If a spherical shell is compressed to half it's radius without any change in it's shape and mass how would the gravitational potential at the center of the sphere change?

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anonymous · Answer

This all i got ... Hop its work ... you are welcome

anonymous · Answer

Let's assume you're talking about a sphere somewhere in space. The total mass and shape remain intact but the size decreases. If I would be floating around in space, the force by which the sphere pulls me towards me, is the same as for the larger sphere, because the mass is the same.

$$Ḟ=-(GMm/r ^{2})ṙ$$

G is the gravitational constant, M is the mass of the sphere, m is my mass, r^2 is the distance (squared) and ṙ is the unity vector. See, the force remains the same.

The gravitational potential is how much energy it would cost to get my mass to a certain position, on which you'll want to know the potential. I'll bring the mass in from infinity and experience an attractive force, so I wouldn't have to do any work. The potential is the integral of this force on my body over all smalls steps to the point of interest. Because, work is force times distance, right?

$$U=-\int\limits_{\inf}^{Point}Ḟ•dṙ$$

$$U=-\int\limits_{\inf}^{Point}-(GMm/r^2)•dṙ=-GMm/r $$

So, the experienced force is the same and so the total work to come in from infinity is the same, hence the potential at point of interest.