How could the gravitational pull of a comet affect spacecraft and astronauts landing on it?

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

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?

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.

• 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. Commented Mar 19, 2015 at 15:13

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).