### Where Would Newton Weigh a Newton?

I got into a conversation about units the other day (yeah - some of my friends are as nerdy as me), and it got me to thinking about Newtons. They're named after the guy, but not based on him in any physical sense. But what is the physical relationship? Where would you have to go for Newton to weigh a Newton?

This is actually a pretty simple calculation. If you think back to your high school or college physics days, the force due to gravity is:

F = G*m1*m2/r^2

where G is the universal gravitational constant, m1 and m2 are the masses of the two objects attracting each other, and r is the distance between their centers of mass. G is known thanks to science, as are the masses and radii of various bodies in our solar system. That leaves just one more unknown - the mass of Sir Isaac Newton himself.

So just how big was Newton? To tell the truth, I doubt that anyone knows for sure. Doing a google search on "How much did Newton weigh" didn't yield anything concrete. But I did come across an interesting article on the blog, And Now You Know:

How tall was Isaac Newton? 5 feet 6 inches, perhaps shorter

So, the title of that article gives the answer in itself. Newton wasn't very tall by today's standards. In fact, he wasn't even very tall by the standards of his day. John Conduitt, who knew Newton personally and saw him on a regular basis (he was married to Newton's niece), described him as "he was short of a \middle/ stature & in \plump/ \in/ his later years inclining to be fat."

So assuming Newton was 5'-6" and on the 'plump' side, how much would he have weighed? Here's an interesting chart from the UK's NHS, Height/weight chart. For someone 5'-6", the middle range for 'overweight' is just over 12 stones (who actually weighs themselves in stones?). So, let's round that up to 12.5 stones, or 175 lbs.

Okay, so now we've got Newton's weight on Earth as 175 lbs, which is equivalent to 79.5 kg. With that in hand, let's go through one example calculation for Earth, just to double check that we're doing everything correctly.

F = G*m1*m2/r^2
F = (6.67e-11 m³/kg-s²) * (5.97e24 kg) * (79.5 kg) / (6,371,000 km)^2
F = 781.3 N

In normal units, that's 175.65 lbs - close enough to my original estimate once you account for rounding errors, so it looks like everything's being done correctly. But that means, on Earth, Newton would have weighed far more than a Newton. The moon's smaller. What about it? Well, once you go through the calculation for the Moon, it turns out to 29.06 N - still too much. Even the dwarf planet of Pluto has too strong of gravity. Below is a table showing various bodies in our solar system, and how much Newton would weigh on each one (I also included pounds for the people like me who don't have a good feel for Newtons). Just so you know, those last three bodies are moons of Saturn, and they're all more or less round.

 Body Mass, kg Radius, m Newton'sWeight, N Newton's Weight, lbs Earth 5.97E+24 6,371,000 781.34 175.65 Moon 7.35E+22 1,737,100 129.28 29.06 Pluto 1.31E+22 1,153,000 52.12 11.72 Enceladus 1.08E+20 252,100 9.02 2.03 Mimas 3.75E+19 198,200 5.07 1.14 Janus 1.9E+18 89,500 1.26 0.28

So, you have to get down to something as small as Janus, which only has a diameter of about 180 km, before Newton would weigh roughly a Newton. That's not a lot of force.

Image Source: Photoshopped from IGS.net and Wikimedia Commons