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Does acceleration due to gravity increase when an object at a said radius is doubled? Like for instance, if I said the radius of the earth is $5\, \mathrm{km}$ (not to scale obviously) if I doubled that, so $10\, \mathrm{km}$ will acceleration due to gravity double? Or maybe $g/2$? Or would the acceleration stay the same no matter distance?

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The force on a particle of mass $m$ on the surface of a spherically symmetric mass distribution (the Earth) is $$F=\frac{GMm}{R^2},$$ where $G$ is a constant, $M$ is the body mass and $R$ is the radius of the distribution. By Newton's second law, that force equals $ma$, where $a$ is the acceleration of particle $m$. As you can see, this acceleration, which we call gravity acceleration, reads $$a=\frac{GM}{R^2}.$$ If you double the radius (keeping the mass $M$ fixed), acceleration is decreased by a factor of four.

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Does acceleration due to gravity increase when an object at a said radius is doubled ? Like for instance, if I said the radius of the earth is 5km (not to scale obviously) if I doubled that, so 10km will acceleration due to gravity double ?

This depends on exactly what you mean by this.

Acceleration due to gravity is given by :

$$g=\frac {GM} {r^2}$$

And note that $G$ is a universal constant, $M$ is the mass doing the attracting and $r$ is the distance from the object's center of mass.

Let's assume you simply move twice as far away as $r$, then the gravitational acceleration you feel will be quartered - that power of two !

But if you made the Earth twice and big kept it's density the same - so you filled the extra space, what would happen ?

Well :

$$M=\frac 4 3 \pi r^3 \rho$$

Where $\rho$ is average density.

That makes :

$$g = \frac 4 3 G \pi \rho r$$

So in this case ( doubling the radius and filling the space ), the gravitational acceleration at the new radius would be twice as large - a completely different result from the first interpretation of your question.

So, as always in physics, you need to be very precise in describing what you mean.

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