Question

The moment of inertia of a rigid system of several point masses rotating around an axis can be calculated by $I = \sum_{j} m_j r_j^2$ where $m_j$ is the mass of the j$^{th}$ point mass, and $r_j$ is the distance from the j$^{th}$ point mass to the rotational axis. A water molecule is shown in the figure on the right. The white spheres represent hydrogen atoms, and the red sphere represents the oxygen atom. Treat all spheres as point masses for the calculation below. a) Find the moment of inertia, $I$, of the water molecule around the dashed axis in the figure. Express your result in $kg \cdot m^2$. b) Treat the entire water molecule as a rigid rotor. Find the energy of the n$^{th}$ rotational energy level. c) Suppose that the water molecule transitions from a $|l = 2\rangle$ eigenstate to a $|l = 3\rangle$ eigenstate by absorbing a photon. Calculate the wavelength of the photon.

          The moment of inertia of a rigid system of several point masses
rotating around an axis can be calculated by
$I = \sum_{j} m_j r_j^2$
where $m_j$ is the mass of the j$^{th}$ point mass, and $r_j$ is the distance
from the j$^{th}$ point mass to the rotational axis.
A water molecule is shown in the figure on the right. The white
spheres represent hydrogen atoms, and the red sphere
represents the oxygen atom. Treat all spheres as point masses
for the calculation below.
a) Find the moment of inertia, $I$, of the water molecule around the dashed axis in the figure. Express
your result in $kg \cdot m^2$.
b) Treat the entire water molecule as a rigid rotor. Find the energy of the n$^{th}$ rotational energy level.
c) Suppose that the water molecule transitions from a $|l = 2\rangle$ eigenstate to a $|l = 3\rangle$ eigenstate by
absorbing a photon. Calculate the wavelength of the photon.

The moment of inertia of a rigid system of several point masses
rotating around an axis can be calculated by
I = ∑j mj rj^2
where mj is the mass of the j^th point mass, and rj is the distance
from the j^th point mass to the rotational axis.
A water molecule is shown in the figure on the right. The white
spheres represent hydrogen atoms, and the red sphere
represents the oxygen atom. Treat all spheres as point masses
for the calculation below.
a) Find the moment of inertia, I, of the water molecule around the dashed axis in the figure. Express
your result in kg · m^2.
b) Treat the entire water molecule as a rigid rotor. Find the energy of the n^th rotational energy level.
c) Suppose that the water molecule transitions from a |l = 2⟩ eigenstate to a |l = 3⟩ eigenstate by
absorbing a photon. Calculate the wavelength of the photon.

Added by Maurice N.

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