I was told that objects with the same mass but different sizes (or densities) will have a different gravitational pull. Is this true? If so, why?
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Different objects with the same mass but different sizes will have different gravitational pulls at their surface, which is probably what you heard. But that is only because their surfaces are in different places. If you compare the gravitational pulls of different objects with the same mass at a fixed distance away from their centers, you will find that their gravitational pulls are the same. (As long as the objects are basically spherical.)
You can see this easily from the mass: in the formula for Newtonian gravity,
$$\vec{F} = -\frac{Gm_1m_2}{r^2}\hat{r}$$
which applies to spherical objects, the size of the object does not actually show up anywhere in the formula.
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$\begingroup$ The reason why this formula holds, is because of Gauss's law (assuming that the density is constant at the same radius). $\endgroup$ Commented Jun 15, 2013 at 18:29