Dimensional formula of distances in certain formulas

[Mathivanan]

Homework Statement

Dimensional formula for 'the square of the distance between two bodies' in universal gravitation and 'distance from the axis squared' in moment of inertia. Is L^2 is the dimensional formula for both the distances in the above two cases?

Homework Equations

F=Gm1m2/d^2; moment of inertia=mass*distance from the axis squared

The Attempt at a Solution

 

[haruspex]

Is L^2 is the dimensional formula for both the distances in the above two cases?

Yes, except that in the gravitational case it acts as a divisor, so becomes L^-2.

 

[Mathivanan]

Yes, except that in the gravitational case it acts as a divisor, so becomes L^-2.

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]

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.

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]

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.

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.