# Tennis Racket Half-Turn on Flip

Years ago in college, I was between classes in the Physics Bldg, and saw this exam problem posted on the bulletin board:

“When you flip a tennis racket (holding the handle, one complete revolution, so you catch the handle) it does a half-turn in the axis perpendicular to the long axis. Why?”

Over the years, from time to time, playing badminton or tennis, for example, I’ve thought about this problem. Unfortunately, I was reading the “Electric Kool-Aid Acid Test” when I we were doing that chapter in Physics 101, so I don’t know the precise equations and principles which need to be applied to solve this.
So, you Physics scholars out there, Why? I suspect it has something to do with torque, or angular momentum, conservation therof, etc, etc. Please, no responses from illiterates with their “pyschic reasons” why there is a half-flip.

And why a half-flip, no more, no less?

Made an error in the wording of the question,

The half-turn is about the long-axis (not perpendicular to it) perpendicular to the axis of the flip.

I’ll be dipt. I just spent about 5 minutes flipping, and if I could figure out why it’s doing it, I might be able to get it to stop doing it. No such luck!

Gah. It only works when you hold the racquet face parallel to the floor. Hold it perpendicular and no half-turn.

I suspect one side of the racquet head is always heavier than the other, but I don’t quite know how that would account for it either…

I’ll have to look up my notes from Structural Dynammics; I’ll probably use the LaGrangian method. Hopefully, somebody will beat me to it!

Even if I do get the equations worked out, I may have to email them to you in a Word document. The thought of trying to type them in ASCII make me shudder!

Initial observation follows:

First, do this. Flip the racket as described before, but with the plane of it perpendicular to the floor instead of parallel to it. No rotation about the longitudinal axis.

When the racket is perpenicular to the floor, its mass moment of inertia* about the “flip” axis is at a minimum. When the racket is parallel to the floor, its mass moment of inertia* about the “flip” axis is at a maximum value. When you start off with the racket in a horizontal position and flip it, the racket will position itself in such a way so that the moment of inertia is minimized while the handle is passing the highest point in the flip (i.e. it’s following the path of least resistance). Conservation of energy causes the racket to finish its half rotation before you catch it.

[extraneous pondering]OK, the extra mass (the handle) that far away from the c.g. PLUS the difference in the two moments of inertia cause the half rotation.[/extraneous pondering]

All right, I’m still thinking about this and am away from my Structural Dynamics textbooks until Tuesday, so bear with me. I hope you understand a little of what I’m talking about, and perhaps can add to it.

*Mass moment of inertia has to do with why a skater spins faster when her limbs are tucked in than when they’re outstretched.

Initial observation follows:

First, do this. Flip the racket as described before, but with the plane of it perpendicular to the floor instead of parallel to it. No rotation about the longitudinal axis.

When the racket is perpenicular to the floor, its mass moment of inertia* about the “flip” axis is at a minimum. When the racket is parallel to the floor, its mass moment of inertia* about the “flip” axis is at a maximum value. When you start off with the racket in a horizontal position and flip it, the racket will position itself in such a way so that the moment of inertia is minimized while the handle is passing the highest point in the flip (i.e. it’s following the path of least resistance). Conservation of energy causes the racket to finish its half rotation before you catch it.

[extraneous pondering]OK, the extra mass (the handle) that far away from the c.g. PLUS the difference in the two moments of inertia cause the half rotation.[/extraneous pondering]

All right, I’m still thinking about this and am away from my Structural Dynamics textbooks until Tuesday, so bear with me. I hope you understand a little of what I’m talking about, and perhaps can add to it.

*Mass moment of inertia has to do with why a skater spins faster when her limbs are tucked in than when they’re outstretched.

Same thing happens with a TV/VCR remote. Try it, you know you want to.

Strainger writes:

This is close, though conservation of energy is not really relevant and the minimum and maximum moment statement is inaccurate. I forget all the exact equations from my classical mechanics class, but the important thing to realize is that the racket has not 2 but three ways to rotate. Hold the racket with one hand on the handle and one hand on the head and spin it along the axis of the handle and you’ll see it won’t twist that way either.

