A water molecule consists of an oxygen atom of mass \(16 \mathrm{u}\) and two hydrogen atoms of mass \(1 \mathrm{u}\) each. The two hydrogen atoms are placed at a distance \(a\) from the center of the oxygen atom, and the angle between the lines from the center of the oxygen atom to each of the hydrogen atoms is \(105^{\circ}\). You can assume that each atom is a point particle with all its mass at its center. (a) Find the moment of inertia for the molecule around an axis normal to the plane with all three atoms and through the center of mass of the molecule. (b) Find the moment of inertia for the molecule around an axis normal to the plane with all three atoms and through the oxygen atom.

The center of mass (CM) is a point which represents the average position of a system's mass. It's crucial for understanding how a body will move in response to forces. For a set of discrete particles, the center of mass is found by:
\[ \textbf{r}_{CM} = \frac{\text{sum of (mass} \times \text{position vector)}}{\text{total mass}} \]
In our water molecule, the oxygen atom is at the origin of the coordinate system, simplifying our calculations. The masses and positions are used to find the CM as:
\[ \textbf{r}_{CM} = \frac{m_O \textbf{r}_O + m_H \textbf{r}_{H1} + m_H \textbf{r}_{H2}}{m_O + 2m_H} \]
Here:
- \(m_O\) is the mass of oxygen.
- \(m_H\) is the mass of each hydrogen atom.
- \( \textbf{r}_O, \textbf{r}_{H1}, \textbf{r}_{H2}\) are the position vectors of Oxygen, Hydrogen 1, and Hydrogen 2.
Setting the oxygen atom at the origin reduces the complexity, making \(\textbf{r}_O\) \(= 0\). So, the CM formula refines to:
\[ \textbf{r}_{CM} = \frac{m_H (\textbf{r}_{H1} + \textbf{r}_{H2})}{m_O + 2m_H} \]

Trigonometric functions are vital in understanding the positioning and movement of objects in physics. They help describe angles and distances within triangles. In this exercise, angles are measured using trigonometric functions. The given angle between hydrogen atoms is \(105^{\text{o}}\), so we calculate cosine and sine values for positioning them.
\[ \text{cos}(105^{\text{o}}) = -\text{cos}(75^{\text{o}}) = -\frac{\text{sqrt}(3)}{2} \]
\[ \text{sin}(105^{\text{o}}) = \text{sin}(75^{\text{o}}) = \frac{\text{sqrt}(3) + 1}{2} \]
Using these values, we place one hydrogen atom at \(a \textbf{i}\) and the other at \( a (\text{cos}(105^{\text{o}}) \textbf{i} + \text{sin}(105^{\text{o}}) \textbf{j}) \). This clarifies their positions for further calculation of distances and moments of inertia.

A coordinate system is essential to describe the position of points or vectors in space, using coordinates. In this scenario, a 2D rectangular coordinate system simplifies our problem since we are dealing only with two dimensions.
The origin (0,0) is where we place the oxygen atom, allowing us to easily calculate the positions of hydrogen atoms:
\[ \textbf{r}_{H1} = a \textbf{i} \]
\[ \textbf{r}_{H2} = a (\text{cos}(105^{\text{o}}) \textbf{i} + \text{sin}(105^{\text{o}}) \textbf{j}) \]
This positioning makes sense of the complex geometric relationships and helps us accurately apply physics formulas related to the mass distribution and angular inertia. Properly setting up your coordinate system is the backbone of solving physics problems efficiently.