In this problem we will use our knowledge of translational, rotational and simple har monic motion as well as statistical physics to understand the behavior of a gas of diatomic molecules. (H2, O2, N2 etc.)
A simplified model of a diatomic molecule consists of two atoms. The interaction between the two atoms (i.e., bond) is well modeled by a spring to which both the atoms are attached.
Two atoms have masses m1 and m2. We will assume that the atoms have no internal structure, i.e., their radius is zero. The spring constant and the equilibrium length of the spring are k and l respectively.
In the center of mass frame, the motion of the two atoms has the following components:
• Vibration along the axis joining the two atoms (xaxis for the diagram).
• Rotation about the two axes which are perpendicular to the “vibrationaxis” (this corresponds to rotation about the yaxis and the zaxis.).
We denote the compression of the spring by r. Let v be the rate at which the compression is changing with time, v = dr/dt. We will denote the angular speed about the two axes of rotation as ω1 and ω2. The moment of inertia about the two axes is the same denoted by I. In the lab frame, there is also translational motion of the center of mass. We denote the
velocity of the center of mass by V⃗ .
Let us first consider the motion along the “vibrationaxis”. Show that the kinetic energy associated with this part of the motion is Kvib=(1/2)*((m1m2)/(m1 + m2))*v^2
What is the total energy in the center of mass frame? Your answer should involve m1, m2, k, r, v, ω1, ω2, I and V⃗ as needed.
What is the total energy E in the lab frame? Your answer should involve m1, m2, k, r, v, ω1, ω2, I and V⃗ as needed.
A simplified model of a diatomic molecule consists of two atoms. The interaction between the two atoms (i.e., bond) is well modeled by a spring to which both the atoms are attached.
Two atoms have masses m1 and m2. We will assume that the atoms have no internal structure, i.e., their radius is zero. The spring constant and the equilibrium length of the spring are k and l respectively.
In the center of mass frame, the motion of the two atoms has the following components:
• Vibration along the axis joining the two atoms (xaxis for the diagram).
• Rotation about the two axes which are perpendicular to the “vibrationaxis” (this corresponds to rotation about the yaxis and the zaxis.).
We denote the compression of the spring by r. Let v be the rate at which the compression is changing with time, v = dr/dt. We will denote the angular speed about the two axes of rotation as ω1 and ω2. The moment of inertia about the two axes is the same denoted by I. In the lab frame, there is also translational motion of the center of mass. We denote the
velocity of the center of mass by V⃗ .
Let us first consider the motion along the “vibrationaxis”. Show that the kinetic energy associated with this part of the motion is Kvib=(1/2)*((m1m2)/(m1 + m2))*v^2
What is the total energy in the center of mass frame? Your answer should involve m1, m2, k, r, v, ω1, ω2, I and V⃗ as needed.
What is the total energy E in the lab frame? Your answer should involve m1, m2, k, r, v, ω1, ω2, I and V⃗ as needed.
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