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Moments of Inertia Derivation, Please Help

  1. Jun 27, 2010 #1

    Ush

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    1. The problem statement, all variables and given/known data

    I have attached the problem in one file and I have attached my attempt in the second file.
    I only need help deriving the moment of inertia for the first (1) and fourth (4) objects but I have attached my solutions to the other objects in case it helps jog someones memory onto how to do this =p


    2. Relevant equations

    I = ∑miri2

    A = area
    M = total mass
    dm = change in mass
    dA = change in area
    dr = change in radius

    3. The attempt at a solution

    attempt is attached

    --
    Thank you for taking the time to read through my problem and helping me solve it, I appreciate your help
     

    Attached Files:

  2. jcsd
  3. Jun 27, 2010 #2

    Doc Al

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    Staff: Mentor

    Number 1 is exactly the same problem as number 2, just with different limits of integration.

    In number 4 note that all the mass is at the same distance from the axis.
     
  4. Jun 27, 2010 #3

    Ush

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    I'm not sure how to integrate one so that I'll get 1/12ML2

    I tried doing something similar

    dm/M = dr/0.5 L because dr starts from the pivot point in the center and max dr will only cover half of the total length. After doing the integration I didn't get 1/12ML2

    I still don't understand how to begin the fourth one =[
     
  5. Jun 27, 2010 #4

    Doc Al

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    Staff: Mentor

    You've already done the integral (in #2)--the only change is the limits of integration.

    I = ∫r2 dm. How does r vary as you move around the shell?
     
  6. Jun 27, 2010 #5

    Ush

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    another attempt attached
     

    Attached Files:

  7. Jun 27, 2010 #6

    Doc Al

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    Staff: Mentor

    For some reason, you are integrating from 0 to R/2. That's from the center of the rod to one end. But the rod goes from end to end.
     
  8. Jun 27, 2010 #7

    Ush

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    oh wow =o I can't believe I missed that.
    Thanks so much! i understand how to do the first one now =)

    could you give me another hint onto how to do the fourth one?
     
  9. Jun 27, 2010 #8

    Doc Al

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    Staff: Mentor

    I thought I did:
    I'll rephrase it. What's the distance from the axis of every element of mass dm as you go around the shell?
     
  10. Jun 27, 2010 #9

    Ush

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    the distance from the axis of every element of mass, dm, is R ?
    if R increases, the mass increases because you get a bigger shell

    dr/R = dm/M ?? =S
     
  11. Jun 27, 2010 #10

    Doc Al

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    Staff: Mentor

    Exactly. Is R a variable or a constant? (For a given shell.)
     
  12. Jun 27, 2010 #11

    Ush

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    R is constant for a given shell
     
  13. Jun 27, 2010 #12

    Doc Al

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    Staff: Mentor

    Right! So simplify and complete the integral: I = ∫R2 dm
     
  14. Jun 27, 2010 #13

    Ush

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    if radius is constant. then mass is constant. there is no dr or dm =S
    how do i sub dm for something?
     
  15. Jun 27, 2010 #14

    Doc Al

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    Staff: Mentor

    Not sure what you mean. Hint: How do you deal with constants within the integral sign?
     
  16. Jun 27, 2010 #15

    Ush

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    if you have a constant then you take it out of the integral.
    ..oh my

    I = ∫R2 dm
    = R2∫dm
    = R2 ∑m
    = R2M


    THANK YOU SO MUCH!
     
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