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A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :?
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A diatomic molecule is formed by two atoms which may be treated as mas...
Understanding the Moment of Inertia of a Diatomic Molecule
The moment of inertia (I) quantifies an object's resistance to angular acceleration about a given axis. For a diatomic molecule made of two atoms, the calculation involves the masses of the atoms and their separation.
Parameters Involved
- **Atoms**: The two atoms are designated as m₁ and m₂.
- **Distance**: The distance between the two atoms is r, and the center of mass (CM) is located at a point that divides the distance r in proportion to the masses.
Center of Mass Calculation
The center of mass (CM) of the diatomic molecule can be found using the formula:
\[
\text{CM} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1 + m_2}
\]
Where \( r_1 \) and \( r_2 \) are the distances of m₁ and m₂ from the center of mass, determined by:
\[
r_1 = \frac{m_2}{m_1 + m_2} \cdot r
\]
\[
r_2 = \frac{m_1}{m_1 + m_2} \cdot r
\]
Moment of Inertia Calculation
For an axis passing through the center of mass and perpendicular to the rod, the moment of inertia is given by:
\[
I = m_1 \cdot r_1^2 + m_2 \cdot r_2^2
\]
Substituting the values of \( r_1 \) and \( r_2 \):
\[
I = m_1 \left(\frac{m_2}{m_1 + m_2} \cdot r\right)^2 + m_2 \left(\frac{m_1}{m_1 + m_2} \cdot r\right)^2
\]
This simplifies to:
\[
I = \frac{m_1 m_2 r^2}{(m_1 + m_2)}
\]
Thus, the moment of inertia of the diatomic molecule about the center of mass is:
Final Expression
\[
I = \frac{m_1 m_2 r^2}{(m_1 + m_2)}
\]
This expression highlights how the mass distribution around the center of mass affects the rotational inertia of the molecule.
The moment of inertia (I) quantifies an object's resistance to angular acceleration about a given axis. For a diatomic molecule made of two atoms, the calculation involves the masses of the atoms and their separation.
Parameters Involved
- **Atoms**: The two atoms are designated as m₁ and m₂.
- **Distance**: The distance between the two atoms is r, and the center of mass (CM) is located at a point that divides the distance r in proportion to the masses.
Center of Mass Calculation
The center of mass (CM) of the diatomic molecule can be found using the formula:
\[
\text{CM} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1 + m_2}
\]
Where \( r_1 \) and \( r_2 \) are the distances of m₁ and m₂ from the center of mass, determined by:
\[
r_1 = \frac{m_2}{m_1 + m_2} \cdot r
\]
\[
r_2 = \frac{m_1}{m_1 + m_2} \cdot r
\]
Moment of Inertia Calculation
For an axis passing through the center of mass and perpendicular to the rod, the moment of inertia is given by:
\[
I = m_1 \cdot r_1^2 + m_2 \cdot r_2^2
\]
Substituting the values of \( r_1 \) and \( r_2 \):
\[
I = m_1 \left(\frac{m_2}{m_1 + m_2} \cdot r\right)^2 + m_2 \left(\frac{m_1}{m_1 + m_2} \cdot r\right)^2
\]
This simplifies to:
\[
I = \frac{m_1 m_2 r^2}{(m_1 + m_2)}
\]
Thus, the moment of inertia of the diatomic molecule about the center of mass is:
Final Expression
\[
I = \frac{m_1 m_2 r^2}{(m_1 + m_2)}
\]
This expression highlights how the mass distribution around the center of mass affects the rotational inertia of the molecule.
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A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :?
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A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :?.
A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A diatomic molecule is formed by two atoms which may be treated as mass points m, and m₂ joined by a massless rod of length r. Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is :?.
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