Gyro Prank?

Question

Here's an advanced-level question: An often recounted tale tells of the physicist who hid a large spinning flywheel inside his suitcase. The hotel porter took the suitcase and proceeded to walk around a corner to the left. What happens to the suitcase?

a) The suitcase simply pivoted around the corner in the way the porter wanted it to go, with the front of the case pivoting left and the back of the case pivoting right.

b) The case tended to pivot in a way exactly opposite to the way the porter wanted it to go, with the front of the case pivoting right and the back of the case pivoting left.

c) As the porter went around the corner the case pivoted top over bottom, with the top of the case pivoting left and the bottom of the case pivoting right.

d) As the porter went around the corner the case pivoted top over bottom, with the top of the case pivoting right and the bottom of the case pivoting left.

The flywheel is rotating vertically.

The flywheel also rotates forwards in teh direction of travel.

Yes, the porter is carrying the suitcase in his left hand when he is viewed from behind.

The answer is d. Gyroscopic motion is quite complicated, but we can simplify it if we picture the spinning flywheel as a round pipe or tire or doughnut in which some heavy liquid is circulating. Next, picture the ring made into a square ring. The liquid circulates by flowing up one side, then horizontally over the top, then vertically down the other side and then horizontally back on the bottom.

Next, picture the "square" flywheel pivoting from a position in line with the porter at his side (we'll call this position 1) to a position in which the front angles to the left and the back angles to the right as the porter goes around the corner to the left (we can call this position 2). The part of the liquid flowing up and down in the vertical arms of the pipe does not change its direction of flow as the square flywheel or loop pivots from psition 1 to 2 -- the vertical sides remain vertical. But the liquid flowing in the horizontal arms does change its direction of flow.

For example, the liquid in the top might begin flowing in a counter-clockwise direction forward as the porter walks with the case and finish in a counter-clockwise direction but angled to flow towards the new direction of motion as the porter pivots the case to turn.

A portion of liquid flowing in the top pipe or bottom pipe is actually forced to travel in a curve as the porter turns the loop. Now, when a thing goes through a pipe which is turning it exerts a force on the side of the pipe. (Remember my Carousel question?). Why? Because the thing tends to go straight, but is forced to turn. The direction of force on the sides of the pipe would be counter-clockwise. The curves on the top and bottom would be opposite and so the dirction of the force on the top and bottom are opposite. The force on the pipe is the same as the force on the flywheel and that force is communicated to the suitcase. The top tilts to the right and the bottom to the left.

Answer 1

Assuming the flywheel is rotating vertically, the answer would be "b". That would have been fun to watch.

Answer 2

First Draft (ignorable)

Four physicist each gave their suitcases to the porter, each contained gyroscopes.

The suitcase first suitcase simply pivoted around the corner in the way the porter wanted it to go,. (gyro axis is pointing up)

The second case second tended to pivot in a way exactly equal and opposite to the way the porter wanted it to go. (This assumes that the porter was unable to cause a significant deviation in the orientation of the gyro)

The third case pivoted top over bottom, with the top of the case pivoting left and the bottom of the case pivoting right.

The fourth case pivoted top over bottom, with the top of the case pivoting right and the bottom of the case pivoting left.

At which point the porter, who was an off duty policeman, pulled out his gun and promptly shot the last three suitcases.

And to think I have problem with my car pulling when I step on the gas. All of the above are possible answers depending on the rotation and orientation of the gyroscopes :-)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Edit 1 (ignorable)

Ah, new details emerge, the flywheel is has assumed a state of rotating vertically, ...................

I also realize now, that the problem probably has to do with the suitcase hanging from the porter's hand.

(Is this right? I had initially assumed that the porter had placed the suitcase on one of those carts they push around or was carried it under his arm. Basically, no torque on the flywheel frame in a direction perpendicular to vertical and the answers I gave, above, reflect that thinking)

If this is the situation, the answer is "d" where the flywheel is rotating counterclockwise when viewed from above and "c" when rotating clockwise. (I will have to double-check tonight) The suitcase will also 'tend' to stay upright as opposed to tilting due to the centripetal force.

As an aside, with a vertically rotating gyro, under the situation I initially imagine, (the porter not swinging the case from his arm), the only effect I can think of is the temporary acceleration or deceleration of the gyro. My gut feeling would be "d," but I have to think this through and the effect should be minor. And I don't think this is the question. Back to my daytime job.

>>>>>>>>>>>>>>>>>>>> Edit 2 (ignorable)

Funny, I pulled out my low speed 700 mm flywheel (bike wheel) last night and ran a couple of tests. The result is that my intuitive flippant answer, stated above, for a vertical flywheel was correct.
The flywheel remains vertical. The centripetal force is so small compared to gravity that it caused minimal deviation rounding the corner. (Not very surprising). So I tried it at a higher speed and got a small inward tilt (towards the center of the curve) with a slight twist counter-clockwise (from above) and a slight rollover with the top front edge going backwards. The flywheel was spinning counter clockwise when viewed from above.

The result was pretty much what I expected. The most noticeable result for the observer, however, was not any rollover effect, which was minimal, but the rigidness of the flywheel in maintaining its vertical orientation.

