Challenge Questions

Single flip about the head is horizontal, not vertical.

I have a problem interpreting the question. Is the question about the effects of different moment of inertia in the directions of rotation, or is it about rotations about orthogonal axes?

I suspect the intention is the latter, in which case the "flip" might mean a half-rotation about the stem (or handle). Making this assumption, most throws-and-catches result in N+1/2 rotations about a horizontal axis perpendicular to the handle. Depending on your viewpoint, that second rotation will always reverse top and bottom, but not necessarily left and right.

Given the appropriate viewpoint, the deliberate left-right swap would be maintained, whereas the vertical swap would be subject to an additional swap - i.e no net swap at all.

That said, I very much doubt that I personally could toss-and-twist a hammer to produce such similar results twice - at least not without practice. Perhaps that is why you trained John to toss the hammer both times. (This appears contradictory to the remainder of the context, but I suppose it's makes sense)

Perhaps the question should have been about turning mattresses (regular half-turns about alternating orthogonal axes supposedly prolongs their lives).

Firstly, I tried this with a hammer and it does indeed do what is stated in the challenge and is consistent - claws alternate from right to left on each throw. When you toss the hammer starting with the claws facing upwards, it does not rotate from left to right or up and down.

I am betting that it has to do with the fact that the centre of gravity is off-centre in the plane of "attemted" rotation (Vertical line in diagram) when the claws are sideways on, whereas the COG lies on the plane of rotation when the hammer is thrown with the claws upright (Picture the hammer rotated 90 degrees about the vertical axis in the picture - The COG will align itself with the vertical axis due to symmetry of the hammer in this plane).

Though why you only get a half rotation about the vertical axis during a full rotation about the horizontal axis I'm not quite sure - Any ideas???

I think you nailed it. I think the half rotation might be partly luck and partly because the handle shape (oval cross section) makes it more likely to stop there when you grab it.

GA!

The reason must be the relationship between the position of each CofG and the values of the MofI relative to the holding position. If you broke the claws off you might get a full turn.

That will be the reason that I didn't see this when trying it with my old claw hammer - the head is somewhat smaller and the handle longer than on your (presumably standard) example.

P.S. This could also be the reason why as a child I found it harder to hit nails in straight in the school workshop than I had usinfg my tools at home; the further forward you place the centre of gravity, the greater will be the tendency for any slight offset in initial position to turn into twist. And to think that all this time I was blaming it on the bench height being set for the older children...

Being one who likes to beat on things until they fit, I have a good selection of hammers. I tried this with a couple of claw hammers, and then a ball peen hammer. All reacted the same way. (The ball peen hammer had a significantly larger flat head side than round head side.) This could lend support to the displaced CG theory.

However, I then tried the experiment with a hammer with a symmetrical head. I used a relatively handsome hammer with a nice patina on the wooden handle, a chromed-plated center head section bearing the name "Snap-on", with two soft faces, each the same size and translucent amber in color. These faces are replaceable, I think, although I have never had occasion to actually do the replacement... but then the hammer is only 35 years old.

The CG of this hammer is on the centerline of the handle, about 2" down from the centerline of the head. Experimenting with this hammer gave me the opportunity to become fairly adept at hammer flipping (and made me think I could become a successful slacker, whiling away hours flipping hammers when I should be hammering nails.) I became so adept at hammer flipping, that I could focus my attention on just the head with only secondary attention given to the handle, which would fall readily to hand despite my inattention upon it.

The sad truth: the symetrical hammer acted just like the asymmetrical ones, with the head not moving very far through space and making a half rotation, and the handle end going much faster through space. With practice, I was able to do consistent double flips, with the head doing a full rotation.

My vote goes to those who mention precession.

Never beat anything to fit unless you intend to paint it to match.

And, don't forget, professionals file off the parts that don't fit completely before painting to match.

You're exactly right. That's how my avatar vehicle ended up with three wheels -- the fourth one didn't fit, so I filed it off. A little paint, and no one's the wiser. It actually looks like I intended it to have three wheels.

You say there is a third wheel, but how do we know you didn't take the hammer collection to it?

I give you a good answer for this.

One was a 20 oz framing hammer and one was a 12 oz finishing hammer ?

dadw5boys: Take a careful read. They used the same hammer.

When Tom tossed the hammer its heavy head was concentric to the arc of the loop

One word: Precession.

Precession came to my mind also. That may explain why it repeats so well since precession would be dependant to a large degree to orientation change. Possibly what Wikipedia calls "torque free" precession.

Precession is the answer. An object is most stable rotating about its minor axis. When called upon to rotate about a larger axis, it's less stable, and will precess, or nutate (I'm still not quite sure if these two are synonymous) in response to any energy dissipation, which is abundant in the presence of friction, due to air resistance or John's hand. (This is why airplanes and satellites love to go into a "flat spin.")

The hammer's longest axis is a line axial to the handle (passing from the end of the handle up through the head). Its semi-major axis would be the line passing through the CG from the claws toward the nailing surface. This is the axis John rotated it through the first time, when it precessed. And its smallest axis, about which it rotates most stably (least tendency to precess), is a line passing through the CG from one flat side to the other. This is the axis about which John rotated it the second time, when he stole Tom's turn.

