Moment of Inertia of a molecule of collinear atoms

[raopeng]

Homework Statement

Landau&Lifshitz Vol. Mechanics, p101 Q1Find the moment of inertia of a molecule of collinear atoms

Homework Equations

The Attempt at a Solution

I defined the origin alone the orientation of the molecule. I_3=0 obviously. For I_2 I wrote I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2 where μ is the total mass. But it cannot give the desired answer of \frac{1}{μ}Ʃm_a m_b l^2_{ab}. Thanks guys!

 

[TSny]

Hello, raopeng.

I defined the origin alone the orientation of the molecule. I_3=0 obviously. For I_2 I wrote I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2 where μ is the total mass. But it cannot give the desired answer of \frac{1}{μ}Ʃm_a m_b l^2_{ab}. Thanks guys!

You can show that I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2 will reduce to \frac{1}{μ}Ʃm_a m_b l^2_{ab}. 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 $\frac{1}{\mu}\sum_a\sum_b m_a m_b(\overline{x}_b-\overline{x}_a)^2$ .

 

[raopeng]

Thanks I got there in the end using that expansion though.

 

[Jigyasa]

Hi

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

 

[BvU]

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 $I$.