So I was in the shower today...

[thegrogen]

This is a bit off-topic, but consider for a second that a massive object, bigger than Earth, passed really close to Earth, within a few kilometres. Wouldn't all loose objects, including people, be pulled towards that object? And if it wasn't directly above you, you could "fall" sideways.
Earth would be pulled too, albeit more slowly.

If it really passed that close, maybe. But in reality, an object that close would rub atmospheres with us and the two worlds would be sucked into each other, making a gigantic peanut world with an even higher gravity.

Farther away, that wouldn't really happen because since you are closer to the Earth, the Earth's gravitational pull is stronger on you than the other one. The other object will still pull on you, of course. It'll cause tides and whatnot, just like the moon does. The tides would be stronger, that's all, and you might even feel them tugging at you.

Also, we know that the Earth's core is solid because earthquakes can't pass through it very easily, because it is so dense. We have found that earthquakes on one side can't be felt directly opposite the earth from the epicenter, so therefore there must be something dense in it that the waves cannot pass through.