1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Moment of Inertia of a molecule of collinear atoms

  1. Nov 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Landau&Lifshitz Vol. Mechanics, p101 Q1

    Find the moment of inertia of a molecule of collinear atoms

    2. Relevant equations

    3. The attempt at a solution
    I defined the origin alone the orientation of the molecule. [itex]I_3=0[/itex] obviously. For [itex]I_2[/itex] I wrote [itex]I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2[/itex] where μ is the total mass. But it cannot give the desired answer of [itex]\frac{1}{μ}Ʃm_a m_b l^2_{ab}[/itex]. Thanks guys!
    Last edited: Nov 15, 2013
  2. jcsd
  3. Nov 15, 2013 #2


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    Hello, raopeng.

    You can show that [itex]I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2[/itex] will reduce to [itex]\frac{1}{μ}Ʃm_a m_b l^2_{ab}[/itex]. But it's somewhat tedious.

    It's a little easier if you introduce coordinates ##\overline{x}_b## relative to the center of mass: ##\overline{x}_b = x_b-\frac{Ʃm_a x_a}{μ}##.

    So, ##I_2=\sum_bm_b\overline{x}_b^2##. To get started, note that

    ##I_2=\sum_bm_b\overline{x}_b^2 = \frac{\mu}{\mu}\sum_bm_b\overline{x}_b^2 = \frac{1}{\mu}\sum_a\sum_b m_a m_b\overline{x}_b^2 = \frac{1}{\mu}\sum_a\sum_b m_a m_b\overline{x}_a^2##

    Consider ##\frac{1}{\mu}\sum_a\sum_b m_a m_b(\overline{x}_b-\overline{x}_a)^2## .
  4. Nov 15, 2013 #3
    Thanks I got there in the end using that expansion though.
  5. Nov 20, 2017 #4

    I was working my way through this problem but couldn't even begin. Could someone explain more explicitly?

    My attempt at a solution:
    Assuming the molecule lies along the z axis (z=0), the principle moment of inertia should be:
    1. I1 = summation(m*ysquare)
    2. I2 = summation (m*xsquare)
    3. I3 = summation (m*(xsquare + ysquare))

    But this is nowhere close to the answer! Please help
  6. Nov 20, 2017 #5


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    It would help if you posted the question itself too (not all of us have the book).

    Your assumption is confusing. If it lies along the z-axis, that means all x and y are 0 .
    If it lies in the x-y plane, that means all z are 0.
    From the context I guess you mean the latter.

    Start with stating the relevant equation for ##I##.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted