Would the gravitational force of a paper plate the size of our solar system crush someone standing at the middle of the plate?

Well, calculate it by integration. Say, a paper plate weighs 10 gram per square meter. Do not forget that there is symmetry, so the force components parallel with the plate cancel.

Calculate your integral.

I’m not going to calculate it, a paper plate is pretty thick compared to even the sun on that scale. Yes, you would be a human smoothie on that plate and even more so on the rim.

On second thought you specified that the plate has the mass of the earth which hardly seems feasible but whatever. You would not be able to stand still enough not to float away either in the center or on the rim.

I was thinking that his paper plate was as thin as it is now, but enormously long. It would or would not provide enough gravity to pulp you. Hence, homework to calculate the integral for the boy.

Oh, just any old disc would make more sense, though once again that’s not what he said.

The earth stretched to a solar system sized pizza (damn, now I did it myself) would be a lot weaker than the earth on the rim because of the high r for most of the matter. In the center the matter in every direction would cancel eachother out for the most part, you would still be able to fly and choke at the same time.

A paper plate the mass of the entire solar system could not exist. It would collapse into its own gravitational well till its internal temperature reached a sufficient level to induce a thermonuclear reaction between the carbon and oxygen in it. This would blow it apart in a supernova. The bottom line is the poor sap standing on that plate would be toast, any way you slice it. :-)

http://en.wikipedia.org/wiki/Shell_theorem

I would assume that a paper plate is approximately a cross-section of a sphere.

Take any off-center bit of the plate. There is a bit of the plate 180 degrees off from it. Basically, a lot of it is going to cancel out. It would cancel out completely if you could contrive to have the person reside at the center in the plate rather than on the plate, as @phaedryx knows.

No, you’d probably be close to weightless, actually, since the gravitational vectors would cancel.

The gravitational vectors in a slice of a sphere would no more cancel than the vectors in a complete sphere cancel. If the plate were rotating at a great speed, then the centrifugal force would work to counter the gravitational pull toward the center. But if its rotation were insufficient to counter gravity, it would collapse into a sphere with a mass totaling that of the entire solar system. With that mass, the sphere would continue to collapse toward zero volume till its internal temperature reached a sufficient level to trigger a supernova explosion of the carbon and oxygen in the paper.

@ETpro I’m talking about a second body (i.e. a person) in relation to the plate on top of the center, as @bricklayer asks, and assuming the plate remains stable somehow. Of course, the plate will indeed want to collapse in on itself.

If you were “on top” of the plate at some point off-center, you would slide towards the center, accelerating, but the acceleration decreasing as you went, until you hit the center, where the acceleration would drop to zero. You would carry on in a straight line from inertia, now decelerating at an increasing rate as you travel farther from the center. Without friction you would stop for a moment at the opposite spot across the plate from your starting position and fall back again and repeat this endlessly. With friction, you would stop some distance closer to the center than your previous distance with each oscillation until you were stationary at the center.

Again, this is all assuming the plate remains stable.

@hiphiphopflipflapflop However, if the plate were made of paper, unless it were rapidly spinning, it could not possibly remain stable. Since the plate would rapidly collapse to a sphere the mass of the entire solar system, our intrepid plate astronaut would still be in deep kimchee.

You wouldn’t be crushed as most of the gravitational forces would cancel each other out. Your centre of gravity would be drawn to the centre of gravity of the plate so you would be pulled through the plate until the plate was around your chest. The plate is then going to collapse and you would then be crushed by tons of falling paper. If you were at the outer rim of the plate gravitational forces would make you fall towards the centre.

@flutherother Unless the plate is spinning, then you’ll be flung out towards Pluto.

OK, but no one said the plate was spinning and it would depend how fast.

@flutherother I know. I was just amused by the mental picture of someone being flung off a huge spinning solar system sized dinner plate.