# What would be the mathematical formula to figure out a person's weight on a world bigger or smaller than Earth?

Okay, trying to figure out this math guys! This may be simple math to some of you, but not me!!!

I googled various ways of asking this question and got some good results, but I'm looking for a simple formula to use.

If I weigh 200 pounds on Earth, how much would I weigh on a planet that's 90% smaller than the Earth, or how much would I weigh on a planet that's 25% bigger than the Earth. (I don't know the MASS of the moons/planets)

Can you tell me the mathematical formula to figure this out so I can change the weight of person/object and see what they would weigh on other moons/planets (without knowing the MASS of those objects)?

I found this Weight = Mass x Surface Gravity but what if I don't know Mass or I don't know Gravity? I just know smaller/bigger? Not possible?

• en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation – nzaman Dec 8 '18 at 13:20
• updated question... what if I don't know the mass of the moon/planet ... just that it's bigger or smaller than the Earth? – J.J. Hartly Dec 8 '18 at 13:29
• Assume the same density as the earth, then calculate the mass based on the volume – nzaman Dec 8 '18 at 13:31
• @nzaman Mass doesn't change only weight based on gravity I guess. So I have to figure out the moons/planets gravities first? – J.J. Hartly Dec 8 '18 at 13:41
• There ISN'T a formula that depends on just the size. It depends on the mass of the planet, and how that mass is distributed. For an extreme case, a planet made of lead is going to have much higher gravity than one made of styrofoam. – jamesqf Dec 8 '18 at 19:01

I'll assume that your world is spherical and not rotating fast enough for centrifugal force to make a difference. The surface gravity then depends on two things: The planet's mass and its radius. Let M be the planet's mass in units of the Earth's mass (i.e., Earth has mass 1.0.) Let R be the planet's radius in terms of the Earth's radius (i.e., we're using units of about 4000 miles for radius.) So Earth has M=1.0 and R=1.0.

For another planet, the surface gravity would be M/R2 (in units of the Earth's surface gravity.)

So for a planet twice the mass of Earth, but the same size, the surface gravity would be 2.0/1.02 or 2Gs. For a planet the same mass as the Earth, but twice the radius, the surface gravity would be 1.0/2.02 of 0.25Gs. And so forth.

Looking at your two specific questions, you are asking about two planets, one 90% smaller than Earth, and one 25% bigger. H have to know the planet's mass, so I'll assume that your planet has the same density as Earth (it's reasonably close to correct and much easier to figure out.)

Begin with the formula for surface gravity in terms of the mass and radius of a planet, M/R2. Assuming that all planets have the same density as Earth, the mass, M, would be R3 since a sphere's volume scales as the cube of the radius and the mass of a constant-density sphere is proportional to its volume. (We're still dealing with M and R measured in terms of the Earth's mass and radius.)

Plug this into M/R2 and we get R3/R2 which reduces to simply R. In other words, the surface gravity of a planet the same density as Earth is simply proportional to its size.

So surface gravity for your planets, one 90% smaller than Earth, and one 25% bigger, would be 10% of Earth's surface gravity and 125% and people would weight 10% as much and 1.25 times as much.

(As an addendum, it's worth noting that the assumption of all these planets having the same density is not bad, but it is not exact, either. First of all, it applies only to rocky planets. Earth's density is around 5.5 (in the most common units). In those same units, a gas giant like Jupiter has a density of only 1.3, so the constant density formula would drastically overestimate its surface gravity. The Moon is rocky, but because it's less massive (and also because it contains less iron) its density is only 3.3. As a good rule of thumb for rocky planets, the more massive a planet is, the denser it is because the planet's own gravity compresses the rock somewhat. So the simple result that the surface gravity is proportional to R will tend to underestimate the gravity of larger planets by a bit and overestimate the gravity of smaller ones by a bit, also. But not by a huge amount.)

• So ... 200x.1 (10%) = 20 pounds or 200x1.25= 250 pounds? So if the planet is 10% smaller than Earth... 200x.9 (90%) = 180 pounds ... this seems to be what I was looking for, but I'll see what other answers pop up ... thanks! – J.J. Hartly Dec 8 '18 at 14:07
• I just wanted to point a small mistake in Mark Olson's math: while the mass for the new planet is indeed proportional to R³, when swapping m/r² (small planet's) for R³/R², we're mixing up r² (small planet's) with R² (Earth's), yielding a wrong result. Pluggin the 0.1 and 1.25 values into the original M/R² formula, we get, respectively: $0.1/(\sqrt {0.1²}) = 0.464G$ and $1.25/(\sqrt {1.25²}) = 1.077G$ As for the cube root above remember that, since M scales with R³, R scales with $\sqrt {M}$. – Paulão Tresperna Oct 17 '19 at 21:33

It is worth mentioning that we tend to talk about weight and mass as if they were the same thing (which they aren't), and it's important to avoid that here. If your mass is 75kg, then on Earth your weight – the force you exert on the ground, measured in newtons – is $$W=mg = 75\mathrm{kg}\times 9.81\mathrm{ms^{-2}}=736\mathrm{N}$$ on a planet with 10% of Earth's diameter, it's 74N, and on a planet 25% bigger than Earth it's 920N. But your mass is still 75kg wherever you go.

The problem, of course, is that no one thinks of weight in Newtons, so those numbers are useless for describing what things weigh. You could say:

I weigh 10% of my Earth weight on the mini planet

or

I weigh the equivalent of of 7.5 Earth kilograms

– just don't say "I weigh 7.5kg on the mini planet".

• I weigh the equivalent of 7.5 Earth kilograms THAT makes total sense to me and since I'm a writer, that's really what I wanted to know... how to properly relay my characters weight on other planets! I just didn't know how to ask the question properly I think :-) the formula you show doesn't make sense to me and is why I was looking for something simpler – J.J. Hartly Dec 8 '18 at 14:23
• Ah OK, so in that case the number you want is (Earth weight) x (proportional size of planet), per Mark Olson's answer. For something that weighs 100kg, on a planet 90% the size of Earth, the number is 10 x 0.9 = "the equivalent of 90 Earth kilograms" – bobtato Dec 8 '18 at 14:38