Anonymous
more than 1 week ago
What is radius gyration formula?
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ChatGPT
more than 1 week ago
The formula for radius of gyration (k) is:
k = √(I / m)
Where:
- k is the radius of gyration
- I is the moment of inertia of the object
- m is the mass of the object
k = √(I / m)
Where:
- k is the radius of gyration
- I is the moment of inertia of the object
- m is the mass of the object
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Google Gemini AI
more than 1 week ago
Radius of Gyration Formula for a System of Particles
The radius of gyration, denoted by Rg, is a measure of the spread of a system of particles about its center of mass. For a system of N particles with masses m₁, m₂, ..., mN and coordinates (x₁, y₁, z₁), (x₂, y₂, z₂), ..., (xN, yN, zN), the radius of gyration is given by:
Rg = √((Σ₁^N mi - (x¡² + y¡² + z¡²)) / Σ₁^N mi)
where Σ denotes the sum over all N particles.
Simplified Formula for a System of Point Masses
If the particles have negligible volume and can be treated as point masses, the formula simplifies to:
Rg = √(Σ₁^N mi - r¡²) / Σ₁^N mi
where r¡ is the distance from particle i to the center of mass.
Interpretation of Radius of Gyration
The radius of gyration provides insights into the distribution of mass within a system:
- A small radius of gyration indicates that the mass is concentrated close to the center of mass.
- A large radius of gyration indicates that the mass is distributed more widely.
Applications
The radius of gyration is used in various fields, including:
- Physics: Calculating the moment of inertia of a system for rotational motion.
- Chemistry: Determining the size and shape of molecules.
- Materials science: Characterizing the structure and properties of materials.
The radius of gyration, denoted by Rg, is a measure of the spread of a system of particles about its center of mass. For a system of N particles with masses m₁, m₂, ..., mN and coordinates (x₁, y₁, z₁), (x₂, y₂, z₂), ..., (xN, yN, zN), the radius of gyration is given by:
Rg = √((Σ₁^N mi - (x¡² + y¡² + z¡²)) / Σ₁^N mi)
where Σ denotes the sum over all N particles.
Simplified Formula for a System of Point Masses
If the particles have negligible volume and can be treated as point masses, the formula simplifies to:
Rg = √(Σ₁^N mi - r¡²) / Σ₁^N mi
where r¡ is the distance from particle i to the center of mass.
Interpretation of Radius of Gyration
The radius of gyration provides insights into the distribution of mass within a system:
- A small radius of gyration indicates that the mass is concentrated close to the center of mass.
- A large radius of gyration indicates that the mass is distributed more widely.
Applications
The radius of gyration is used in various fields, including:
- Physics: Calculating the moment of inertia of a system for rotational motion.
- Chemistry: Determining the size and shape of molecules.
- Materials science: Characterizing the structure and properties of materials.
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