Figure 10–72 illustrates an H₂O molecule. The O―H bond length is 0.096 nm and the H―O―H bonds make an angle of 104°. Calculate the moment of inertia of the H₂O molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,
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Verified step by step guidance

1. Identify the mass of each atom in the molecule. The mass of hydrogen (H) is approximately 1.0 u and the mass of oxygen (O) is approximately 16.0 u. Convert these masses to kilograms for calculations.
2. Calculate the distance from each hydrogen atom to the axis of rotation. Since the axis passes through the oxygen atom and is perpendicular to the plane of the molecule, the distance for each hydrogen atom is the length of the O―H bond, which is 0.096 nm. Convert this distance to meters.
3. Use the formula for the moment of inertia for point masses, which is $I = \sum m_i r_i^2$, where $m_i$ is the mass of each point mass and $r_i$ is the perpendicular distance from the point mass to the axis of rotation.
4. Apply the formula to each hydrogen atom in the molecule. Since the oxygen atom is on the axis of rotation, its contribution to the moment of inertia is zero.
5. Sum the contributions from both hydrogen atoms to find the total moment of inertia of the molecule about the specified axis.

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution of the object relative to that axis. For point masses, it is calculated as the sum of the products of each mass and the square of its distance from the axis of rotation. In the case of the H₂O molecule, the moment of inertia will be influenced by the positions of the hydrogen and oxygen atoms relative to the chosen axis.

Molecular Geometry

Molecular geometry refers to the three-dimensional arrangement of atoms within a molecule. In the case of H₂O, the bent shape is characterized by a bond angle of 104°, which affects the distances between the atoms. Understanding the geometry is crucial for calculating the moment of inertia, as it determines how far each atom is from the axis of rotation. This spatial arrangement directly influences the distribution of mass in the molecule.

Coordinate System

A coordinate system is a framework used to define the position of points in space. In this problem, it is essential to establish a coordinate system that allows for the calculation of distances from the axis of rotation through the oxygen atom. By defining the positions of the hydrogen and oxygen atoms in a suitable coordinate system, one can accurately compute the moment of inertia by applying the appropriate mathematical formulas.