The discussion revolves around the dimensional formulas for distances in the context of universal gravitation and moment of inertia. Participants are examining whether the dimensional formula L^2 applies to both cases and the implications of this in terms of physical interpretation.
The conversation is ongoing, with participants providing insights into the nature of dimensional analysis and its limitations. There is acknowledgment of the potential for misunderstanding when dimensional formulas are applied to different physical contexts.
Participants express concerns about the definitions and interpretations of dimensional formulas, particularly in distinguishing between distance and area. There is a recognition that dimensional analysis does not capture all nuances of physical relationships.
Yes, except that in the gravitational case it acts as a divisor, so becomes L-2.Mathivanan said:Is L^2 is the dimensional formula for both the distances in the above two cases?
Thanks. However, I have one more doubt. The dimensional formula for area is also L^2. By definitions, in the above two formulas, they represent distances rather than area. The dimensional formulas mislead me.haruspex said:Yes, except that in the gravitational case it acts as a divisor, so becomes L-2.
The dimensional formulas only capture that aspect of an expression. They don't care whether the two distances represent an actual area or have some other relationship. E.g. surface tension can be thought of as energy per unit area or force per unit length. In some cases, quite different physical entities can have the same dimension: torque and energy are both force x distance; action and angular momentum are both ML2T-1. It doesn't catch all the distinctions you'd like to make.Mathivanan said:Thanks. However, I have one more doubt. The dimensional formula for area is also L^2. By definitions, in the above two formulas, they represent distances rather than area. The dimensional formulas mislead me.
Thanks for your answer. I thought that distance should have dimensional formula of L, be it square of the distance or distance from the axis squared. The core concept in the definition is distance in both the formulas.haruspex said:The dimensional formulas only capture that aspect of an expression. They don't care whether the two distances represent an actual area or have some other relationship. E.g. surface tension can be thought of as energy per unit area or force per unit length. In some cases, quite different physical entities can have the same dimension: torque and energy are both force x distance; action and angular momentum are both ML2T-1. It doesn't catch all the distinctions you'd like to make.