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The Physics Of Forces In Judo:
Making The Weak Equal To The Strong

By Jearl Walker

Judo is a martial art that demands an intuitive understanding of the physics of forces, torques, stability and rotational motion. This article examines a few of the basic throws of this form of combat. The grace that each throw requires is not easily conveyed, but each throw can be broken into components that can be examined in terms of classical physics. The experiments I shall describe call for actual performance of the throws, but you should do them only under expert supervision since Judo if performed incorrectly can be dangerous to you and your opponent.

In judo the main goal is to overcome your opponent’s stability. The skill lies in the anticipation of his movements and the timing of your response. The idea is to avoid forcing your opponent into a firm resistance to your throw that would pit your strength against his. A small but skilled budo player has a distinct advantage over a larger but unskilled opponent if the contest of strength is avoided.

The Basic Hip Throw

The basic hip throw in Judo

Probably the best example of this advantage is the basic hip throw, which is most effective against a taller and slower opponent. In the normal judo competition you face your opponent with your hands grasping the lapels or shoulders of his uniform. To execute the throw you step forward with your right foot to a point in between his feet, pulling him downward and toward your right. The throw works well if you have caught your opponent just as he has stepped forward with his right foot. He is still stable against a pull directly toward you, but is considerably less stable against a pull to your right because of the position of his feet.

The footwork in the hip throw

During your step forward you curve your body forward so that your head is at your opponent’s shoulder level. Next you rapidly turn your left hip backward while pulling him into your right hip. This should be the first body contact during the movement. If you continue the pull with your hands and turn to the rear with your left hip until you are facing in the same direction as your opponent, he will be rotated over your right hip and into the mat.

Since you do not want to hurt your opponent in the sport, you maintain your grip on his uniform during the fall so that he lands on his left side and can slap the mat with his left arm during impact. The slap spreads the impact force over a larger area so that the stress on his ribs is not enough to hurt him. Part of the early training in judo involves timing the slap on the mat to coincide with the impact. The only time I have been hurt in the sport is when I failed to slap properly.

Timing and smooth execution are essential to the hip throw, but an understanding of the physics, particularly of the torques and the center of the mass, is also necessary. Your opponent’s center of mass is the geometric center of his mass distribution. It can be regarded as the point where the gravitational force acts on the body as a whole, which is why it is sometimes called the center of gravity. Your opponent is stable as long as his center of mass remains over the support area outlined by his feet. When he stands upright in a normal posture, his center of mass is approximately between his spine and his navel. Therefore he is stable until you force or trick him into moving his center of mass or into losing part of his support area.

An instability due to the weight vector

Suppose during a throw you manage to move your opponent’s center of mass forward of his feet. Even if you then no longer aid the throw, the gravitational pull on his center of mass creates a torque that might make him fall. To calculate a torque one must multiply two items: the force acting to bring about a rotation and the lever arm between the pivot point and the force. The lever arm is the perpendicular line from the pivot point to a straight line through the force. If you have made your opponent unstable, the force that can rotate him about his feet to the mat is the gravitational pull on him. I represent this force, which is merely his weight, by a vector going straight down from his center of mass. Here the lever arm is the horizontal distance between the pivot point at his feet and an extension of a vertical line running through the weight vector. The torque on an unstable opponent is the product of his weight and the lever arm. When your opponent is upright the lever arm for his weight vector is zero and so the torque is zero too. When he is caught with the center of his mass forward of his feet, the lever arm is no longer zero and the resulting torque causes his rotation. The longer the lever arm (the more he is leaning), the greater the torque.

One of the objectives of judo is to trick your opponent into an unstable position so quickly that recovery is impossible. Once he is unstable you can continue the throw by applying another torque to him, one that will bring him to the mat long before he can even attempt to regain his stability.

During the hip throw you initially pull on your opponent’s uniform to make him unstable. If you pull directly toward you, you will not easily cause this instability because his center of mass would be moved over his forward foot. He could then maintain his balance by bending that knee. To make him unstable you have to move his center of mass a relatively great distance until it is beyond his forward foot. The motion requires a strong and prolonged pull, which he (the opponent) can counter quickly.

The forces in a hip throw

An easier way to make your opponent unstable is for you to pull to your right, because in that direction his center of mass must be displaced only a relatively short distance before it no longer lies over the support area. He will probably be unable to counter such a pull before the instability is established. Thereafter he will not be able to counter the continued rotation involved in the hip throw.

Your pull has an additional purpose. It curves your opponent’s body so that his center of mass is brought forward to his navel or just outside his body. This new position will aid you in rotating his body over your right hip. Once body contact is made a new pivot point is established at your hip and your pull creates a new torque on the opponent, one that will cause the rotation over your hip.

As mentioned above, the torque is calculated by multiplying the force on the opponent by the lever arm between the line of that force and the pivot point. This time the force is your pull and the pivot point is your hip. Thus the hip throw gives rise to two torques on your opponent, one torque due to his own weight and unstable position and one due directly to your pull you are exerting on him. The throw begins with the first torque so that you can set up the second one without resistance from him.

