Here is a puzzle to amuse you. It’s a smallish question on which you might like to test your math. My mathematical skill is low, or slow; I work things out cautiously and am in danger of missing a step. Perhaps this is a case where I did.
John Lash is a philosopher of comparative mythology (http://www.metahistory.org/siteauthor.php). He says he has made long use of my Astronomical Companion, is particularly interested in the motion of the Moon, and is fascinated by the thing called the barycenter. We think of the Moon as revolving around the Earth, but, as described by me in this paragraph,
“A truer picture… is that both Moon and Earth revolve around the barycenter (center of mass) of the Earth-Moon system; and it is this barycenter that revolves in a smooth orbit around the Sun. The Earth has 81.3 times as much mass as the Moon. So their barycenter is 1/81.3 of the distance from the center of the Earth to the center of the Moon. This proves to be inside the Earth, 4728 km from the center and 1650 km (1025 miles) below the surface. There is no one particle that is the barycenter: the Earth rotates, so the barycenter keeps traveling (at an average speed of XXX km/hr) through the Earth’s mantle, staying always below the Moon. If you happen to be in a tropical country, then at some time in the month the Moon will pass over your head, and at that moment you can tell yourself that the barycenter of the Earth-Moon system is a thousand miles under your feet, gliding through the rock at the speed of a fast jet plane.”
Detail from the Astronomical Companion illustration. The Moon, about 60 Earth-radii away. is about three picture-widths off to the left.
“XXX” stands for the speed I gave. John says it can’t be correct, in fact is vastly too great. Since I didn’t keep a note of how I arrived at my figure, whereas he has devoted hours or years to pondering such matters, I’m predisposed to expect that he’s right.
So how about this? Calculate for me the speed of the barycenter through the solid Earth.
These are the figures I think you need:
–The Moon goes 360 degrees around the sky, that is, returns to the same place against the starry background, in 27.32 days on average (its sidereal period).
–The Earth rotates once (that is, turns 360 degrees) in 24 hours. Actually that’s the solar day; the sidereal day, in which a point on Earth comes around to facing the same direction among the stars, is 23.93 hours; but I think that if you use 24 your answer will at least be of the right order of magnitude.
–The Earth’s equatorial radius is 6378 kilometers.
–The barycenter is 4728 kilometers from the center of the Earth. (Another average, varying slightly with the Moon’s distance, but that doesn’t matter.)
–The circumference of a circle is the radius multiplied by 2 multiplied by pi (3.1416).
If you want to convert kilometers to miles (as John did), divide by 1.609344.
Let us know your answer!