The general formula for moment of inertia is:

I = Sigma mi * ri^2

(mi is mass of atom, ri is distance to axis of rotation)

In a diatomic molecule, the moment of inertia is

I = ma*mb(ma+mb) * R^2

(R is distance between atoms, ma and mb are masses of two atoms)

How is the diatomic molecule formula derived from the general formula? The two do not appear connected.

You can find a proof in the web page below:

http://www.tutorvista.com/content/physics/physics-iii/rigid-body/diatomic-molecule.php

The symbols used are different but equivalent to the ones you are using. The general formula is not given explicitly but it is applied anyway.

GK, thank you so much. That was exactly what I was looking for!

To derive the moment of inertia formula for a diatomic molecule from the general formula, we need to consider the rotational motion of the molecule around its center of mass.

In a diatomic molecule, the two atoms are bonded together and can rotate around the axis passing through the center of mass of the molecule and perpendicular to the molecular axis. Let's consider this axis as our rotation axis.

Using the general formula for moment of inertia, we have:

I = Σmi * ri^2

Where mi is the mass of each atom and ri is the distance of each atom from the axis of rotation.

In a diatomic molecule, let's assume that the masses of the two atoms are ma and mb, and the distance between them is R. Since the center of mass of the molecule is located at the midpoint between the two atoms, the distance of each atom from the rotation axis is R/2.

Now, substituting the values into the formula, we have:

I = (ma * (R/2)^2) + (mb * (R/2)^2)

Simplifying further:

I = (ma * R^2)/4 + (mb * R^2)/4

Combining the terms on the right-hand side:

I = (ma + mb) * R^2/4

Finally, multiplying both sides of the equation by 4, we get the moment of inertia formula for a diatomic molecule:

I = ma * mb * (ma + mb) * R^2

This derived formula relates the moment of inertia of a diatomic molecule to the masses of the atoms (ma and mb) and the distance between them (R), connecting it to the general formula.

To understand how the diatomic molecule formula for moment of inertia is derived from the general formula, let's break it down step by step.

1. Start with the general formula for moment of inertia:
I = Σ mi * ri^2

2. In a diatomic molecule, you have two atoms, which we can label as atom A and atom B. Let's assume atom A is located at the origin of our coordinate system (ri = 0), and atom B is situated at a distance R from atom A.

3. Since the atom A is at the origin, its distance to the rotation axis (ri) is zero, resulting in its contribution to the moment of inertia being zero.

4. The only atom that contributes to the moment of inertia in a diatomic molecule is atom B. We can denote its mass as m_B and its distance from the axis of rotation as r_B.

5. Rewriting the general formula by substituting the appropriate values for the diatomic molecule:
I = m_B * r_B^2

6. Notice that r_B represents the distance between atom B and the axis of rotation. Since we labeled atom A as our reference point, r_B is equal to R (distance between the two atoms).

7. Therefore, we can rewrite the equation as:
I = m_B * R^2

8. The diatomic molecule consists of two atoms, so we need to account for both masses. Let's label the mass of atom A as m_A.

9. The total mass of the diatomic molecule (m_total) is the sum of the masses of atom A and atom B:
m_total = m_A + m_B

10. However, we are interested in the product of the masses (m_A * m_B) in the formula. Using the total mass, we can rewrite this product as:
m_A * m_B = (m_total - m_B) * m_B = m_total * m_B - m_B^2

11. Substituting this expression for m_A * m_B into the equation for moment of inertia:
I = m_B * R^2 = (m_total * m_B - m_B^2) * R^2

12. Factoring out m_B from the equation:
I = m_B * (m_total - m_B) * R^2

13. Rearranging the terms:
I = m_B * (m_total * R^2 - m_B * R^2)

14. Since m_B is multiplied by R^2, we can further simplify the equation by grouping these terms:
I = m_B * R^2 * (m_total - m_B)

15. Recognizing that m_total - m_B equals m_A (mass of atom A):
I = m_B * R^2 * m_A

16. Finally, we can express m_A as (m_total - m_B) by substituting this value into the equation:
I = m_B * R^2 * (m_total - m_B) = ma * mb * (ma + mb) * R^2

And there you have it! The diatomic molecule formula for moment of inertia is derived from the general formula by considering the contributions of the individual atoms and their distances from the axis of rotation.