Radius of Gyration: Dist to Axis & Mass Concentration

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Discussion Overview

The discussion revolves around the concepts of radius of gyration and center of mass, exploring their definitions, differences, and implications in physics. It includes theoretical considerations and conceptual clarifications regarding mass concentration and moment of inertia.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the radius of gyration is related to the distance from an axis to a point where mass is concentrated, questioning if this is equivalent to the center of mass.
  • Others argue that while both concepts relate to mass concentration, they are distinct; radius of gyration involves integrating ##\int r^2 dm##, while center of mass involves ##\int \vec{r} dm##.
  • One participant asserts that there exists a single point where the entire mass could be considered concentrated, but this is challenged by others who clarify that this point is hypothetical in the context of radius of gyration.
  • A later reply emphasizes that the radius of gyration is defined as the distance to a point mass that would yield the same moment of inertia as the actual object, introducing the formula r = sqrt(I/m).
  • Some participants discuss the implications of symmetrical objects, noting that the center of mass may not coincide with the radius of gyration.
  • There is a suggestion that there may be two points of interest, but this is contested, with one participant stating that the center of mass is a real point while the radius of gyration is an imaginary concept.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between radius of gyration and center of mass, with no consensus reached on whether they can be considered equivalent or if they represent distinct concepts.

Contextual Notes

The discussion highlights the complexity of definitions and the mathematical relationships involved, with participants pointing out the need for careful consideration of the terms used and their implications in physical contexts.

Zubair Ahmad
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For radius of gyration we say distance from axis to a point where would mass is concentrated. isn't that center of mass?
 
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Not exactly. Both center of mass and radius of gyration are measures of where the mass is concentrated, but they are different. To compute radius of gyration, you have to integrate ##\int r^2 dm## where dm is the mass. To get center of mass you have to integrate ##\int \vec{r} dm##. Center of mass has three components; it is like a vector. Radius of gyration is a scalar (a simple number, not a vector).
 
But physically we have a single point where we say whole mass is concentrated.
 
Zubair Ahmad said:
For radius of gyration we say distance from axis to a point where would mass is concentrated. isn't that center of mass?
No, that is not the correct definition of radius of gyration. The radius of gyration, r, is the perpendicular distance to the axis of rotation of a point mass whose moment of inertia (I=mr^2) is the same moment of inertia of the actual object having that same mass. So say an object having 10 kg of mass with an I of 1000 kg-m^2 is equivalent to a point mass of 10 Kg located10 meters away from the axis, so r = 10 m. In other words, r = sq rt (I/m).
 
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Zubair Ahmad said:
But physically we have a single point where we say whole mass is concentrated.
No, we have a single point where IF the whole mass was concentrated there, the moment of inertia would be equivalent to the moment of inertia of the object. A symmetrical object for example has its actual center of mass at the center of it at the rotation axis, but its radius of gyration is not 0, it is sq rt (I/m)
 
So can we say physically there are two such points
 
Zubair Ahmad said:
So can we say physically there are two such points
No, sir. The center of mass is real; the radius of gyration is imaginary
 

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