Re: [Phys-L] pendulum in free fall

[John Denker <jsd@av8n.com>]

On 1/5/25 10:27 PM, stefan jeglinski wrote in part:
>  Punchline: an accelerating cycloid means the bob is under a non-uniform tension.

Carl Mungan posted the right answer very early in this thread.
And thinks to Phil Keller for the nifty graphic.

There remains the pedagogical issue: Even if you now why the
right answer is right, you still have to explain why the wrong
answer is wrong. That involves some psychology, and also some
fundamental physics.

My guess is that we are dealing with a classic case of misapplying
a formula. In the equation a = v^2/r, the "r" in question is not
the length of the string; it is the local radius of curvature of
the trajectory. As you can see in the graphic, when the elevator
velocity has become large, the trajectory is quite stretched out
(as seen in the lab frame) so even though the v^2 is large the
v^2/r remains the same.

Here's another super-fundamental yet often neglected physics
fact:  The gravitational acceleration g depends solely on
what the reference frame is doing, not on what any of the objects
are doing. Similar words apply to centrifugal force, for the same
fundamental reason: Centrifugal force exists in the rotating frame
and not otherwise. It depends solely on what the frame is doing,
not on what any of the objects are doing.

So the elevator and its contents are weightless in the frame
comoving with the elevator. Meanwhile, they are absolutely not
weightless in the lab frame. So the curvature you see in the
graphic is not due solely to the string. The string gets help
from g.

Also (!): If the string doesn't break in the elevator frame, it's
not going to break in any other frame. It's the same physics, no
matter what frame you choose. This is perhaps the most fundamental
principle of all

And yes, we are assuming a super-massive elevator falling without
friction.

People find accelerated frames confusing. Prof. de Coriolis didn't
work out the equations for a rotating frame until 1835, more than
150 years after the Principia. To this day you can find grossly
wrong derivations of the Coriolis force all over the internet.