As noted, this is a trick question. The simple approach is to note that mass goes up with the cube of radius or diameter, but the force of gravity goes down with the square of radius, so the increase in "g" would be linear with the increase in radius.

Thus, the new Earth would be 10/9.8, or 2% larger in radius. Not much.

But the trick is this. The meter itself was first defined as being 1/40,000,000 of the circumference of the Earth. Thus as you
increase the size of the planet, the people on it would change their definition of the meter, so to them their planet would
always be the same size in meters, up to a point. And so "g" would always be 9.8. It's a function of the density of the planet, not the size.

Today the meter is not even a fundamental unit. It started by defining pole to equator as 10,000 km. Then it became the length between two marks on a bar. Today, the second is the defined fundamental unit (based on cesium vibrations) and the speed of light is defined to be exactly 299,792,458 meters/second -- and thus the meter is calculated from that.

Thus to get exact, since a second is 9,192,631,770 periods of the particular transition in a cesium atom, a meter is defined as how far light goes in 30.664419 of those vibrations.

This is a tricky puzzle question I thought up some time ago, but I figured I would blog it.

As people who study physics know, the acceleration a falling body undergoes if dropped (in a vacuum) at the surface of the earth is known as “g”, or 9.8 meters per second per second.

This is so close to 10 that most students and people doing back of envelope calculations often use 10 as the value of a “g”. It’s easy. Fall for one second and you’re going 10 meters/second.

So I got to wondering, how much would the earth have to change to make “g” equal to 10 instead of 9.80665?

So here’s the puzzle. By what ratio would you have to increase the diameter of the Earth so that the people on this bigger planet would have “g” equal to 10? Assume the average density of the Earth remains the same (5.46 g/cc).

Then click to the Puzzle Answer