The challenger says that John tries to launch the hammer identically on both occasions. The only difference is the initial position of the head of the hammer (vertical or horizontal). Any intentional rotation is the same for both cases. Can the effect of air resistance really be that large? It is quite a massive object moving a rather small distance.

I think it wants to precess anyway -- energy dissipation is what tips it over the edge. I don't claim that it would not precess in a vacuum.

But in any case, I'm starting to like the whole misaligned CoG theory, although I don't know how we became so all-fired sure the CoG is off-center.

I tried it with my own hammer and it didn't work, so I assumed that John had launched the hammer in a way that caused it to twist. But I wasn't very happy about it, because the only way you might then observe something similar to what is described is if the hammer only rotated 180^O about the minor axis (which is what I would describe as a half-flip).

Following MPM's revelatory posting, I've gone back to my originally uncooperative hammer. Based on his clue, I looked at the balance, and its centre of gravity seems lie close to the major axis of the handle.
Then I wound a little thick solder wire round the claws of the hammer*, held it in place with a little duct tape, and tried again. There was some sign of twisting, so I added some more. Now it behaves exactly as the challenge suggests.

So, it seems it only happens when the CoG is a suitable amount off-centre. Perhaps the heads of claw hammers are made to a standard design these days?

*Obviously this unbalances it in the opposite direction to MPM's diagram - but it's relatively easy to implement, and it demonstrates the principle.

I'm starting to like the whole misaligned CoG theory

However, this works equally well with a symetrically headed hammer. I think your original explanation is the best.

I think that all this metal-bashing has made your wrists too firm, and you are basically seeing the effects of asymmetry in your arms. I mean by that that you are imparting a very consistent twist to your hammers as you swing them upwards. The reason I say this is that you say you are seeing the same effect with hammers that you say are symmetrical about the plane in which you throw them - and that would pretty-much preclude a consistent rate (or even direction of) precession.
However, that would mean that you should also apply a twist to the vertically-held hammers. (On which subject, I didn't try the vertically-oriented throw on the hammers with the solder weights, because I doubted I had maintained the symmetry)
Enlightenment please...

If you have a rubber band, you can use it to hold a book closed. You can then try flipping the book about its three main axes. When it is rotating around its middle length axis (typically the one parallel to a line of type about halfway down a page) it is unstable, but when rotating about the other two axes, it is pretty stable. The hammer (even with a symmetrical head) is fairly stable when rotated (like a top) around the handle centerline and when rotated around the axis perpendicular to the hammer head's length. But when rotated around the axis that is parallel to the hammer head's length, it tumbles in its flight.

Just to check, I took a book (and the hammer) into to my vacuum room (the one with the frictionless walls and floor) and the behavior seems almost the same with or without air friction (although I'd have to admit that making observations while continually bouncing off the walls was tricky).

You can do a similar experiment with a tennis racket.

Thanks for the reference. Nice video and explanation.

So, it seems that the answer for this case is a combination of the intermediate moment of inertia and the initial rotation not being truly around any of the principal axes. I would like to give you a G.A. for pointing us to the "intermediate axis theorem"*, but no description of the principle appears in your actual contributions (and I can't for the life of me see how the term "precession" is any sort of explanation for this)
*though I have so far been able to find anything mathematical on the topic. If I can't find a reference I will have to try and recreate it for myself to see why and how it predicts this effect.

Having returned from vacation, I tried the throw with a tennis racket - and it behaves convincingly as described.

But I was still unable to demonstrate the effect using my old hammer unless I forced the CofG off-axis. Presumably the combination of a relatively wide symmetrically-weighted head and a shaped handle resulted in the rotation being close enough to the principal axis that I did not see anything significant.

I also took my courage in both hands and tried it with my other hammers. The very small pin hammer was all over the place whichever way I threw it. The wide-headed club hammer bruised my hands, but I could see a different in final twist between the two orientations; however, the difference was less than a quarter-turn, so I wouldn't be able to swear what was spin and what asymmetric catching. On the other hand, I borrowed a neighbour's claw-hammer, and it behaved like John's hammer.

As a complete change, I tried the same experiment with an old plastic 50-cm ruler that is flat one side and domed with a groove on the other (the edges are almost sharp). I went for a two-handed half flip. When flipped from a flat position it appeared to rotate mainly about the intended axis, so the top and bottom were interchanged as one might naively anticipate, When flipped with the major surfaces vertical, it performed an "anomalous" rotation, so the flat edge changed sides.
I attribute the first behaviour to the ease of flipping the flat ruler symmetrically. The second behaviour would be down to the offset of the centre of gravity from the edges along which the torque might be applied. This appears to validate MPM's insight: given adequate initial asymmetry, the effect will occur when spun about the axis with the largest moment of inertia.

I think it's full of prunes! Maybe it's in my wrist technique, but when I tried this, one full turn from either orientation brought the thing back to its original position. Nothing was positionally reversed either way 'round. Suggestions?

