Amaze your friends with racquet tricks and the explanations behind them.
By Rod Cross
Somersaulting and flipping racquets
There are some things in life that are almost totally useless, such as rock music, which some people find important. The “Tennis Racquet Theorem” is another good example. The theorem says that if you toss a tennis racquet (starting with stringbed parallel to the ground) into the air to perform a somersault before you catch it, it usually also flips over in a half-twist and lands upside down in your hand. This can be good or bad depending on circumstances.
If the racquet happens to be a cat falling several floors out a window, this flipping-over effect is good for the cat. If the racquet happens to be a fresh piece of buttered toast with jam on top, then the flipping over effect is bad, especially for the carpet. If you are an Olympic diver or gymnast, flipping over several times while doing a few somersaults can earn you extra points, perhaps leading to fame and fortune.
As far as tennis players are concerned, the Tennis Racquet Theorem is similar in significance to whether bath water rotates clockwise or counter-clockwise in the Southern Hemisphere when the bath plug is pulled out. But it is a fascinating occurrence, fun to watch, and apparently fun to do, since so many players are constantly flipping and spinning their racquets between and after points or during simple idle moments about the club. It is doubly fascinating since the flip does not occur when you toss the racquet to rotate in any other way. If you begin the somersault by holding the racquet face perpendicular to the ground, no twist occurs. If you spin it about the long axis, it doesn’t also do a somersault, but if you toss it in a somersault, it will also spin.
How it flips
A tennis racquet, like any other solid object, has three different axes at right angles to each other. A racquet can be made to rotate about any of the three axes separately or it can rotate about all three axes simultaneously, which is the usual situation in any tennis stroke. One can even measure the ease or difficulty of rotation about each axis, and it can be given a number specified by the swingweight (moment of inertia) about that axis. The three swingweights for a tennis racquet are typically about 15, 120, and 135 kg·cm2, when each of the three axes passes through the balance point of the racquet. More commonly, the middle swingweight most associated with a tennis stroke is measured about an axis near the end of the handle, in which case the typical value that is quoted is around 330 kg·cm2.
The easy axis of rotation is the one passing along the handle up to the tip of the racquet. It is easy to spin a racquet about this axis, and many players do just that between points either as a nervous habit or to distract their opponent. They can even get the racquet rotating up to about 10 revolutions per second or 600 rpm. Interestingly, the racquet doesn’t flip around and whack them in the back of their hand when they do this, even though it is a simple spin around an axis just like the somersaulting racquet above.
Similarly, players sometimes stick their finger in the hole in the throat section and twirl (cartwheel) the racquet around the index finger. Again the racquet only rotates about one axis without any flips about the others. This cartwheel axis has the largest swingweight, and we can call it the “hard” axis since it is hardest to rotate at high speed around this axis. Andy Roddick especially loves doing this. It is just as easy to toss the racquet in the air and spin it around the “hard” axis, and the racquet doesn’t flip over. It is only when a racquet is rotated (somersaulted) about the third axis that it tends to flip over. It is the same with a book or a box of cereal. Only one of the three axes causes flipping, and it is not the one with the smallest or the largest swingweight, but the one with the medium size swingweight.
Suppose you toss a racquet with the strings starting in a horizontal plane, parallel to the ground, as shown in Figure 1. Toss it fast enough so that it rotates through a complete circle and then catch the handle. Most times, the racquet lands upside down as shown in Figure 2, having completed half a twist around the easy spin axis. It doesn’t always do that. If you are really careful to make sure the racquet has no twist when you toss it, then you might get it to rotate without flipping over. But any slight twist at the start will grow rapidly and cause the racquet to flip around the easy axis. Sometimes it appears that there is no twist at all, and then it suddenly appears “all at once” at the very end of the somersault.
It might seem that the racquet flips only around the easy axis. However it also flips around the hard axis just as rapidly. Suppose the racquet has reached its halfway point in mid-air, having completed 90 degrees of its 180-degree twist or flip. The racquet is still rotating in somersault fashion in the direction it was tossed, so it arrives at a point where the racquet is rotating edge-on, as shown in Figure 3. At this point the racquet is rotating in cartwheel mode, as opposed to a somersault, and it is also in the middle of its twist and is still twisting around the easy axis. In between these two modes, the rotation is part somersault, part cartwheel, and part twist.
The tricky part of all this is to visualize which axis is which. There are three axes at right angles that we can visualize as being attached to the racquet. The problem is the racquet is rotating so the axes are also rotating while the racquet itself rotates about each of the three axes.
To prevent our heads spinning as well, it is common to imagine that the three axes are attached to the ground rather than the racquet. In that case the racquet rotates only about two of the ground-based axes, doing a 360-degree somersault about one axis, a 180-degree twist about another axis and no rotation at all about the third axis. But the racquet really does rotate edge-on during part of its journey, so it really does rotate about the hard axis just as fast as it does about the easy axis.
Why it flips
It is possible to toss the racquet in a somersault without it flipping, but very difficult. More likely is that the racquet makes it about 180 degrees without any visible twist, and then it does its flip all in the last 180 degrees. That’s because the influence of the initial twisting force of your hand grows exponentially with time. For that reason, if you try to do a double somersault, the racquet will flip several times, not just two half flips, one on each somersault. The time to flip will depend on the initial twist rate and the three swingweights. During the entire acrobatic act, the energy gets channeled from one axis (about the medium somersault axis) into the other two axes. The rotation speed about these axes increases exponentially with time. But, as noted before, this only happens when the rotation begins around the medium somersault axis.
So there you have it — interesting, odd, probably not the focus of your next tennis lesson, but, nonetheless, not totally frivolous or irrelevant. You see, it helps to explain how a player can twist a racquet around the long axis while it is simultaneously rotating rapidly about the usual swing axis, especially during a kick serve. Biomechanists like to use the word “pronation” to describe twisting of the forearm. Obviously, arm muscles are needed to twist the arm, but a racquet can twist around all by itself, even though it doesn’t have any muscles of its own. Just toss it in the air, and it will happily flip over in front of your very eyes. So the arm doesn’t have to do very much pronation work at all. The arm just needs to exert some control over the process so the racquet doesn’t twist in the wrong direction or at the wrong rate. Even if you threw a racquet at a ball, it would probably deliver a pretty good kick serve with about the right amount of slice and topspin.
See all articles by Rod Cross
About the Author
Rod Cross retired in 2003 as an honorary member of the Sydney University staff and continues to work on the physics of sport and forensic physics. He is a physicist and co-author of The Physics and Technology of Tennis available from the USRSA.
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