Equivalence Principle

Galileo famously demonstrated, or at any rate talked about demonstrating the equivalence principle by dropping heavy and light balls off the leaning tower of Pisa. Absent friction, the balls should fall at the same rate and reach the ground at the same time. WB has recently posted a puzzle on this subject. Since I found the puzzle hard, mainly because of what I considered miscelleaneous complications, and because Wolfgang didn't like my solution, I am posting my own more idealized, and hence slightly different, version here.

Consider a spherically symmetric matroyoshka Earth, with a very small inner sphere at its center, and a still smaller sphere at the center of it. The total mass of the Earth is M, the mass of the middle spherical shell (not counting the innermost sphere) is m, and the mass of the innermost sphere by itself is m1. Consider the following two experiments: (1) The spherical shell of mass m is separated from the Earth and raised to a distance between its center, and the center of the Earth, still occupied by the innermost sphere, is R. It is then dropped from rest, and the time until contact recorded. (2) The spherical shell, including its inner sphere (mass m + m1) is similarly raised to the same distance of separation and dropped, and its time recorded.

Note that the spheres dropped differ in mass, but not in diameter.

Will the two spheres have exactly the same time of fall? Assume no friction or external forces. For hints and answers consult WB's post and attached comments.

Comments

Post a Comment

Popular posts from this blog

Anti-Libertarian: re-post

The most pointed critique of social security and liberalism in general is the libertarian critique. The problem with libertarians, for a liberal, is that we have too much in common. We both believe in individual rights, tolerance of individual differences, and dislike government prescription of religion. The basic difference, it seems to me, is the different answers we give to Cain's (with thanks to the Captain - my original version had the corpse asking the question!) famous question: "Am I my brother's keeper?" That's not really my basic bitch against the libertarians though. My real complaint is the same as my complaint against most religion - its premise is a fraud. For those who can't stand to wait for the punchline, I believe that trying to implement libertarian principles leads to tyranny or social disintegration. Demonstrating that takes some historical (and pre-historical) context. For all but the last 15,000 or so of the 100,000 years the human spec
Read more

Uneasy Lies The Head

Uneasy lies the head that wears a crown ........... Shakespeare Henry IV, Part II. So it is with us and our slightly distant primate cousins , so it seems. A 9-year study of wild baboons in Kenya by Princeton University and the Institute of Primate Research may turn conventional assumptions of alphas males upside down. The study found that alpha males have high testosterone levels, which allows them to dominate other baboons, get access to more food, and attract better mates. This is expected. Surprisingly, though, the alpha males also have high levels of stress hormone glucocorticoid, which scientists measured from their feces samples. In fact, the stress of alpha baboons was on par with low-ranking baboons. The convention wisdom is that access to food and females provides life security and thus less stress for dominant males. Conversely, weak males who are constantly threatened by starvation and physical violence from stronger males are thought to be more stressed. The st
Read more

We Call it Soccer

The US played its best, and maybe only good game of World Cup, but lost thanks to brilliant goalkeeping by the Swedish keeper and the bizarre method soccer uses to settle ties when three US stars could not put the ball on frame from the 12 yard spot.  At the end it came down to a blocked shot that spun in by maybe a couple of centimeters. Well, that's football - too much like life.   Which brings me to a modest proposal.  Replace the thirty minute overtimes with as many 7-player 15 minute periods as needed to settle it.  Scoring should be much easier with fewer players.
Read more