You need the same design of claw hammer that was used by John and by Maths_Physics_Maniac. It didn't work for mine either...

My guess is that one thrower is left handed, the other right handed.

This difference would mean that although they both do "the same throw" the hammer is rotated differently about it's CoG (as per excellent diagram in post #3) causing the difference in orientation after the toss.

Although the question isn't that clear as to the orientation (or plane) in which each throw rotates the hammer.

Tom was not the one tossing the hammer, John was.

Because Tom flip the tool in the way the head is balanced for its intended use.

John flipped the tool where center of balance of the head is not in line with axis of the handle could be any where depends on who manufactured the tool and type of claw hammer. This caused by the off centered weight of the head dropping as the hammer spins around the center of the flip

The obvious answer is center of gravity relative to the handle. If held with the claw or head facing you, CofG is co-linear with the centerline of the handle, resulting in a straight flip. If held with the side of the head facing you, CofG is no longer co-linear with the handle, resulting in a rotation of the head when it is tossed up. A more likely reason is that Tom didn't buy any beer for John, since he's been doing all the work on Tom's deck all day. When John flipped the hammer, it hit Tom's head on the way down, turning the claws around.

good answer .....

and it is true .... "women go crazy for the sharp dressed man" ....

What we are looking at here is accelerated angular motion, from an applied torque

The "Moment of Inertia" of a three dimensional body has 9 components.

Ixx = ∫_area x² dm. Iyy = ∫_area y² dm Izz = ∫_area z² dm which are the three principle M of I, but it also has components known as cross Moments of I

Ixy = ∫_area xy dm Ixz = ∫_area xz dm etc

The familiar angular acceleration equation T = I.α is really a simplified form of

T = I.α where T is a 3 component vector; I is a 3x3 matrix; α is a 3 component vector.

Thus applying a torque in the x direction will generate accelerations in the x,y and z directions unless the Ixy and Ixz are zero. A similar argument applies to the y and z axes.

So when the hammer is flipped about its x axis, it will be accelerated in the x axis by Ixx AND also accelerated in the y axis by Ixy and the z axis by Ixz. The primary spinning will be Ixx with secondary motion depending upon the relative values of Ixy and Ixz. Changing the starting position of the hammer and flipping it about its y axis will bring Iyx and Iyz into play and the motion will be different.

Usually one cross M of I is significant and the others are zero or near to it. The effect is easily seen if you spin a tennis racquet. You can spin it quite stably around the handle axis (many players do whist waiting for a serve); you spin it easily in the plane of the strings; but if you try to spin it in the plane perpendicular to the strings, it starts to tumble and the motion is clearly different from the two previous cases.

With regards to the challenge question, the two different tosses are attempts to spin the hammer about two different axes. In the first case, there is enough cross coupling to cause the hammer to rotate around the handle axis by ½ revolution before it is caught, in the second case the cross coupling is zero (hammer is symmetrical in plane of spin) and there is no rotation around the handle axis.

OK, I voted you a GA because it sounds like a good, logical, and probably correct explanation. However, I still haven't got a hammer to twist about that way. Maybe I have a defective hammer...

The rotational shift is because the head is pulling down (pushing the claws up) and when you toss the hammer, the resulting friction imparts the horizontal twist.

When you toss a hammer with the head and claws in line with the main rotational direction you don't have the rotational friction.

Most people will toss the hammer with the head going head up and rotating toward you. Now try to toss the hammer with the handle rotating up and away from you, Different animal altogether?

Ok I am going to go out on a limb here and and offer a different suggestion.

I would like to suggest that if you toss the hammer as you would grip it for the purpose of hammering them the release of the fingers would be enough to start the hammer spinning about it's axis when held in the crossed position, as it is more ready to receive being acted upon by an outside force. Thus the offset of the CoG accentuates the reaction.

However when the alignment of the hammer is that which puts the CoG in full alignment(claws away and head towards you) the the applied force of the release has little effect overcoming the CoG.

If both situation are correct then tossing the hammer with the handle running from fingertip to palm and being held in place with the thumb would rule out any causal effect of the grip and prove the theory.

I shall try this and report the findings on four different hammers. Meanwhile please feel free to beat me up on this theory.

<be gentle>

Ok I grabbed four hammers, 3 that were completely different on one similar to one of the others.

Implements of destruction:

20 oz framing hammer (long claws heavy waffle faced head)

12 oz claw hammer (basic hammer you see every day)

10 oz ball-peen hammer (solid steel construction head and handle)

3 pound drilling hammer (well balanced hammer side to side short handle)

I tossed all hammers multiple times first changing only the grip. I even dropped them straight to the floor.

Conclusion:

Grip has an effect on the rate of spin but did not prevent it from spinning. However I did take time to check the balance of each hammer and the only hammer that did not spin when held in the flat position with the extended finger and thumb grip was the 3lb drilling hammer. It had no spin what so ever.

One thing I did notice is that when I tossed all the hammers they always spun in the direction of the heavy side of the hammer. In all cases (with the exception of the drilling hammer) the driving or flat surfaced head was always the heaviest and the rotation always followed the movement of the head.