I think now the correct view of the problem as laid out with the additional information is that the porter walks down the hall. As he starts to make his turn, the suitcase which is on the inside, decelerates, and as the bottom swings forwards, the case twists slightly inward towards the porter, but the big thing the porter notices is that the bottom twists forward less than expected (the suitcase noticeably remains vertical). The porter than more or less rotates around the suitcase which rotates freely. The porter then begins to walk down the new hall, there is a slight acceleration the suitcase swings back and the top of the suitcase pulls away from the porter slightly but if he notices anything, he notices is that the suitcase tends to remain vertical when it swings back.

So the answer, given the fact pattern, remains "a."

But let me add that the flywheel will behave almost identically to the front wheel of a bike wheel (at least while turn towards the left, assuming a counterclockwise rotation). If the centripetal force of the turn on the case causes the bottom of the case to swing in slightly towards the right, just like a bike, the suitcase will pivot gently left . Because of the feedback loop between the swing and the turn, the porter will hardly notice the action. The centripetal force caused by the porter's own turn will be what causes the pivoting of the suitcase. "A" it is "a" it remains.

But just wait until the porter tries to put the suitcase on one of those suitcase stands!

>>>>>>>>>>>>>>>>>>>>>>>> Edit 3 (read)

I'll be very interested in linlyons's results. I reran my low speed flywheel experiment once more -- and with more sensitivity.

I attached about 18 inches of string to my bike wheel. Rotated it counterclockwise (I presume that by "forward," you mean the flywheel is rotating counter clockwise) guessing ~ 180 rpm. Just could not get it to do anything at walking speeds -- particularly with it towards the center of rotation (me pivoting around it). It fit answer "a" exactly.

So I then flung it out having it rotate counter clockwise around me for about 180 degrees (my garage is only so big and I didn't want to embarrass myself by stepping outside). It looked beautiful. It made this nice inward tilt (~ 20 degrees), but the tilt was far less than the angle of the string and the angle one would expect if it were rotating around me with no angular momentum. To me, it looked like the axis was angled directly at the center of rotation (but the sensitivity in this department, alas, is not great, and I am sure that it was actually angled 'ahead' of the center of rotation.) (When I tried rotating in the clockwise direction [opposite the flywheel] the axis tried to point outwards)

What I actually think is happening is that the orientation of the 'vertical' axis of the flywheel is pointing ahead of center of rotation (the center of rotation being me). This creates a self-correcting phenomena, which, I think, is the key to understanding what is happening. There is some instantaneous point up and in front of the center of rotation that the flywheel is preceding around. What this means is that the torque I apply and the perpendicular output force are both corrective in moving said instantaneous point of precession so that the effective rate of precession matches the angular velocity I am applying and the flywheel maintains the same inward slant and forward slant (relative to my rotation of it). Pretty cool -- I think I'm right, although the terminology might be a little off.

This self-correcting phenomena is not too surprising. Bicycles make turns and they do so smoothly. (See, generally, http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics ) It is pretty rare for a rider to crash and burn. The counterclockwise rotating flyweel will display most of the same qualities as a bicycle making a left hand turn.

I note, however, in this matter, there may be some stability problems given the high angular momentum to mass ratio in the flywheel suitcase combo. Basically a lag in pivoting right to left and an overly vertical orientation. The so called instanteous point of precession would be too far in front of the center of rotation (around myself or the porter). Note: Precession is slower with greater angular momentum.

Generally, this would mean is that suitcase would tilt inwards (but not as much as centripetal acceleration would cause if there were no angular momentum) and rotate right to left. But the rotation right to left (counterclockwise) would trail the overall rotation by a fixed angle such that the 'vertical' axis of the rotation of the flywheel would point in front of the axis the porter is rotating around. (This, again assumes counterclockwise rotation of both the flywheel and the porter). But when the porter finished his turn, it would again self correct. (If the flywheel were going clockwise all of the above would be true except that the 'vertical' axis would point down)

THE FINAL ANSWER:

Given that the suitcase is near the center of rotation and that the acceleration will be much less than g and fairly high angular momentum to mass ratio, my money remains on "a" -- but as the suitcase moves farther from the center and experiences greater inward forces, and assuming both the flywheel and the porter's rotations are counterclockwise, it will begin acting like "c" (top over bottom with the top rotating towards the center of rotation (left)), but the angle of tilt will be less than if the flywheel were not rotating (i.e., it will remain more vertical).

Additional note, the more angular momentum compared to mass in the system, the greater the tendency for the suitcase to remain upright and simply pivot left to right, again, "a" is the answer even if it is swung 'hard' by the porter.

Notwithstanding, the porter would find disconserting both (i) the tendency for the suitcase to stay upright and (ii) the way the suitcase lags behind his own pivoting right to left. Here, the porter would need to apply a right to left twist with his hand in order to make the system work.

(As an aside, I must confess that I read answers "c" and "d" too quickly and I presumed they described a motion significantly different then they actually do. This is why I didn't immediately jump on "c" as the alternative to "a" in my second edit, above, even though I described "c" to the tee (the paragraph ending ("'A' it is 'a' it remains") and I mixed up "c" and "d" in my first edit above. My error and my wasted time)

Note to Dr. H: Wonderful problem, got me thinking of things I have not thought about for a long, long time. Had a great time trying to figure it out.