# Homework Help: Simple pendulum in a lift

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1. Apr 14, 2015

### philk

1. The problem statement, all variables and given/known data
This is a 'random discussion' that I had today with a student; it is not out of a textbook, nor does the solution carry any weight at all (pls excuse pun).

A simple pendulum is happily swinging back and forth attached to a pin in the wall of the lift, where the pin is more than the length of the pendulum, below the lift roof.
Suddenly the lift cable snaps and the lift goes into free fall (ignore air resistance). 2 questions:
(i) what is the subsequent motion of the pendulum as viewed by a person in the lift
(ii) are there any videos that verify the (surprising) answer to (i) ?
I believe that the pendulum, depending on the state when the cable snaps, either remains absolutely stationary or goes in full circles at a constant angular speed .. becoming a perfect analogue clock.

2. Relevant equations
none

3. The attempt at a solution
It seems to me that under free fall the bob becomes (effectively) weightless. This would seem to suggest the solution above but does that really happen? It seems surprising.

2. Apr 14, 2015

### RaulTheUCSCSlug

Well how about doing a free body diagram first. What forces are acting on the bob when it is suspended, and what forces act on the bob when it is in free fall. I tried to reply to your other thread, but I am assuming it just got moved to here. Draw a free-body diagram and tell us what you think.

3. Apr 14, 2015

### RaulTheUCSCSlug

Perhaps NASA may have a video of what happens to a pendulum in space, this might be the same situation.

4. Apr 15, 2015

### philk

Thanks
Yes this is how I arrived at my conclusion. I am using a 'weightless paradigm' inside the free falling lift.
If the bob is at a point of maximum displacement then T = W cos α, since v=0 so centripetal acceleration is 0. In free fall I am assuming it is correct to take the view that W=0, giving T=0 so the bob stays exactly where it started, i.e. inside the lift it just "hangs" motionless but at an angle to the vertical.
If the bob is moving when the cable snaps, then T - Wcosα = mv^2/r, and -Wsinα = mr(theta double dot) so if W=0, theta double dot = 0 so transverse speed is constant; presumably T also changes (instantaneously) so that it provides exactly the force needed to maintain centripetal acceleration.
If we shift away from the 'weightless paradigm', and view the bob from an inertial frame outside the lift, it seems to me that the analysis is hard as the pendulum is no longer moving in a circle. I don't wish to go there!

5. Apr 15, 2015

### philk

yes that would be pretty much the same except that the pendulum would have to be started by hand (or it just sits there motionless). However my question refers to a pendulum in motion and the subsequent motion; this could not be simulated in an orbiting station because simple pendula don't exist there.

6. Apr 15, 2015

### RaulTheUCSCSlug

Seems to me that outside the frame the pendulum would make almost like a over lapping circle type of motion, perhaps it could be described within a four dimensional scenario, but you are correct, it will either stay perfectly still, or spin in full circles. Although a video of this happening in space wouldn't show exactly what you want, it would show the centripetal motion that occurs when it is attached to a string and pushed in weightlessness.

7. Apr 15, 2015

### philk

Thanks for confirming that Raul.
I am now minded to do an analysis from an inertial frame to see if the path of the bob confirms this. If I succeed I will post the details here!

8. Apr 15, 2015

### RaulTheUCSCSlug

Yeah, message me if you do, I would be rather interested to see it!