John picks up a claw hammer and holds it with the claws pointing to the left. He tosses it up in the air giving the hammer a single flip about the head. When he catches it, the claws are now pointing to the right. John grabs the same hammer, holds it with the claws up and does the same type of toss and catch. The claws are still up. Why did the hammer orientation change for John and not for Tom?

Where is Tom? Did Tom do the same two tosses?

Tom is now dead he has a hammer stuck in his forehead and will not be joining us in any further experiments.

Thank you,

Since they were taking a break from building a deck (relevant information), the hand not throwing the hammer for Tom had a beer that was half full while John's beer was half empty.

After playing with this a while, I rotated the hammer about 45° and got either a full rotation or a quarter (well, maybe 3/4 - when I tried to watch to see which, I missed and the hammer hit my foot). Lot of interesting stuff in this question!

The reason for the rotation then the head is sideways and not when the head is vertical had little to do with the center of gravity. It has more to do with the principal axes of the moment of inertia tensor. By symmetry we may safely assume that the principal axes are parallel to the axes of the handle and head, with the third axis orthogonal to both. Now we must consider which principal moment is largest and which is smallest. Without getting into integral calculus, I think most here will agree that the smallest value is for the axis parallel to the handle, as the distance of any element of the hammer is shortest from this axis. The largest value would correspond to the axis that is perpendicular to the head and handle axes, as more mass is further from this axis than any other. This of course leaves only the middle value which corresponds to the axis parallel with the head of the hammer. This axis is the one about which our first dear thrower causes the hammer to rotate. His friend, the second thrower, causes the hammer to rotate about the axis normal to the plane of the hammer, and this is the axis with the largest principal moment of inertia. Now I hear what your say, where the heck am I going with all this... Due to conservation of angular momentum, if a body is rotated about it's largest or smallest principal axis of inertia, the direction of the rotation vector cannot change as these principle values are the extremes. However if the object is rotated about the middle principle axis of inertia, then it can conserve its total angular momentum while altering its rotational velocity vector as the moment of inertia projection in the direction of its velocity vector can either increase or decrease since it is not an extreme value. (It's nice to know my graduate studies in classical mechanics come in handy sometimes.)

Here's the fun part. Try this for yourself with a baseball cap (or a shirt box). First determine the alignment of the principal axes of rotation, then which is smallest, largest, and middle. The flip your hat about each of these three axes and observe the results.

- Sully

Now that I've chipped the floor twice, banged my toe once, and dented the handles of several hammers, I'm pretty sure I know what's happening.

First, the "sideways" orientation (John) is the easiest starting point, and an "off center cog" hammer is the easiest to work with. And, the half flip left to right is the most common end result. I've been able to start at various angles up to about 60° from horizontal, use hammers with what looks to be perfect symmetry, and get complete 360° flips, even with a claw hammer.

When the hammer is first spun, it has angular momentum about what John might call the x axis. It also has angular energy. Both will be conserved.

For an off-center cog hammer, the initial spin clearly imparts some rotation about what John sees as the z axis. Even if there is no off-center component, any slight wobble (or even breeze) is enough to start the process. As soon as the head rotates just a teeny bit about the z axis, the moment of inertia with respect to the x axis is increased.

In order to conserve angular momentum about the x axis, ω_x must slow slightly. But, that means the angular energy must also be reduced (remember we're in the air and have no convenient earth to soak up momentum or energy) and that can only be done by having the head spin faster about the z axis. But that makes the problem worse. And, so on.

If you start at the extremes of I (Tom), a little wobble makes total I a little smaller and the system need to speed up to conserve angular momentum, but there's no place to get any energy to keep that in balance. So, the hammer can't rotate that way.

John should quit horsing around and get to work.

By the way, if you cut off about 60% of the hammer handle, it quits flipping.

Here is a partial explanation of what should be happening with the hammer (and the tennis racket). My thanks to Blink for pointing me to a reference that set me thinking a bit more clearly.

When you flip the hammer, you deliberately impart spin about a horizontal axis. At the same time, you accidentally impart a small spin about the vertical axis.
. However, as the hammer spins about the intended axis, the handle goes through the vertical position. At this point the moment of inertia about the vertical axis becomes rather small, so conservation of angular momentum requires that the angular velocity about the vertical axis is increased in the ratio of the momenta of inertia in the two orientations.
. Of course, this effect will occur whether the racket head is horizontal or vertical. First thoughts were that the difference was the size of this effect, combined with changes in the rate of rotation about the intended axis. But that would be to miss an essential secondary factor: with a full flip, the handle goes through two vertical orientations, and an initial quarter-turn about the handle in the first vertical orientation ought to be cancelled by the quarter-turn in the inverted vertical orientation.
. So something more complex needs to happen for net rotation to be observed when the racket is finally caught. I haven't got to the bottom of this - maybe interaction with rotation about the third axis is required*? However, I doubt that I will be able to describe this even if/when I do**. Suffice it to say that there must be some "stability" to the amount of this overturning for it to be observed over such a wide range of objects
*N.B. that the exercise books might provide a clue. I found that I could throw exercise (and other) books where all ends were cut together so that they came back to their original orientation if thrown approximately symmetrically, whereas I was generally unable to avoid inverting books where the covers projected beyond the internal pages.
**Maybe someone could provide a slowed-down film showing multiple frames, or even an illustration from a CAD system?

Incidentally, the reversing of direction of rotation relative to the axis of the handle is even stronger when applied to MPM's explanation (which I initially accepted): initial rotation abut the handle would be reversed during the initial half-turn and again during the second half-turn. As illustrated by the exercise book, that does not necessarily imply that initial rotation about the handle is irrelevant.

**Maybe someone could provide a slowed-down film showing multiple frames, or even an illustration from a CAD system?

I've been thinking along just those lines. I find it quite difficult to accurately judge even the direction of rotation, let alone precisely how much rotation really occurs. With the symmetrical hammer, it seems that the rotation is very close to half a turn each time, which seems too predictable, given that the slight rotation that is imparted accidentally should vary in both magnitude and direction.

Another thought which just popped up. The hammer handle is acting as a pendulum about the mass of the hammer head. Therefore, the handle accelerates and decelerates (in the plane of intended rotation). Although air drag could not cause enough deceleration to cause gyroscopic precession, this pendulum effect could.

Just as I was convincing myself, seconds ago, that this might be a key, I realized that the same effect shows up in flipping books and boards, which, once released, would rotate about a central CG. (But maybe there is something in the transition from rotating about one end -- at the release -- to rotation about the CG. With a biggish hammer, 30 fps might be enough to be helpful.

Just a couple days ago, I read an ad for a digital camera which could do 1000 frames per second in video mode, and 40 fps in "still" burst mode (for capturing a still of a sporting event etc.) Let me think... why might my wife want such a thing?

"Gosh, honey, I've been thinking about how hard it is to get great pics of our daughter playing soccer...."

As it works with the tennis racket, we should be able to throw that up about 3' for a nominal "single flip". That gives about 0.4-seconds to return to your hand. I imagine that most of the anomalous turn will happen in a relatively short time, but we can possibly manage without that - what we need to understand is how the racket approaches and leaves the near-vertical positions. Perhaps even 50 fps would be adequate for this, though 180fps would be safest. Help - anyone?

Regards.

Today is Nov 4 but answer is not here yet.

Is it due to Presidential Elections?

Have a fine day today !!!

North of the equator- South of the equator. It's that toilet flushing trick again.

Sorry about the confusion about who tossed the hammer the second time. I should have let the typing set for a couple of days so I could read what I typed and not what I intended to type.

I offered this challenge question partially because it has interested me for some time now and I thought it might interest others. The other reason was that I really don't understand much of mechanics behind why this phenomenon exists. I can without much trouble get a complete 180 degree spin of the hammer head on every toss if oriented with the plane of the head horizontal. The type of hammer does not matter. While a claw or ball peen hammer is the most obvious, I get the same behavior with symmetric hammers as well. I've even made up well balance sections of PVC pipe in a tee and get the same result. I do believe that I am in fact quite trainable and the phenomenon has as much to do with my response as that of the object being tossed.

Following is a description of the physics as I understand it. When you develop the equations of motion of the rigid body, you get an equation that is of the form

d²λ/dt² + ω²λ{(I_yy-I_zz)(I_zz-I_xx)/(I_yyI_zz)}^1/2 = 0

where ω is the angular velocity about one of the principal axes, the nominal angular velocities about the other two principal axes are small and one of these angles is λ. I_xx, I_yy, and I_zz are the three principal moments of inertia. Depending on the relative values of I_xx, I_yy, and I_zz, the term inside the square root is either positive, negative, or zero. If positive, the equation solution for λ is a simple harmonic equation and is bounded. If the term is zero (two or three moments of inertia are equal), the equation solution is dλ/dt is a constant (constant velocity). If the term is negative the rotation is nominally about the axis with the intermediate moment of interia and the solution if of the form

λ = Ae^kt + Be^-kt

This leads to the unstable situation where the perturbation grows exponentially with time. However, this is for a small displacement solution so its validity disappears at some amount of deviation.

For a much clearer discussion please see http://farside.ph.utexas.edu/teaching/336k/lectures/node75.html in particular the sections on Rotational Stability.

While this does give some insight into the phenomenon, it still leaves me a little cold. How unstable is unstable? Why do I always get almost exactly half a turn? Is it me or the laws of physics? Does the rotation about the handle axis happen quickly or increasing with time as implied by the equation above? I also have trouble watching the hammer, or racket, or other object spin in the air and understand what the angular rotations really are versus time. Maybe some high speed video would provide some insight. Maybe we need to send up a rubber hammer on the Space Shuttle and have them do the test at slow speed in a weightless environment. Does anyone know of a three dimensional simulation tool that could be used to do numerical experiments?

Thanks,

Jim

I think that John tossing the hammer twice makes perfect sense - but might be expressed as Tom challenging him to make the second toss.

I think you will find that k is purely imaginary during periods when no external force is applied to the body, and the reason for the unexpected half twist is that the small unintended angular velocity about a vertical axis (that you impart accidentally) represents a significant rate of spin when the handle of the hammer is vertical. That part of the explanation seems relatively straightforward. What is difficult is working out when and why any initial twist is not reversed when the hammer goes vertical for the second time. as this appears to happen for a wide range of people and of tossed objects, I suspect that approximately half-integral twists occurs over a relatively wide range of starting conditions.

Given that the theoreticians appear state that the rotation is cyclical*, it may even be that only integral numbers of half-twists can occur.
*Of course, it is more than possible that I have misinterpreted their statements, as I am far from knowledgeable in this area.

I too am bothered by those half-integers. I flipped a fairly symmetric hammer perhaps hundreds of times. For one, two, or three flips, I got half a turn, one turn, and one-and-a-half turn. When I did half flips, I sometimes got a half turn and sometimes no turn, depending on whether the flip had enough speed to have gotten a full flip if left alone.

I don't understand the physics of this.

I managed to flip a symmetrical hammer once or twice without any rotation about the handle axis - I think I was lucky on these occasions not to impart any rotational torque on release of the hammer. I think this does tally with the intermediate axis theorem, if you read it carefully.

Remember the theorem says that an object is more stable rotating about the major and minor axis and less stable rotating about the intermediate axis - it does not say an object cannot rotate about the intermediate axis. So the upshot of this is, you can rotate any object around any of its rotational axes and it will continue to rotate about that axis unless acted on by external forces. The force required to knock it off it's intermediate axis is obviously a lot less than that required to knock it off it's major or minor axis.

That's my understanding of it anyway...

Regarding zero rotations - as you say, there is no reason why not. I suspect the reason we generally see the half rotation is related to the mechanics of our natural throw - including the structure of our wrists and elbows.

Regarding interpretations - I agree that you require to impart less force when the prime rotation is about the intermediate axis. But perhaps we should not consider this as "upsetting" a rotation, more as setting up a complex motion. And I think the more important (and perhaps surprising) feature is that the required secondary rotational velocity (even at the end of the handle) is much smaller when the primary rotation is about the intermediate axis and the secondary rotation is around the axis of greatest inertia.

To see if I could get a better handle (sic) on the experimental observations, I tried some throws using a piece of wood that I had available (36x15x0.6cm³ - wear gloves). I chose this because its size and symmetry should allow me to understand what I was doing better than with the hammer. I was able to use two-handed throws so that any asymmetries could be deliberate. It was quite easy to attain zero 'accidental' spin; and rotating my body while throwing (to impart a small spin about the vertical axis) gave the reported inversion. I could also produce the full-inversion you report, but the movements required were sufficient that I suspect that I was also applying twist about the long axis of the plank.

So that pretty much agrees with the "official" answer, eh? I still have not replicated the reported results, in fact, I'm not flipping any more flipping hammers. Maybe my technique is substantially flawed. But then, I can't get a decent hook bowling, either...

"So that pretty much agrees with the "official" answer, eh?"

Not quite, because the asymmetry of the hammer will only cause the throw to produce a twisting around the axis of the handle, and that is the one direction where rotation will not be enhanced by the so-called instability. So far as I can see, this will also not contribute to a final positional rotation, because the direction of the twist it imparts reverses at 1/4 and 3/4 of a turn around the "principal" direction of rotation, which seems too even for other smaller twists to prevent the partial twists cancelling each other (it might contribute to changes with time in the effect of the twists around the vertical axis, but observations with symmetrical tennis rackets would seem to suggest that the natural structure of a human throw provides all the twists that are needed)

As I described previously, I initially saw no consistent twist with my hammer. It may be that the design of some hammers has insufficient difference between its non-major moments of inertia, or that my throwing action is overly symmetrical (or that the shape of the handle is just right to counteract the normal tendencies). However, my one-handed experiments flipping tennis rackets (of different generations, but all still playable) produced a pretty consistent half-twist.

I think I'll just have to accept that I've been a flipping tosser...

Yes. I found a description (the URL of which I then lost - a student walked in and I didn't want to seem to be loafing during office hours, so I closed the site too quickly) of a juggler who claims he learned to flip such objects without flips after three years of practice.

I found this thread quite fascinating, and have a suggestion which I hope can be implemented by someone with access to gyroscopes who can report the results. It might throw some light on the topic which is so far shrouded in mystical mathematics.

The flywheel of a gyroscope can be replaced by an object like a rectangular block with different edge lengths, which could be mounted with any of the three principal axes as rotation axis. If made to spin and then released, the subsequent behaviour with regard to stability should be visually evident. The experiment could be repeated with long and short square prisms (Ixx=Iyy<Izz and Ixx=Iyy>Izz).

I had a look at the ...utexas.edu... lecture page but could only glean a qualitative picture of what is involved.

Unfortunately I have never had the opportunity to handle a gyroscope of any sort, or even observe one with any understanding, so my knowledge of 3-D motion is limited to some dimly-understood notions from mechanics texts. Though I am a mechanical engineer, I've not really had to deal with rotations about more than one axis. I'm not even sure if my suggestion is practical. I would like to try out the experiment myself, but lack the material means, and have my hands full otherwise.

Even my visits to the CR4 challenge threads are managed at infrequent intervals, because of my problems in coping with computers and internet! I'm usually too late to join in any discussion and must generally remain content with reading the threads off-line. But I'm hoping that some more posts with useful inputs will appear on this topic.

Regards, =TeeSquare=

I do like the concept of flywheel or gyroscope. I believe it does provide some insight into the problem. Since the hammer or other tossed object (ignoring losses to the air) has a constant rotary kinetic energy and a constant rotary momentum, the magnitude of these quantities does not change. As far as I know, energy does not have a direction, but the momentum does. I believe the initial rotary momentum is a vector whose three components are the angular velocity times the angular moment of inertia, each for a principal moment of inertia. While the individual angular velocities can change, the magnitude and direction of the angular momentum vector does not. This vector is a fixed quantity.

Now if I could just figure out how to use this to translate from the equations for angles fixed to the body and get angles in the global coordinate system, maybe I could do a numerical simulation that made sense.

The problem with talking about precession and gyroscopes is that the gyroscopic precession that is generally discussed is caused by a torque. In contrast, the effect we are looking at here is observed while the object is moving freely under gravity - i.e in the absence of torque. Similarly, the familiar form of gyroscopic precession is equally prevalent when the main axis rotation is the principle axis inertia*.
I therefore believe the situations to be entirely different - but they do have in common that in both cases conservation of angular momentum causes objects to move in ways that are not as might immediately be expected.

*By way of illustration - the Parisian Nobel-laureate physicist Jean Perrin pulled the following practical joke sometime around 1925.
He packed a high-speed gyroscope into a suitcase, set the gyro spinning and left the suitcase in a railway station. A porter picked up the case and walked off with it. All went normally until the porter came to a corner. Instead of turning with him as he rounded the corner, the case moved towards the horizontal. The porter then attempted to force the bag to point in the new direction he wanted to travel, but it rotated anywhere other than in the intended direction. Dropping the bag in fright, the porter ran off, shouting that "Le diable soi-meme soit etre la dedans" (The devil himself must be inside).

Mea culpa for bringing gyroscopes into this discussion. Here's a quote-in-quote from KenFry (post 18 in the "Spinning Wheel, Spinning True" challenge thread, blogentry 1871 -- 05/08/07). I wasn't able to follow that discussion fully either since I lack a clear understanding of gyroscopic behaviour. But I think the period in question is earlier than Jean Perrin's prank (circa 1925) which you quoted (also referred to by a guest in post 33 of that thread).

"Arthur Webster of Clark University played pranks like this. The excerpt below is from this site.

Indeed, physicists always enjoyed meeting Webster on his arrivals at railroad stations for Webster liked to play practical jokes of a scientific nature. Webster, with his expertise on the gyroscope, had constructed a portable, battery-powered gyroscope housed in a suitcase. As his train would come into the station he would start the gyroscope. Once the gyroscope was up to speed he would hand this suitcase to a porter with instructions to take good care of it. Webster would then walk briskly down the station platform, making abrupt turns as he went. The suitcase, however, would not follow these turns, shooting off into space with an alarmed porter hanging on desperately and with the assembled greeters responding with hilarity."

But coming to the problem on hand, I think what you are implying is that any talk of gyroscopes is relevant only when one of the angular velocities (or angular momenta?) involved is at least an order of magnitude greater than the others -- that is trying to change the axis direction of something which is already spinning 'fast'. I don't know enough to argue with that. Nor do I know anything about 'precession' other than having observed a spinning top (or is that 'nutation'?). Maybe you are quite justified in concluding that the hammer toss phenomenon is something quite different from what is 'generally understood' as gyroscopic behaviour.

My interpretation of some statements in the utexas lectures is that a rod made to spin about its axis and then released in gravity-free space (or in free fall) would not be stable, but start waggling or even perform some crazy antics (which could perhaps be mathematically determined, though not by me, ever). Perhaps that amounts to saying that a useful gyroscope cannot be constructed using a long cylinder as spinning body.

All this is strange uncharted territory for me, but I do have some sort of intuitive feeling that a spinning rod will be unstable, though that is probably a false analogy drawn from the whirling of slender shafts and other strange phenomena encountered in engineering, often caused by forced or self-excited vibrations near resonance conditions -- another subject whose mathematics was way beyond me (a lot of stuff involving Bode plots, Nyquist criteria, and what not).

SlideRuler's analogy in post 76 is revealing, but doesn't seem altogether satisfactory as I can't see why a rectangular base pyramid is considered, rather a block with all rectangular faces, in fairness to all three axes. In which case there would still be SOME stability when resting on the small end. There is something peculiar about the intermediate inertia-value axis, not the minimum one.

