I've been a Dragon Ball kick lately. This post had me floored. There's analyzing a series you like, and then there's analyzing a series you like. This falls firmly into the latter category. I found it while reading leavemywife's let's play of Dragon Ball Z: Attack of the Saiyans, which you should read. It's a good let's play of a good game.
"Seriously, you folks start talking about physics but don't even take your calculator out?
To do calculations with the gravity we need to know the gravity of the planet and the size.
The surface gravity is simple. 10g, or 10 x 9.8 m/s^2 = 98 m/s^2.
For the size, I went to the Dragon Ball Wiki. It says that King Kai's car is a 1957 Red Chevrolet Bel Air. According to Wikipedia, that car has a length of 195.6", or almost 5 meters.
The scale seems to differ a bit between the anime and the game, but I'll go with a picture from this game. It won't change much anyway.
As you can see, the diameter of the planet is about six and a half times the length of the car. That makes 32.5 meters. The radius is half the diameter, 16.25 meters. By the way, that's a tiny house.
Now, we can use the formule g = GM / r^2, with g being the surface gravity, G the gravitational constant, M the mass of the planet in kg, and r the radius in meters. Let's rewrite that. M = g*r^2 / G.
Filling it in: M = 98 * 16.25^2 / 6.674E-11 = 3.88E14 kg
That's a rather big number, but it doesn't tell us that much by itself. But we can use it to calculate a mean density of the planet. Density is simply mass divided by volume.
The volume of a sphere is given by (4/3)pi*r^3. For King Kai's planet, that is 17974 cubic meters.
The density is 3.88E14 kg / 17974 m^3 = 2.16E10 kg / m^3 or roughly 21.6 billion kg / m^3. For the Americans, that's about 1.35 billion pounds per cubic foot. Imagine lifting a block of that stuff.
Anyway, that density is in the same order of magnitude as a white dwarf. That's a very dense, relatively small star that has almost burned out. As it completely burns out, which can take hundreds of billions of years, it might turn into a black dwarf, with similar properties except that it doesn't glow. Its matter is in a 'degenerate' state, meaning that the atoms are so close together they don't act in a way we're used to. The only reason a white/black dwarf is stable is because it is about as heavy as our sun. This gives it so much gravity that it can keep itself together.
King Kai's planet weighs many, many, many orders of magnitude less than our sun. This means that in reality, this planet would immediately blast apart in an enormous explosion.
Of course you could try to prevent this by using a different kind of mass distribution. Put something much denser in the center, and make the outer layers a bit less dense. For instance, put a neutron star or a black hole in the center, those are both much denser than a white dwarf.
The thing is, this wouldn't be stable either. Either the black hole is too strong, the gravity is way higher than 10g, and everything is sucked in, or it's not strong enough and the planet explodes (or if it's exactly in between, the inner part would be sucked in while the outer part would be blasted away).
The only way I can think of to make a planet like this keep together is by using some magical "forcefield" that prevents it from flying apart. The force required for such a thing would be incomprehensibly large. And you need to keep applying it every nanosecond.
Oh by the way, if King Kai's planet were possible, the small size distorts gravity so much that Goku's head would experience significantly less gravity than his feet. Enough to make anyone feel a bit... lightheaded."