There's a classic problem in physics textbooks which asks you about astrology. It's sometimes said - the problem will tell you - that the gravitational pull of the doctor delivering you is stronger than that of Jupiter, therefore it's unlikely that the planets are exerting a whole lot of influence on your life. The problem asks you to check this.
Of course gravity is not generally purported to be the conduit of the supposed influence of the zodiac, but it's an interesting problem. And in fact it will turn out that Jupiter generally has the doctor beat, but not by that much. It's a neat little order-of-magnitude demonstration of the how gravity scales.
Still, I'm not a huge fan of the problem for different reasons. The problem almost always just uses Newton's equation for the gravitational force due to a point object. A doctor isn't - his head is farther from the baby than his hands, and so are his feet, and so on and so forth. As such it's pretty hard to figure out what the "distance" is from the doctor to the baby. Using the center of mass would be a very good approximation if the distance was much longer than the doctor's height, but it's not.
So I propose we use this classic problem to learn a little more about approximations. What if we assume that instead of being a point mass, we assume the doctor's mass is evenly distributed along a line? Sure it's still a "spherical cow" type of situation, but it's a lot better than the previous point approximation. The situation will look something like this, with the blue dot representing the baby and the black line representing the doctor. The distances are labeled:
Each little bit of mass on the line will contribute a bit of gravitational force. I've picked out one representative bit and labeled its distances specifically, and the sum of all the forces from all the bits will be the force on the baby. From Newton's law of gravitation for a point mass, the force from one bit will be equal to the gravitational constant, times the mass of the first object, times the mass of the second object, divided by the square of the distance between those objects. So calling the baby's mass M, the mass of that bit of the doctor dm, and plugging in the square of the distance:
However, this is the force toward that mass dm. Because I've drawn the situation symmetrically with the baby even with the doc's center of mass, the upward component of the force from dm will be exactly canceled by the downward component of the force from the bit of mass located at -h. So only the horizontal forces contribute since they're all pulling in the same way. To get that force, we multiply by the cosine of the angle between the line r and the hypotenuse. Cosine is equal to the length of the line opposite the angle divided by the length of the hypotenuse, so plugging that in we see that the total horizontal force is:
It is in general pretty difficult to add up this quantity mass by mass. We'd prefer to do it length by length, so let's express dm in terms of the mass per length - ie, the density λ = m/L so that dm = λ dh, where dh is the tiny height of that bit of the mass. Doing that and collecting terms:
Now add up all the little lengths dh from the bottom to the top by integrating:
It would be a tremendous pain to typeset the solution to this integral, as it's somewhat involved. I'll just present the solution, but for those interested in trying it yourself the best method is trig substitution. Let r = h tan(u) and go from there. After the fireworks are done, you'll get:
So assuming an 8 pound baby and a 180 pound doctor with a height of 6 feet standing 1 foot from the baby, the total gravitational force is about 3.36 x 10-8 newtons. Way too small to detect, but it's a real number. At its average distance from earth, Jupiter would produce a force of about 7.6 x 10-7 newtons, or a little more than 20 times more force. Twenty times pretty much nothing is still pretty much nothing, but still - Jupiter wins.
I still can't say I'd recommend going to an astrologer though...
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But the Moon exert a force a few order of magnitude greater than the good Doctor or Jupiter. So, the astrologic effects of the Moon should dwarf the effects of any other "planets", except the Sun.
Quite true, but oddly the newspaper horoscopes never bother with the moon. Go figure.
If gravity is so important in astrology, why is it ignored? For example, distance to Mars varies between 0.5 and 2.5 AU with a two year cycle. That is a 25-time difference that repeats regularly. And yet, astrologers only talk about the constellation behind the planet.
"And yet, astrologers only talk about the constellation behind the planet."
Actually, they talk about constellations behind the Sun, or, more precisely, the constellation that was behind the Sun the month before.
The force from Jupiter you calculated can't, even in principle, have any effect. It will pull on the baby, the crib and the entire Earth by the same amount, accelerating the entire planet slightly towards Jupiter. In the local frame of the baby this just isn't detectable.
If you want some gravitational effect on the baby you will have look at tidal effects, that is, how different parts of the baby are pulled with different force causing a slight stretching effect. Tidal forces drop off as 1/r^3, which means that the doctor will have far, far more effect than Jupiter.
My argument is that all matter subject to gravity is in constant connection, as evidenced by gravity's infinite speed.
Nonetheless I feel compelled to take in other points of view. How many examples do we have of a proven gravitational effects, in those cases where neither body is rotating?
When I took a course in geophysics in the mid 1950's the professor told us of a gravity meter which could detect differences in gravity at points one cm apart. How about that?
Right. The mining industry uses airborne gravity meters of some sort, as a way of building evidence for what may be in the deep ground below. They do a flyover of the territory and then they go and analyze what the gravity meter has told them.
Graeme Bird: The speed of gravity is not infinite. It propagates at the speed of light. Rotation is irrelevant for spherically symmetric objects, and we can measure the gravitational force between masses (rotating or not) in the laboratory.
"The speed of gravity is not infinite. It propagates at the speed of light."
No thats not right. You are probably lying but you may be ignorant. Prove it.
