Science & Nature Archive

Friday, June 7, 2013

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

Thursday, May 23, 2013

Tuesday Boy Problem Solved by Simulation

The other day, I came across a logic/math problem I hadn't heard before, The Tuesday Birthday Problem. It goes like this:

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

This puzzle was apparently first presented at a convention for mathematicians, magicians and puzzle enthusiasts (yeah, that's a pretty specialized convention) by Gary Foshee. Immediately after giving the puzzle, he followed up with this.

The first thing you think is 'What has Tuesday got to do with it?' Well, it has everything to do with it.

I know my first inclination was to dismiss that extra fact. How could it have any effect on the probability of the sex of the other child. I first read this puzzle late at night when I was tired, so I didn't feel like putting too much thought into it. Instead, I just read the explanations of how that extra bit of information alters the odds. But I still wasn't ready to buy those explanations just yet. But rather than try to think through the explanation that night, I decided to tackle it from a different angle. Instead of trying to figure out the odds, I'd just program a simulation and see how it played out.

In fact, this is a very simple simulation. I didn't program it in the most efficient manner, but it got the job done. Here's what I did. I created a 4 x 10,000 element array. That is, 10,000 sets of kids, with four pieces of information to designate sex and birth day of the week for each kid (sex 1, day 1, sex 2, day 2). Then, I randomly assigned sex and birth day to each of the kids. Next, I created a couple variables that would be filled in in the next stage. First was a variable keeping track of the number of sets where at least one was a boy born on a Tuesday - that is, the number of sets where the father would have given his first statement. The other variable was the number of sets with a boy born on a Tuesday and another son - the sets fulfilling the second statement. With the array and variables in place, I went back and did some if statements to simulate the father's conditions, increasing the totals of those variables as appropriate. When that was done, I simple divided the number of sets with kids with a boy born on a Tuesday and another son by the number of sets with at least one boy born on a Tuesday.

After running this program a few times, I found a small problem. 10,000 sets wasn't enough. The fraction was varying by several percentage points each time I ran it. So, I added one more feature to allow the program to keep a running average every time it ran.

Oh, and just to be sure I was doing things properly, I added a similar set of calculations to calculate the probability for a simpler puzzle:

I have two children, one of whom is a son. What is the probability that I have two boys?

This is much easier to understand, so it was my control to make sure the algorithm was working properly.

Warning: Don't read on if you want to solve the problem on your own, first.

Well, guess what I found out. After running the simulation on 100,000,000 sets of kids, I got a probability of 0.4813391 for the Tuesday boy problem, and 0.3333046 for the simpler boy problem. Those are very close to the actual odds of 13/27 (0.481481481...) and 1/3 (0.33333333...). It's pretty counterintuitive, but I guess those eggheads know what they're talking about, after all.

Image Source: Wikimedia Commons

Anyone interested in checking this out for themselves can download my program below:

Thursday, September 27, 2012

Mars Curiosity Rover - Is It Worth the Price Tag?

Here's a short article I got started on back when the Curiosity Rover first landed, but then kind of forgot about and let linger. But, it's still relevant, so I've decided to finish it off and post it.

Whenever there's any type of science project in the news that doesn't seem to have immediate practical applications, some people inevitably ask why the research is being done. And when the price tag seems high, then even more people pose the question and lament the 'waste' of money.

I've written on this subject a couple times before. In this entry, Knowledge for Knowledge's Sake, I made two points defending science. First, as the title of that post suggested, that knowledge in and of itself is enough of a reason for some of us. "In the same way that some people may find beauty in a painting, others can find beauty in a deeper understanding of the mysteries of our universe." The other point was more pragmatic, that we don't always know where research will lead, and that there may actually be practical applications that we can't anticipate right now. Do you think that Albert Michelson and Edward Morley had any idea that their experiments looking for aether were one link in the chain that would eventually led to the GPS in my iPhone? My other entry on this subject, Why Study the Higgs Boson?, was mostly linking to other people making the same points, but more eloquently than I could. For example, I quoted Steven Weinberg, in reference to 19th century experiments on electricity, "If these physicists had limited themselves to work of obvious practical importance, they would have been studying the behavior of steam boilers."

So, those same points hold for the Curiosity Rover. But what about the price? The mission cost on the order of $2.5 billion (that's the American billion, or$2.5 thousand million for those of you using the long scale). That's a lot of money. Is knowledge for knowledge's sake really worth that much?

