Today I posed a puzzle that Isaac Newton posed in 1679 in a last writing to Robert Hooke. I mentioned that the answer is counterintuitive and also gave them the hint that it might have something to do with figure skating. Here is the problem again, with its solution. The Fall of the Tower The first known diagram of the puzzle, by Claude Dechales Cursus seu Mundus Mathematicus (1674). An object is dropped from a high tower. Taking into account the rotation of the Earth itself, where will the object land? slightly to the west slightly to the east exactly at the bottom Solution 2. slightly to the east. The Earth rotates from west to east. Our first thought might be that the object falls slightly to the west, on the basis that as the object falls, the Earth will turn slightly to the east. But such an argument ignores the fact that the object originally, like the tower itself, rotates with the Earth. Our second thought might be that it lands exactly at the bottom of the tower. Wrong again! The object lands slightly to the east because, as it falls, it spins around its axis a little faster, essentially using the same mechanism that causes an ice skater to spin faster when he pulls on his arms. Thus, the object lands slightly to the east. Illustration from Wonders in Motion by David Acheson The effect is small, but not absurd. If the Eiffel Tower were at the equator, the deviation to the east would be about 11 cm. In more technical details, the result is derived from the conservation of angular momentum, which is a measure of the momentum of an object rotating around an axis. When an object falls from a tower, gravity pulls it toward the Earth’s center of mass. However, the object also rotates in a circle around the center of the Earth. As the object falls, its angular momentum remains the same. Which means that the expression wr2 – where w is the angular velocity of the object and r is its distance from the axis of rotation (in this case, the center of the Earth) – remains the same. Since the value of r decreases during the fall, the value of w must increase, which means that the object spins faster. If it rotates faster than the Earth, it will land slightly east of the position directly under the tower. Thanks to David Acheson for today’s puzzle. He is the author of a new book, Wonders in Motion, The Story of Dynamics, which, among other things, explains in more detail the background of Newton’s puzzle. I hope you had fun. I’ll be back in two weeks. Wonders of Motion by David Acheson is on sale at Guardian Bookshop. I’ve been doing a puzzle here every other Monday since 2015. I’m always looking for great puzzles. If you would like to suggest any, please email me.