Secondly I'm sure this alternative theory of gravity is wrong. But I assure you that the lab and all the objects in it are rotating. Although I do see that as a minor point.
You are laughable Brennan. You are a kook. A nut. You just said something entirely wrong, with utter certainty, based on nothing. You are typing checks your brain cannot cash. Scroll up crank. You will see that I've proved you wrong in advance.
We've already had the argument and I won it.
Scroll up? No. Got to the last thread instead. Sorry for being excessively insulting. I lost track of which thread I was on and so thought that you were merely repeating the mindless dogma of the last thread.
"The speed of gravity is not infinite. It propagates at the speed of light."
But this is a very silly thing to say.
Graeme, getting flustered are we? Please explain this Gaede Rope theory that you have been blathering about. What is the evidence for gravity having infinite speed?
I'm sorry people. I seem to have dragged along an internet stalker. Go to the last thread stalker.
When does asking questions, being insulted, having my posts changed and deleted on your blog constitute "stalking"?
You have no credibility, so your claims have little validity.
You were the one spamming scienceblogs weren't you?
To Matt Springer and others, I don't want to mess up your blog, but I just wanted to paste this one comment in from Graeme Bird's blog to show how deluded he really is. He posted this over a comment of mine that he deleted. I'll leave it that.
DAWKINS IS A VERY SMART MAN. BUT IâM AT LEAST AS SMART AS DAWKINS. IN FACT HIS ANTI-RELIGIOUS ARGUMENTS ARE PRETTY OLD HAT TO ME. THEY WERE THE SAME ARGUMENTS I MADE AS A TEENAGER. I MIGHT EVEN BE THE ONE TO SHOW HIM WHERE HE IS COMING OFF THE BEAM.
FOR EXAMPLE, ITS JUST A FACT THAT THE BIG BANG AND EVOLUTION ARE NO LONGER COMPATIBLE. THIS YOUNG EARTH CREATIONISM THAT BIG BANG THEORY REPRESENTS, PRESENTS TO US A UNIVERSE WAY TO YOUNG TO EXPLAIN THE COMPLEXITY WE HAVE VIA EVOLUTION. NOW IT OUGHT TO BE PRETTY EASY TO EXPLAIN THIS TO DAWKINS. BUT THE FACT IS THAT HE DOES NOT SEEM TO HAVE NUTTED THIS OUT FOR HIMSELF.
PLUS HE DOESNâT SEEM TO HAVE FIGURED OUT THE IMMENSE HARM OF VARIOUS SECULAR-RELIGIONS. OTHERWISE HE WOULDNâT BE SO NUTTY WHEN IT CAME TO CHRISTIANITY. AND I THINK IâD BE QUALIFIED TO MAKE THIS CASE TO HIM.
IT HARDLY MATTERS IF HE THINKS IâM A NUT WHEN IâM PROBABLY SMARTER THAN HIM.
Just ignore this stalker. He's a real kook but hopefully he'll go away.
It's pretty well-known, Electric, that Bird is a completely deluded psycho who's arguements on just about any topic are to be traeted with the mocking derision they deserve.
The guy's a nut.
Stalkers in stereo. You know this fellow was able to access maybe 50 IP Addresses in just two or three hours.
The gravitational force shouldn't matter much, since the baby and the table and the whole Earth are freely falling toward Jupiter with the same acceleration. It's the tidal force that you should be looking at. Then the doctor really dwarfs the effect of Jupiter.
All that nonsense from people who allegedly read your article and not one of them noticed that you have a strange graphic typo in it? Your first equation is some sort of left-over from your earlier article about orbital speeds, not the force equation you said it was.
Happy to help, and I liked your next-level analysis of this classic little homework problem.
Memo to _Arthur:
Astrologers also look at where all of the planets were when you were born, not just the sun.
Thanks CC, I've fixed it.
Numerology : Vedic astrology is an ancient Indian science which explains planetary motions and positions with respect to time and their effect on humans.
The answer to the imprinting of the natal astrology chart (i.e., the location of the planets in relation to the location on earth that the birth takes place) on a newborn infant is Gravity Waves.
The gravity waves are constantly moving and shaping differently around our planet and on surface areas of our planet and when a child is born, apparently the gravity wave shape that surrounds that locale is imprinted on the child, perhaps because of all the water in the body, and the hemoglobin, and the cells with electric charges... it all gets "permanently aligned" if you will, because of the gravity waves, or so this is my latest theory anyway. It made me wonder, is it possible that there is zero-G in the womb? If so, perhaps a person's first exposure to gravity causes this imprinting to occur.
I welcome feedback, so please feel free to email me back, physisists - email@example.com
The effect of the doctor down at the "Business end" of the mother has twice that of Jupiter.
And he moves a lot quicker in angular displacement.
And the strongest gravity wave source (a dead binary star system) has less energy reaching earth in a year than a match struck on the moon.
""The speed of gravity is not infinite. It propagates at the speed of light."
No thats not right. You are probably lying but you may be ignorant. Prove it."
If gravity affected someone before the light reached them, then gravity would have to be a tachyon.
However, it isn't.
It would also cause many problems: all other effects propogate at light speed, so you'd be pulled by a black hole in a place where there is no black hole.
I'm afraid that since YOU determine this rather amazing claim that gravity is faster than light, you need to prove your claim.