I’m just watching Who Wants to be a Millionaire, and one of the later questions was:
How much larger is the Earth’s diameter than the Moon’s? A: 2.5; B: 3.7; C: 5.8; D: 9.2
I blanked briefly. Despite being a science question, and specifically an astronomy question, I had no idea what the correct answer was.
But then I started thinking: can I possibly figure it out? What facts do I know about the Earth and the Moon that I can use? My thoughts turned to eclipses – if I could remember the size of the Sun and the radius of the Moon’s orbit, I could calculate the size of the Moon from similar triangles, and I know the size of the Earth…
But alas I don’t know the size of the Sun offhand. What else do I know about the Moon? It has weaker gravity than Earth… in fact if I remember rightly it has about 1/7 the gravity of Earth. And gravity is proportional to the mass, and inversely proportional to the square of the radius (thank you Isaac Newton). I have no idea what the masses of the Earth and Moon are, but I know they must both have similar densities, being primarily made of rock, so the masses would be proportional to the volumes, and the mass of a sphere is proportional to the cube of the diameter. This means the relative strength of gravity on the surfaces of the Earth and Moon must be roughly in the same proportion as their radii to the power of 3/2.
So if I take 7, and square it to get 49, then take the cube root of 49… let’s see, 4 cubed is 64, so the cube root of 49 must be a bit less than 4… scan the options… the answer must be 3.7. B!
I went through all of this in my head in the 30 seconds that the contestant had to answer the question, and reached my answer before the contestant locked in what was nothing more than a random guess. (She guessed C: 5.8.)
The answer? 3.7. Science wins!
EDIT: Oops! So embarrassing! The moon has a gravity 1/6 that of Earth, not 1/7. And the correct thing to do is divide the radius cubed by the radius squared, leaving the surface gravity varying proportionally to the radius, not the radius to the power of 3/2, assuming the densities are the same. But it also turns out the Earth and Moon have significantly different densities (5.5 compared to 3.3), and multiplying 6 by the ratio 3.3/5.5 gives 3.6, close enough to the answer of 3.7.
In other words, I stuffed up on 3 different facts, and produced the correct answer only by a happy cancellation of errors. Still, I did it in under 30 seconds and got the right answer, so I’m still claiming the money! (Somewhat sheepishly.)
Wouldn’t you expect surface gravity of a spherical body to increase with the first power of the radius? Surface gravity goes like m/r^2. m goes like r^3 (at constant density). So gravity would increase like r^3/r^2 = r.
The reason this estimate doesn’t work in the actual case of the Earth and the Moon is that Earth is significantly more dense–it has roughly 81 times the mass of the moon, at 3.7 times the radius.
Ah crap, you’re right. Okay, so I got the answer through a happy coincidence of sloppy mental calculation. I still did it in under 30 seconds, so I’m not too ashamed.
What? The Australian version must be slightly different than the US – there’s no time limit, though the host can get proddy if they dawdle too long. I would have guessed, just from a mental image of the two of them. I’d have been right (was right reading your post, honest!) but it would have been a guess.
Right, that’s the irony; I would have confidently guessed 5.8 based on the fact that the Moon’s gravity is 1/6 as strong.
“produced the correct answer only by a happy cancellation of errors” – that’s half of science right there, really.
The Ridger: We have a variant on TV here now, “Millionaire: Hot Seat” or somesuch. Dozen or so contestants, time limits, a maximum prize that shrinks with every wrong answer, ability to pass with the hope that enough other people will get questions wrong that you get another chance. I suspect the changes were made in response to the success of Deal or No Deal, but I wouldn’t lock that in.