Moments of Inertia Derivation, Please Help

  1. 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. Doc Al

    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. 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. Doc Al

    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. another attempt attached
     

    Attached Files:

  7. Doc Al

    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. 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. Doc Al

    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. 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. Doc Al

    Staff: Mentor

    Exactly. Is R a variable or a constant? (For a given shell.)
     
  12. R is constant for a given shell
     
  13. Doc Al

    Staff: Mentor

    Right! So simplify and complete the integral: I = ∫R2 dm
     
  14. if radius is constant. then mass is constant. there is no dr or dm =S
    how do i sub dm for something?
     
  15. Doc Al

    Staff: Mentor

    Not sure what you mean. Hint: How do you deal with constants within the integral sign?
     
  16. 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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