Happy New Year, I need help sorting out my thinking on this one. The
problem as posed:
"An elevator cab is suspended from a steel cable. A pendulum inside
the cab hangs from the ceiling of the cab on a string. The pendulum is
set to swinging and has a period T while the elevator cab is stationary.
Suddenly, the steel cable supporting the cab breaks (or is cut) at
precisely the moment when the pendulum bob reaches its *maximum
speed*. Describe the pendulum’s subsequent motion from the point of view
of an observer in the elevator and also of an observer on the ground.
What is the period of the pendulum as the cab falls?"
(We imagine that the elevator roof has a slot or some opening which
allows the pendulum to swing outside of the elevator while it's falling
and then back in. The pivot point is some frictionless shaft bearing
that allows the pendulum to swing in a complete plane)
Me:
Elevator:
When the cable breaks everything goes into free fall (“no gravity”). The
bob drops like every other part of the elevator but it has a horizontal
speed at the moment of the break. The bob moves to the side as it drops
in such a way that the original tension at the break is intact and the
string stays taut at length L (if the bob was/freely/moving its distance
from its pivot would be > L); thus tension is maintained and the bob
moves in uniform circular motion with a tangential speed v = sqrt(2gL).
Ground:
Everything about the pendulum must/look/the same – the pendulum string
can’t go slack for one observer and not the other (yes?); however, the
bob’s path doesn’t look like a circle from the ground – the bob follows
a vertical cycloid that accelerates down. Punchline: an accelerating
cycloid means the bob is under a non-uniform tension.
This is my key issue: if this analysis is correct then we could choose a
value of g for which the pendulum string doesn’t break for the elevator
observer but could break for the ground observer. Or maybe my analysis
is incorrect.
PS the “period” question is interesting. Although the “restoring period"
T ~ sqrt(L/g) goes away (infinity for g = 0), the pendulum still has a
period due to circular (or cycloidal) motion.