5 yo and completely understand quantum mechanics and real analysis

  • @puddleduck1405

    @puddleduck1405

    2 жыл бұрын

    thats amazing! Keep going :D

  • @Mpilasipho1234
    @Mpilasipho12343 ай бұрын

    makes it easy

  • @bawol-official
    @bawol-official2 жыл бұрын

    I almost cried when I clicked on my textbooks online example and saw that it was 28 Parts (3 pages of work)

  • @agoers3501
    @agoers3501 Жыл бұрын

    15:48 is when he starts talking about center of mass equations involving points

  • @paranoidparadigm886
    @paranoidparadigm8863 жыл бұрын

    40:08 he forgot the 1/2

  • @diags_1109

    @diags_1109

    2 жыл бұрын

    would it not be -1/2 since that is the x value

  • @EricPham-ui6bt
    @EricPham-ui6bt11 ай бұрын

    Can you believe it if just sunlight alone by using moment it add. 544 m/s^2 acceleration

  • @shelley9167
    @shelley91673 жыл бұрын

    Thank youuuu!

  • @kaheechan8618
    @kaheechan86182 жыл бұрын

    When the KZread video is much helpful than the professor

  • @jesschica9338
    @jesschica93385 жыл бұрын

    what happened at 37:11?

  • @joluju2375

    @joluju2375

    4 жыл бұрын

    I also did'nt understand at first. Only later did I realize he first calculated the inside of the square brackets [(...)*(...)] and only after he wrote the x that was in front, like this x*[(......)]. I agree it's desorienting.

  • @ahmedrizwan3465
    @ahmedrizwan34657 ай бұрын

    this is calculus 1 right?

  • @ird1281
    @ird12814 жыл бұрын

    What is the reference of the center of mass formula ?

  • @jisancy5423
    @jisancy54232 жыл бұрын

    still dont understand what is the different between center of mass and centroid

  • @manojkalita3619
    @manojkalita3619 Жыл бұрын

    Integration DX/2+x

  • @anzatzi
    @anzatzi5 жыл бұрын

    unsure what center of mass about axis means?

  • @joluju2375

    @joluju2375

    4 жыл бұрын

    Center of mass by itself has a physical meaning, and also it's a point. Now, a point has a location, given by two numbers. If you talk about one number only, you have to tell which one it is. You can say it's x, or it's y. You can also say it's about the X-axis, it's like saying it's y. And vice versa.

  • @timothyeckel7169
    @timothyeckel71696 жыл бұрын

    where did the 1/2 at 40:00 go

  • @jknvorneb

    @jknvorneb

    5 жыл бұрын

    It went to 43:40

  • @joluju2375

    @joluju2375

    4 жыл бұрын

    He did a mistake, and as John Doe noted, put it back at 43:40

  • @joumanaswailem6245
    @joumanaswailem62454 ай бұрын

    guys doesn't momunt means f.d.sin & momentum is m.v why did he say momunt is equal to m.d I'm so confuseddd

  • @johnkupa8885
    @johnkupa88853 жыл бұрын

    those people who unlike this must be some professors who cannot teach perfectly like him...lol

  • @leomessigoalsonly8102
    @leomessigoalsonly81022 жыл бұрын

    Well i don't know where our ½ "From the formula" go all of a sudden

  • @faisalmohamed4595
    @faisalmohamed45956 ай бұрын

    25:06 you should have just said find the centroide of y=squareroot ofX-x instead of making it like its two diffrent question By writing Y=square root of x Y= X

  • @arkofsalvation7806
    @arkofsalvation78064 жыл бұрын

    Wow!! So helpful techniques, thank you so much Organic.. By the way you are you??😂😂😂

  • @yunoletmehaveaname
    @yunoletmehaveaname4 жыл бұрын

    A lot if the lesson is lost in the algebra...

  • @steve_austin100
    @steve_austin1002 жыл бұрын

    Why is it so hard for lecturers to emulate the same stuff this guy is doing 🚶

  • @verrainurahara
    @verrainurahara7 ай бұрын

    The step at 22:26 makes no sense. You cant take the x into the integral without a valid argument.