Let's look at some comparisons. The national budget proposed for 2011 was $3.69 trillion. The defense portion of that was$738 billion. Social Security was about the same. Medicare was $498 billion. So the Curiosity Rover was only .07% of the national budget, .3% of the defense budget (same for Social Security), or .5% of the Medicare budget. We're talking about a miniscule part of the budget. Here's another comparison. Avatar (the movie) grossed$2.78 billion. That single movie grossed more than the cost of the rover. The next highest grossing movie, Titanic, was just about there with $2.19 billion. And several movies over the past two years have grossed over$1 billion. So the cost of the latest Mars rover would be covered by just one or two blockbuster films.

So yes, I think the Curiosity Rover was worthwhile. Whether or not the knowledge it yields will ever lead to practical applications, its overall cost is tiny compared to everything else the nation spends money on. And the cost seems especially reasonable when you consider that people were willing to pay more to watch a movie about visiting another planet than what it cost to actually send a robot to explore another planet.

For some reason, I had this link in the draft copy I'd saved of this entry. Maybe I had some profound point I was going to make, but that I've now forgotten. Or maybe I was using it as an example of why I think planetary exploration is important:
Interstellar Potatoes

Image Source: NASA

Thursday, August 9, 2012

The Roots of Morality

I don't often have posts that are little more than embedded YouTube videos, but this one was too good to pass up. A few months ago, Frans de Waal gave a TED presentation: Moral behavior in animals. I'd highly suggest following that link to watch his full presentation, but one of the videos he showed has been pulled out into it's own YouTube video. The video is of an experiment with capuchin monkeys. These monkeys had been trained to return a rock that a researcher gave them in exchange for a treat. It's important to note that capuchins like certain treats more than others. A piece of cucumber is decent, but they really like grapes. I suppose it would be like the difference in getting a piece of hard candy from your grandmother vs. crème brûlée in a 5 star restaurant (or substitue according to your tastes). In this particular experiment, there were two monkeys involved, each in a separate cage, but adjacent to each other so that they could see each other. The first monkey returned the rock to the researcher, and received a cucumber in return as a treat. The second monkey returned the rock to the researcher, but received a grape in return, which the first monkey clearly saw. Next, the researcher went back to the first monkey, and again gave it a cucumber in return for the rock. Watch the video below to see the monkey's reaction.

This may not be a full sense of morality as developed in humans, but it's certainly a part of it - recognizing an unfair situation. It amazes me just how human like the monkey's reaction is. It reminds me of how a young child with poor impulse control might react.

Now, I know there are dangers in over anthropomorphizing, but really, when we're so closely related to an animal, doesn't it make more sense to think that their thought processes are at least similar to ours, rather than thinking that humans evolved all these brand new and novel characteristics in an evolutionary blink of an eye?

I'll note that I first saw this on Jerry Coyne's Why Evolution Is True. Follow that link to read some good discussion of the video (along with some rather close minded remarks by one particular commenter).

Tuesday, July 17, 2012

Why Study the Higgs Boson?

With the recent news over the probable discovery of the Higgs Boson, I've seen an old question come up again - What's the point of doing this type of research?

I've covered this before on the blog in the essay, Knowledge for Knowledge's Sake. That essay was in reference to dark matter, but it's largely applicable to the Higgs Boson, so I'm not going to repeat myself here. However, I've seen a few good takes from others on this question.

First is an article in the New York Times by Steven Weinberg, Why the Higgs Boson Matters. Jumping to the end, here was his conclusion:

On a longer time scale, the advance of technology will reflect the coherent picture of nature we are now assembling. At the end of the 19th century physicists in England were exploring the properties of electric currents passing through a near vacuum. Although this was pure science, it led to our knowledge of the electron, without which a large part of today's technology would be impossible. If these physicists had limited themselves to work of obvious practical importance, they would have been studying the behavior of steam boilers.

Next is an article by Jerry Coyne, which used Weinberg's article as a starting point, Steven Weinberg on the Higgs boson, and a few words on the value of pure science. Here's an excerpt of what he had to say:

But I wish we could convince the public that there are simple payoffs in understanding. Humans are curious animals: we want to know where we came from, and where the universe came from, and what we and the universe are made of. That is worth something in itself. Even if evolutionary biology had no practical benefits (and yes, there are some, but the vast amount of money given us by taxpayers to study evolution is to promote pure understanding), it would be worth spending money on, just as we subsidize the arts.

And finally, a recent comic on Saturday Morning Breakfast Cereal made the point quite humorously. Here's the first panel from that comic. Click on it to read the whole thing: