What are the dimensions of intensity?

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Intensity is defined as power per unit area, leading to a dimensional analysis that simplifies to mass per cubic time. This reduction lacks physical significance as it omits the geometric aspects of intensity, which relate to energy flow through an area. When intensity is divided by flow velocity, it results in a form that indicates energy density, specifically mc^2 per volume for light. This highlights the relationship between intensity, energy density, and the geometry of energy flow. Understanding these dimensions is crucial for grasping the physical implications of intensity in various contexts.
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Homework Statement


I know that intensity is power per unit area. When I do a dimensional analysis, it reduces down to mass per cubic time. Is there any physical significance to that?


Homework Equations


I=P/A


The Attempt at a Solution

 
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Not really, because that form has removed all the geometry of the situation;
Intensity's geometry comes from the energy flow vector piercing an Area, so a flow Volume is indicated.
On the other hand, if you divide the intensity by flow velocity, you end with m v^2 / length^3 ;
light's intensity divided by light speed leaves you with mc^2 / volume, the energy density in the light.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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