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Example 10.2 The Oxygen Molecule Problem Consider the diatomic oxygen molecule O2, which is rotating in the xy plane about the z axis passing through its center, perpendicular to its length. The mass of each oxygen atom is 2.66 × 10^-26 kg, and at room temperature, the average separation between the two oxygen atoms is d = 1.21 × 10^-10 m. A Calculate the moment of inertia of the molecule about the z axis. B A typical angular speed of a molecule is 4.54 × 10^12 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is its rotational kinetic energy? Solution A Calculate the moment of inertia of the molecule about the z axis. We model the molecule as a rigid object, consisting of two particles (the two oxygen atoms), in rotation. Because the distance of each particle from the z axis is d/2, the moment of inertia about the z axis is the following. I = ?_i m_i r_i^2 = m(d/2)^2 + m(d/2)^2 = md^2/2 I = (2.66 × 10^-26 kg) (1.21 × 10^-10 m)^2 / 2 I = [ ] kg ? m^2 B A typical angular speed of a molecule is 4.54 × 10^12 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is its rotational kinetic energy? We use the equation below to find the rotational kinetic energy. K_R = 1/2 I?^2 = 1/2 I(4.54 × 10^12 rad/s)^2 K_R = [ ] J Exercise 10.2 What If? At a higher temperature T, the average separation between the two oxygen atoms is found to have increased by 4% while the rotational kinetic energy of the molecule increased by 20.5%. How much (in percentage) has the angular speed of the molecule changed? (Positive for increase, negative for decrease.) [ ] %

          Example 10.2 The Oxygen Molecule
Problem Consider the diatomic oxygen molecule O2, which is rotating in the xy plane about the z axis passing through its center, perpendicular to its length. The mass of each oxygen atom is 2.66 × 10^-26 kg, and at room temperature, the average separation between the two oxygen atoms is d = 1.21 × 10^-10 m.
A Calculate the moment of inertia of the molecule about the z axis.
B A typical angular speed of a molecule is 4.54 × 10^12 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is its rotational kinetic energy?
Solution
A Calculate the moment of inertia of the molecule about the z axis.
We model the molecule as a rigid object, consisting of two particles (the two oxygen atoms), in rotation. Because the distance of each particle from the z axis is d/2, the moment of inertia about the z axis is the following.
I = ?_i m_i r_i^2 = m(d/2)^2 + m(d/2)^2 = md^2/2
I = (2.66 × 10^-26 kg) (1.21 × 10^-10 m)^2 / 2
I = [ ] kg ? m^2
B A typical angular speed of a molecule is 4.54 × 10^12 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is its rotational kinetic energy?
We use the equation below to find the rotational kinetic energy.
K_R = 1/2 I?^2 = 1/2 I(4.54 × 10^12 rad/s)^2
K_R = [ ] J
Exercise 10.2
What If? At a higher temperature T, the average separation between the two oxygen atoms is found to have increased by 4% while the rotational kinetic energy of the molecule increased by 20.5%. How much (in percentage) has the angular speed of the molecule changed? (Positive for increase, negative for decrease.)
[ ] %

Example 10.2 The Oxygen Molecule
Problem Consider the diatomic oxygen molecule O2, which is rotating in the xy plane about the z axis passing through its center, perpendicular to its length. The mass of each oxygen atom is 2.66 × 10^-26 kg, and at room temperature, the average separation between the two oxygen atoms is d = 1.21 × 10^-10 m.
A Calculate the moment of inertia of the molecule about the z axis.
B A typical angular speed of a molecule is 4.54 × 10^12 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is its rotational kinetic energy?
Solution
A Calculate the moment of inertia of the molecule about the z axis.
We model the molecule as a rigid object, consisting of two particles (the two oxygen atoms), in rotation. Because the distance of each particle from the z axis is d/2, the moment of inertia about the z axis is the following.
I = ?i mi ri^2 = m(d/2)^2 + m(d/2)^2 = md^2/2
I = (2.66 × 10^-26 kg) (1.21 × 10^-10 m)^2 / 2
I = [ ] kg ? m^2
B A typical angular speed of a molecule is 4.54 × 10^12 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is its rotational kinetic energy?
We use the equation below to find the rotational kinetic energy.
KR = 1/2 I?^2 = 1/2 I(4.54 × 10^12 rad/s)^2
KR = [ ] J
Exercise 10.2
What If? At a higher temperature T, the average separation between the two oxygen atoms is found to have increased by 4% while the rotational kinetic energy of the molecule increased by 20.5%. How much (in percentage) has the angular speed of the molecule changed? (Positive for increase, negative for decrease.)
[ ] %

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