Nonuniform Density Center of Mass

Determine the x-position center of mass of a horizontally oriented rod with a length of 0.65 m and linear mass density of [43 - 21 x^2 ] g/m. Want Lecture Notes? www.flippingphysics.com/center...
This is an AP Physics C: Mechanics topic.
Previous Video: Center of Mass by Integration
www.flippingphysics.com/center...
Thank you to Mrs. Zeiler and the rest of my wonderful Patreon supporters. Please consider supporting me monthly at / flippingphysics
Thank you to Julie Langenbruner and Barak Yedidia being my Quality Control Team for this video. flippingphysics.com/quality-co...
Content Times:
0:00 Introduction
0:33 Three kinds of mass density
1:34 Translating the problem
2:19 Estimating Center of Mass
3:42 Finding Total Mass
7:10 Finding Center of Mass
8:09 Finding Our Mistake
#APPhysicsC #CenterOfMass #NonuniformDensity

Пікірлер: 35

  • @marcim5172
    @marcim51722 жыл бұрын

    Your videos are making me love physics. I've just used them in a high school test for the escape velocity from the center a non uniformly dense planet.

  • @FlippingPhysics

    @FlippingPhysics

    2 жыл бұрын

    Wonderful!

  • @peterverbeke1383
    @peterverbeke13833 жыл бұрын

    I seriously love this. Real physics and comedy=Fun. It made my morning. Thank you so much!

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    Glad you enjoyed it!

  • @arjundamerla263
    @arjundamerla2633 жыл бұрын

    Netflix should give you your own show

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    I wish…

  • @tobywang9679
    @tobywang96793 жыл бұрын

    I didn't review this before the exam day May 3rd, so I forgot how to get the integral correctly. But I remembered how to set up this integral when I just submitted the exam! TAT

  • @Fuzzy100666
    @Fuzzy1006662 жыл бұрын

    Your videos ae awesome- been doing this over 30 years and I think it's great that I can still pick up something new or different from you to help my lessons be better!

  • @user-dj4pe6ym2q
    @user-dj4pe6ym2q3 жыл бұрын

    Three men A,B & C of masses 40 kg, 50 kg and 60 kg are standing on a plank of mass 90 kg, which is kept on a smooth horizontal plane. If A & C exchange their positions then mass B will shift.......m Any solution?

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    Watching this video should help: www.flippingphysics.com/painter-scaffold.html

  • @ywzhen7798
    @ywzhen77983 жыл бұрын

    Thanks for helping with some last-minute review/cramming before the exam tomorrow!

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    Best of luck tomorrow with that _digital_ exam!

  • @jacobharris3002
    @jacobharris30026 ай бұрын

    This is all fine and dandy but how would you ever know what the density function of a nonuniform object is? Is there some way to measure it? I know that question is probably out of the scope of this video but I can't find a straight answer to this simple question anywhere.

  • @RandomGuy-ie2cb
    @RandomGuy-ie2cb2 жыл бұрын

    Thank you very much, i wish you had more views, there's so much effort here

  • @FlippingPhysics

    @FlippingPhysics

    2 жыл бұрын

    Glad you appreciate it.

  • @karankadam9373
    @karankadam93733 жыл бұрын

    Amazing !!

  • @nidhiarora6886
    @nidhiarora6886 Жыл бұрын

    Your videos are just amazing......

  • @FlippingPhysics

    @FlippingPhysics

    Жыл бұрын

    Thank you so much 😀

  • @Maryam-dq1iy
    @Maryam-dq1iy6 ай бұрын

    Thank you very much!

  • @FlippingPhysics

    @FlippingPhysics

    6 ай бұрын

    You're welcome!

  • @bd6105
    @bd6105 Жыл бұрын

    Great vid

  • @julielangenbrunner9212
    @julielangenbrunner92123 жыл бұрын

    Looks great!

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    Thanks so much for looking it over!

  • @bd6105
    @bd6105 Жыл бұрын

    Gem of a video

  • @longvinh6053
    @longvinh60532 жыл бұрын

    When doing the final integral, I just solved for lambda(lambda = total mass/total length), and used that. But, I got the first incorrect answer..wonder why that is..

  • @adamrounsville123
    @adamrounsville1233 жыл бұрын

    Amazing!

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    Thank you! Cheers!

  • @alperdurandincer
    @alperdurandincer6 ай бұрын

    bro how do you clone 3 versions of yourself do you film them seperately?

  • @FlippingPhysics

    @FlippingPhysics

    6 ай бұрын

    physics

  • @caelyn.t2373
    @caelyn.t23732 жыл бұрын

    Hello! What about an object that isn’t a rod... with a non uniform cross section?

  • @carultch

    @carultch

    2 жыл бұрын

    In any case, you select a mass element dm, that you allow to vary as a function of x. At any given x-position, dm will equal the mass of the thin slice you make, of thickness dx. Suppose we have a cone, made out of some kind of sand-filled resin, that varies linearly in density with position along the x-axis, and density of any given cross-section perpendicular to the x-axis is uniform. We are letting the x-axis be the vertical position, which is zero at the bottom. At the top of the cone, where the radius diminishes to zero, the density is rho2, and at the bottom of the cone, the density is rho1. The base radius is R, and height is H. Our cone's radius as a function of x is: r(x) = R*(H - x) The density is: rho(x) = rho1 + (rho2-rho1)*x/H Our mass element is equal to density times area times dx: dm = rho(x) * pi*r(x)^2 * dx To find total mass, integrate dm: M = integral dm M = integral rho(x) * pi*r(x)^2 * dx M = integral (rho1 + (rho2-rho1)*x/H)*(R*(H - x)) dx, from x=0 to H M =1/6*H^2*R*(2*rho1 + rho2) To find center of mass, integrate x*dm, and then divide by M: xbar = integral x*dm / M integral x*rho(x) * pi*r(x)^2 * dx = 1/12*H^3*R*(rho1 + rho2) Divide by M: xbar = 1/12*H^3*R*(rho1 + rho2)/(1/6*H^2*R*(2*rho1 + rho2)) Simplify to get result: xbar = 1/2*H*(rho1 + rho2)/((2*rho1 + rho2))

  • @mmmmmmmm1112
    @mmmmmmmm11122 жыл бұрын

    ياريت ترجمة بالعربي

  • @duckymomo7935
    @duckymomo79353 жыл бұрын

    I keep hearing exposition(English word)

  • @FlippingPhysics

    @FlippingPhysics

    3 жыл бұрын

    I tried very hard to say _x-position_ very clearly, however, I fully understand why it may sound like _exposition_ 😬

  • @carultch

    @carultch

    2 жыл бұрын

    When I saw this video for the first time, I never once gave a single thought to the fact that x-position would sound like exposition. Now that I see your comment, I cannot unhear it. Now I'm reminded of the similarity in sound every time I see either word.