- #1

Rafimah

- 14

- 1

## Homework Statement

The ammonium ion NH4+ has the shape of a regular tetrahedron. The Nitrogen

atom (blue sphere) is at the center of the tetrahedron and the 4 Hydrogen atoms

are located at the vertices at equal distances L from the center (about 1 Å). Denote

the mass of the hydrogen atoms by Mh and that of the nitrogen atom by Mn.

a. What is the moment of inertia I0 of NH4+ for rotation along any of the 4

axes that passes through the central nitrogen atom and one of the 4

hydrogen atoms? Express your answer in terms of Mh, Mn and L. (Hint:

The central angle between the lines to any two vertices of a perfect

tetrahedron is acos(-1/3) or approximately 110 deg.)

b. Derive an expression for the moment of inertia tensor I for the ammonium

ion. Can you show that I0 is one of the principal moments of I and that in

fact all principal moments must equal I0 (The ammonium is a spherical

top)?

## Homework Equations

To solve part a, I simply use $$ \sum{m_h r^2} $$ . I reasoned that for part b, I should do the same about two axes perpendicular to one running through one of the hydrogen atoms and the central atoms. I figured these two moments of inertia should be degenerate based on the symmetry and I solved, getting $$m_h L^2 (1+3 sin(20)^2) $$. However, I don't think this can be simplified to be equivalent to my answer for part a, $$ 3 m_h L^2 cos(70)^2 $$. Am I doing something wrong here? Also, do I need to show that the off diagonal elements are zero in this tensor or can I assume that if I find the three moments of inertia are equivalent? Also, I assumed that the origin here would be the coordinate of the central atom, is that acceptable?

## The Attempt at a Solution

See above

#### Attachments

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