The reason is that a body is stable in rotation when it is rotating about the axis with the maximum or minimum moment of inertia, but not about the other axis. In the case of a tennis racket/vcr remote, rotating about the long axis is the minimum moment, rotating about the shortest axis (i.e. the thickness dimension) is the maximum, and rotating about the medium axis (width) is somewhere in between and thus the body is unstable.

You can think of it like an ice skater spinning - it takes effort to hold her arms in, and if she lets them go out she slows down. Slowing indicates an increase in moment - the tennis racket has no muscles so if an increase in moment is available to it it will be difficult to avoid it.

However, it is possible to flip it about that medium axis if you try hard enough - it’s like balancing a salt shaker on its corner. Not easy, but you can do it. If you experiment, you’ll find that the twist occurs at different points in the flip, so if you are very cautious you can delay the turn until you’ve caught the handle.

Now the question is: can anyone explain that so I can understand it?

I can’t really say I know the answer, but I thought it was related to gyroscopic precession.

It works with a hammer, too.

Well, I’ll bet it turns in the other direction on the southern hemisphere!
::throws a racket and runs for cover::

Seriously, which way does it turn, and why?

Second, if I get this correctly so far, the racket “tries” to get into a different axis of rotation that’s favorable momentumwise (or something like that). To accomplish that, wouldn’t it have to do a quarter-turn (90 deg.)? If so, does it do a half-turn (and more, if you do double/triple/… flips) because it overshoots the mark, so to speak? Is that where conservation of energy comes into play?

Finally, why is it always almost exactly a half-turn per flip, independent of the object you use?

Well this is the last time I attempt a Straight Dope science experiment. I dusted off my tennis racket and immediately flipped it (I was in the kitchen) into my wife’s crystal salad bowl. Both racket and bowl naturally followed the Laws of Motion and Gravity (which, I may point out, work just fine) down to the kitchen floor.
In the future, I will just take the reliable and educated word of other Dopers on experiments like this. Science, in the wrong hands, can be a very dangerous thing.

“…send lawyers, guns, and money…”

`` Warren Zevon``

Nickrz writes:

Whoops - sorry I waxed verbose.
In one sentence: The racket can be spun 3 ways, 2 of which will not flip and one of which will. The “why” of that one sentence involves 3rd year college physics.

TheIncredibleHolg writes:

It turns either way. It’s the same as a pencil balanced on its tip, it’ll fall whichever way it happens to lean first.

Good observation. Indeed, it is like a pendulum, striving for the quarter-turn and overshooting. Conservation of energy and angular momentum do apply. If you were in a little spaceship flying near the head of the racket you’d see it wobble back and forth in half turns.

I’m guessing it’s just the ‘time aloft’. If you spun it off the roof of a 10 story building it should do several half turns before impact.

No comment.

douglips, thanks for finishing that off; I was still knocking it around in my head. I thought of working out the EOMs, but 3 things made me hesitate: 1) They would be damn difficult for most of the Teeming Thousands to understand 2) I was too busy getting ready for my hiking trip this past weekend and 3) reviewing LaGrangian equations and subsequently performing them would’ve been a pain in the ass (emp. on #3).

followed by

Ah-HAH, so the witness has implied that Conservation of energy is, indeed relevant!

I humbly retract my previous ignorant statement and throw myself on the mercy of the Board.

Funny, but I can’t get it to turn the other way? Am I in the wrong universe?

If I remove the tire from my bicycle, spin it, and hold it exclusively by the bolt which would be on the left side if the tire were on the bike and I was going forward, which way would the tire precess? This is not an either/or thing. It will go one way. If you hold the opposite side (or spin it the other way) it will go the other way. (As opposed to falling, which it would do if the wheel was not spinning.)

I never really understood it, but I’ve definitely experienced it.

Rav