The Nonlinear Library LW - The Parable Of The Fallen Pendulum - Part 2 by johnswentworth
Mar 13, 2024
07:34
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: The Parable Of The Fallen Pendulum - Part 2, published by johnswentworth on March 13, 2024 on LessWrong.
Previously: Some physics 101 students calculate that a certain pendulum will have a period of approximately 3.6 seconds. Instead, when they run the experiment, the stand holding the pendulum tips over and the whole thing falls on the floor.
The students, being diligent Bayesians, argue that this is strong evidence against Newtonian mechanics, and the professor's attempts to rationalize the results in hindsight are just that: rationalization in hindsight. What say the professor?
"Hold on now," the professor answers, "'Newtonian mechanics' isn't just some monolithic magical black box. When predicting a period of approximately 3.6 seconds, you used a wide variety of laws and assumptions and approximations, and then did some math to derive the actual prediction. That prediction was apparently incorrect. But at which specific point in the process did the failure occur?
For instance:
Were there forces on the pendulum weight not included in the free body diagram?
Did the geometry of the pendulum not match the diagrams?
Did the acceleration due to gravity turn out to not be 9.8 m/s^2 toward the ground?
Was the acceleration of the pendulum's weight times its mass not always equal to the sum of forces acting on it?
Was the string not straight, or its upper endpoint not fixed?
Did our solution of the differential equations governing the system somehow not match the observed trajectory, despite the equations themselves being correct, or were the equations wrong?
Was some deeper assumption wrong, like that the pendulum weight has a well-defined position at each time?
… etc"
The students exchange glances, then smile. "Now those sound like empirically-checkable questions!" they exclaim. The students break into smaller groups, and rush off to check.
Soon, they begin to report back.
"After replicating the setup, we were unable to identify any significant additional forces acting on the pendulum weight while it was hanging or falling. However, once on the floor there was an upward force acting on the pendulum weight from the floor, as well as significant friction with the floor. It was tricky to isolate the relevant forces without relying on acceleration as a proxy, but we came up with a clever - " … at this point the group is drowned out by another.
"On review of the video, we found that the acceleration of the pendulum's weight times its mass was indeed always equal to the sum of forces acting on it, to within reasonable error margins, using the forces estimated by the other group. Furthermore, we indeed found that acceleration due to gravity was consistently approximately 9.8 m/s^2 toward the ground, after accounting for the other forces," says the second group to report.
Another arrives: "Review of the video and computational reconstruction of the 3D arrangement shows that, while the geometry did basically match the diagrams initially, it failed dramatically later on in the experiment. In particular, the string did not remain straight, and its upper endpoint moved dramatically."
Another: "We have numerically verified the solution to the original differential equations. The error was not in the math; the original equations must have been wrong."
Another: "On review of the video, qualitative assumptions such as the pendulum being in a well-defined position at each time look basically correct, at least to precision sufficient for this experiment. Though admittedly unknown unknowns are always hard to rule out." [1]
A few other groups report, and then everyone regathers.
"Ok, we have a lot more data now," says the professor, "what new things do we notice?"
"Well," says one student, "at least some parts of Newtonian mechanics held up pretty well. The whole F = ma thing worked, and th...
Previously: Some physics 101 students calculate that a certain pendulum will have a period of approximately 3.6 seconds. Instead, when they run the experiment, the stand holding the pendulum tips over and the whole thing falls on the floor.
The students, being diligent Bayesians, argue that this is strong evidence against Newtonian mechanics, and the professor's attempts to rationalize the results in hindsight are just that: rationalization in hindsight. What say the professor?
"Hold on now," the professor answers, "'Newtonian mechanics' isn't just some monolithic magical black box. When predicting a period of approximately 3.6 seconds, you used a wide variety of laws and assumptions and approximations, and then did some math to derive the actual prediction. That prediction was apparently incorrect. But at which specific point in the process did the failure occur?
For instance:
Were there forces on the pendulum weight not included in the free body diagram?
Did the geometry of the pendulum not match the diagrams?
Did the acceleration due to gravity turn out to not be 9.8 m/s^2 toward the ground?
Was the acceleration of the pendulum's weight times its mass not always equal to the sum of forces acting on it?
Was the string not straight, or its upper endpoint not fixed?
Did our solution of the differential equations governing the system somehow not match the observed trajectory, despite the equations themselves being correct, or were the equations wrong?
Was some deeper assumption wrong, like that the pendulum weight has a well-defined position at each time?
… etc"
The students exchange glances, then smile. "Now those sound like empirically-checkable questions!" they exclaim. The students break into smaller groups, and rush off to check.
Soon, they begin to report back.
"After replicating the setup, we were unable to identify any significant additional forces acting on the pendulum weight while it was hanging or falling. However, once on the floor there was an upward force acting on the pendulum weight from the floor, as well as significant friction with the floor. It was tricky to isolate the relevant forces without relying on acceleration as a proxy, but we came up with a clever - " … at this point the group is drowned out by another.
"On review of the video, we found that the acceleration of the pendulum's weight times its mass was indeed always equal to the sum of forces acting on it, to within reasonable error margins, using the forces estimated by the other group. Furthermore, we indeed found that acceleration due to gravity was consistently approximately 9.8 m/s^2 toward the ground, after accounting for the other forces," says the second group to report.
Another arrives: "Review of the video and computational reconstruction of the 3D arrangement shows that, while the geometry did basically match the diagrams initially, it failed dramatically later on in the experiment. In particular, the string did not remain straight, and its upper endpoint moved dramatically."
Another: "We have numerically verified the solution to the original differential equations. The error was not in the math; the original equations must have been wrong."
Another: "On review of the video, qualitative assumptions such as the pendulum being in a well-defined position at each time look basically correct, at least to precision sufficient for this experiment. Though admittedly unknown unknowns are always hard to rule out." [1]
A few other groups report, and then everyone regathers.
"Ok, we have a lot more data now," says the professor, "what new things do we notice?"
"Well," says one student, "at least some parts of Newtonian mechanics held up pretty well. The whole F = ma thing worked, and th...