The aforementioned spinning wheel thread had a lot of interesting inputs (including links provided by Kris, who has inexplicably not surfaced yet in this discussion), in particular the very practical advice regarding the dangers of using heavy rotating hand tools like grinders which may have a large angular momentum about the spin axis (posts 66 onwards).

I'm still hoping that someone with better access to experimental facilities can do some gyroscope tests with different kinds of rotors and report the observations here. One question I have -- possibly quite irrelevant -- is whether the instability of rotation about the intermediate-value MI axis has anything to do with the point lying right inside the Mohr's circle representation for moments of inertia. Can some sort of stability field be imposed on the Mohr diagram?

=TeeSquare=

That is a lot to take in all at once.

The difficulty with gyros lies simply in the definition - they are rotational objects designed and used for their stability, and thus specifically exclude the much more difficult case we are concerned with here (the reason for the difficulty being that we are no longer looking at small perturbations).

Then the rotating cylinder: a thin cylinder rotating about it's axis is theoretically stable. I believe that the awkward practical behaviour of unconstrained rotating rods is caused by the relatively high surface velocity compared with the angular momentum; that means that whenever such a rod comes into contact with a fixed surface the imparted perturbation can be relatively large - and this is, as you suggest, compounded by the easily excited flexural modes of vibration.

Regarding the final paragraph: I haven't myself used the graphical methods I had understood to be covered by Mohr's circle, as I had believed them to be restricted to two-dimensional problems. Given that the angular the resolution formula on which they are based is more general than that, it strikes me I may have been missing something here.

Regards

Fyz

I also read the utexas... lecture and it does address this question precisely, and rigorously. (I admit some of the maths is a bit daunting)

The key to this phenomenon is that it is very easy to visualize the hammer spinning about the xx axis; the yy axis or the zz axis and to imagine (in the absence of gravity and friction) the spin would continue indefinitely: But - the equations which describe these 3 simple motions have varying degrees of STABILITY.

Let me suggest an analogy. Consider a rectangular based right pyramid placed on its base on a table. It will be very stable to small perturbations of rotation about its short edge; and it will be less stable (but still stable) to small perturbations of rotation about its long edge. Now consider if we try to stand it on its apex. Although it is theoretically possible to do (simply place the GofG vertically above the point!), it is clear that this condition has the forces in equilibrium, but is NOT STABLE. The smallest perturbation causes the pyramid to fall.

It turns out that for a spinning body, spin about one of the principle axes is also in unstable equilibrium.

Consider Ixx > Iyy > Izz

The body spins stably about the xx (maximum I) axis - small perturbations cause the xx axis to nutate (wobble)

The body spins stably about the zz (minimum I) axis - small perturbations cause the zz axis to nutate (wobble)

The body spins unstably about the yy (intermediate magnitude) axis - small perturbation increases yy axis displacement exponentially

Finally, if Ixx = Iyy a body will only spin stably about the zz axis.

"Tricky things gyroscopes!"

Reading relatively rapidly through the utexas... notes, I was a little surprised that some of the statements remained unqualified. The one that is relevant to this thread is "In this case, the amplitude of the perturbation grows exponentially in time" - if only...
Naturally, the solutions are only valid when the perturbation from rotation about a principle axis is small.

On the other hand, Mr Fitzpatrick describes most of his notes as being about the motion of a rigid body, and only uses the term "gyroscope" when referring to a device that spins rapidly about an axis of symmetry, and then looks at the effect of a continuous torque (that rotates with the object) on such a device. That is in line with dictionary definitions. Because in that case we are concerned only with small perturbations from the major rotation, I would say that a gyroscope is relatively simple (compared to the problem raised by Jim).

To me, the surprising characteristic of the tossed hammer (or tennis racket) remains not so much the "instability", but the apparent repeatability of the half turn. I suspect that this half turn could be related to the shape. Any offers?

It seems that the amount of turn is governed by the ratio Ixz/Izz

Hope you can read the above. Of course this is all idealized. The hammer is not absolutely symmetrical; you don't apply torque to only one axis etc.

I agree that the instability discussions are a bit of a "red herring" but they do explain why the spin in the y axis remains stable, rather than why rotation takes place in the x and z axes.

One of us has got the wrong end of the handle, I think. My take on this is that the tennis racket is symmetrical about two orthogonal surfaces (the plane of the face and the plane perpendicular to the first that includes the axis of the handle). That symmetry means that, once you choose the obvious three axes, the matrix is diagonal (any contributor to a cross-product has an equal and opposite one at its single reflection, so the cross-products are identically zero)

On the other hand, I believe that shape does enter into this. Clearly, the bulk of the motion must be due to the initial instability at close-to-axial-rotation; after that, I believe that the ratio I_xx:I_yy:I_zz determines how much of any rotation about the axis of the handle is retained after the partial cancellation of effective rotation between the handle-up and handle-down positions. But handling this through the entire rotation looks to me to be extremely tough; there are probably some specific mathematical tools that would hold the complexities in check, but if I ever knew them I certainly can't recollect them now.

Regards

Fyz

Is this an explanation or just a rewording of the problem?

Great input everybody. More Hammer Tossing info available at www.hammertoss.com

No Joke.