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So there are similar questions on this topic, but allow me to ask it slightly differently since I don't have the reputation to comment on the other threads.

I'm asking this as a way to illustrate a concept in a marketing story about what is happening in IT Departments today, not because I'm a scientist. I don't want to get it wrong technically because my audience is technical from a software perspective, albeit not in the area of physics.

So...the question: If I have two objects of same shape and same size, but one has more mass and therefore higher density, does it increase the gravitational pull of that object? Also, to ask my question another way, if there is more atomic activity (more stuff happening) in one object over another, but both have same shape and size, does it have more 'gravitational pull' than the other?

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A mass $m_1$ (spherical for simplicity) will produce a net force on the centre of mass of a second object of net mass $m_2$ as per $$ F=\frac{Gm_1m_2}{r^2}\, . $$ This net force will be greater on a more massive object, but this object also has more inertia so the resulting acceleration $a_2= F/m_2$ will be independent of $m_2$.

The mass distribution of $m_2$ will determine the strength of tidal forces and torques about the centre of mass of $m_2$.

Two objects having identical shapes and sizes but different internal mass distributions, may have their centres of mass at different locations, and may have different principal moments of inertia. Thus they may rotate about different axes and be subject to different torques about their respective centres of mass, even if they appear outwardly identical.

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Yes, an object's gravitational pull is directly proportional to its mass. Doubling the mass doubles the gravitational pull, same with tripling, etc.

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Newtons Law of Universal Gravity says that the gravitational force between any 2 objects is proportional to the product of their masses $M, m$ and the inverse square of the distance $r$ between their centres of mass :
$F=G\frac{Mm}{r^2}$.
That is all. It does not depend on the density of either body, nor on 'atomic activity' whatever that means - perhaps thermal motion (temperature)? radioactivity? chemical activity?

The force between the two bodies is mutual - the same for each. (This is true for all types of force.) If you weigh 700 Newtons this means that the Earth is pulling on you with a force of 700 Newtons. But surprisingly it also means that you are pulling on the Earth with the same amount of force, 700 Newtons.

Gravity is extremely weak if both objects are human-sized. It only becomes significant for mountain-sized objects. Static electricity and magnetism are far far stronger on human scales.

Two extremely dense objects each with a mass of 1000 tonnes placed with their centres 8 metres apart would still only attract each other with a force of 1 Newton - about the same as the weight of an apple.

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