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When I researched the gravity on comets like on 67P/Churyumov-Gerasimenko's surface I found that it was

several hundred thousand times weaker than on Earth

source: www.esa.int/Our_Activities/Space_Science/Rosetta/Frequently_asked_questions

I am asking because I had a dream that I landed on a comet. But its gravitational pull was so strong that we could barely move. What would be the case in which that is possible, perhaps very dense metal core?

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ThePlanMan gives the basics in his answer. In addition: the densest material known to man is Osmium, at $\rho$ = 22 kg/dm$^3$. This compares to water at 1 kg/dm$^3$ and rock at 2-8 kg/dm$^3$. So the densest possible comet is only 22 times as dense as one that consists of water ice. That's not enough to give a gravity of anywhere near 1 G.
Comet 67P/Churyumov-Gerasimenko has a mass of $10^{13}$ kg. To get 1 G, you need 10$^{24}$ kg, or the mass of Earth.
A planet so heavy you can barely move, is either much larger than Earth or contains exotic matter. A neutron star, for instance, but then you're far beyond "can barely move" - a person would be instantly squashed flat.
Could a smaller planet give a 1 G surface gravity? We can calculate the mass of a planet:

$$ m_1 = 4/3 *{\pi r^3} * \rho $$

When I replace $m_1$ with this formula we get the gravitational force for a person on the surface of a planet:

$$F = \frac{G * 4/3 *{\pi r^3} * \rho * m_2}{r^2}$$

Which can be reduced to:

$$F = {G * 4/3 *{\pi r} * \rho * m_2}$$

The gravitational force scales linearly with radius and density. The smallest possible body (solid osmium) that has 1 G gravity is on the order of 1/20 the mass of Earth, i.e. it has a radius of 0.36 times Earth's radius.

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  • $\begingroup$ because of the smaller size, a comet would not have to have nearly the same mass as the earth in order to have 1 g of surface acceleration. See what-if.xkcd.com/68 . You are right about there not being any normal material to achieve such density. $\endgroup$ – Rikki-Tikki-Tavi Mar 19 '15 at 15:13
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A very simple answer is that gravitational force is defined by:

$$F = \frac{G m_1 m_2}{r^2}$$

Here $G$ is the universal gravitational constant, $m_1$ is your mass, $m_2$ is the mass of the comet and $r$ is the distance between the two. So increasing either your mass or the mass of the comet or decreasing the distance between the two are the only ways to get a higher force. If you're on the surface then you're already as close as you can get. So for a comet to have a high gravitational force it would have to have a high mass. This can be either higher density or larger volume (or both, of course).

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