Suppose you do not curve your opponent’s body forward and bring his center of mass out to his navel. Then when you attempt to rotate his body over your right hip, a torque due to his weight will actually counter the torque from your pull on his uniform. Suppose he is still in an upright position when you make the body contact and attempt to rotate him over your hip. The pivot point for the rotation is your hip: I shall apply it in determining the lever arms for both the torques, then acting on the opponent.

An improper hip throw

One of the torques is the product of your pull and the lever arm from your hip to the line passing through the vector of the pull. The other torque is the product of your opponent’s weight and the lever arm from your hip to a vertical line passing through the opponent’s center of mass.

If your opponent is standing upright, the torque from his weight opposes the torque you are applying, since it attempts to rotate him in the opposite direction over your hip. To finish the throw you now must overcome the torque due to his weight, but the time required destroys your advantage in the surprise of the throw. Moreover, you must pit your strength against his.

When the hip throw is properly executed, you bring your opponent’s body forward in a curve, move his center of mass out to his navel and so decrease or eliminate the lever arm associated with his weight. The torque due to his weight is therefore diminished and you have a comparatively easy time rotating his body over your right hip. The throw works better on an opponent who is taller than you are because you can pull him downward into the proper curved posture more easily than you can an opponent who is your height or shorter. You can also more easily slide your right hip under a taller opponent. The lever arm of your pull on the uniform of a taller opponent will also be longer, thereby providing more torque to bring him over your hip.

The Major Outer Reaping Throw

The “major outer reaping throw” (it is called o-soto-gari in Japanese) is somewhat easier to understand in terms of the rotational motions. As your opponent steps backward with his left foot, you step with your left foot just to the outside of his right foot and pull downward on that foot. Your pull will also be toward his right rear so that his body is curved backward. He is already in an unstable position because your pull moves his center of mass to his right rear and away from the support area of his feet. He cannot escape by sliding his feet to the rear and regaining his balance because you have forced him downward. His instability results from the torque his weight creates around a pivot point at his feet, primarily his right foot.

The major "outer reaping" throw

The forces in the throw

The lever arm runs between the pivot point and a line through his weight vector. Placing him in this position sets him up for the next part of the throw, in which you further remove his support area and apply a second torque to bring him rapidly to the mat.

You continue the throw by stepping with your right foot around and behind your opponent’s right foot. Then you sweep your opponent’s hip and leg to your rear while you force his right side downward and to the rear, leaving him with virtually no support base.

The two torques cause him to rotate about a pivot point on your right leg. Even if you did not continue to pull on him after sweeping his leg, he would rotate about your leg because his center of mass is being pulled down by gravity. Your downward pull provides another torque to hasten his fall. In this throw the two torques complement each other.

The Sweeping Ankle Throw

The “sweeping ankle throw” (okuri-ashi-barai) removes your opponent’s leg support in a similar manner. As he is about to place his weight on his right foot in the course of a step forward or backward, you sweep your left foot into that leg just above the ankle. Simultaneously you pull his uniform in the original direction of his travel. Suppose he was moving forward. You pull forward (and so meet no resistance from him) as you sweep his right foot into his left one. Even if he manages to keep his left foot on the mat, his support area is greatly reduced and is swept from under his center.

His weight vector through the center of mass provides a torque that will take him to the mat. If you lower your body while maintaining your pull on his uniform, you will provide another torque that will rotate him to the floor. The pivot point is his left foot, and again the two torques complement each other.

The ankle Sweep

Mat Techniques

How to execute a cross armlock

Judo classes also teach methods of disabling an opponent on the mat. Most of these “hold downs” entail trapping your opponent on the mat with your weight positioned in such a way that he (the opponent) cannot roll over or rise, even if he is stronger than you are. For example, in the “cross armlock” (juji-gatame) your are positioned with part of your weight on the upper torso of your opponent on the mat. He not only is prevented from rising but also probably will not even move for fear of having his arm broken.

The maneuver originates when you are astride your opponent, who is on his back. When he raises his left arm to ward you off, you grasp his right with both of your hands, fall to his left side, throw your right leg across his neck and with your left knee raised drive your left ankle into his side. His left arm is pinned between you legs with the elbow downward. Even a gentle downward pull on his wrist creates tremendous torque on his arm around the pivot point where that arm crosses your right leg. He cannot sit up because your weight creates an overwhelming torque on him as he attempts to rotate his trunk about a pivot point at his hips. He cannot free his left arm even if he is considerably stronger than you are. He could try to counter your torque on his arm by using his shoulder muscles, but they would pull on his arm at approximately the location of the pivot point and so their pull would have a short lever. As in most judo techniques, a person trained in creating the correct torques on an opponent has a tremendous advantage even if the opponent is much stronger.

In the second article in this two-part series we will examine the application of the physics of force to another combat system: aikido.

Article ©2002 Jearl Walker
Artwork ©2002 Michael Goodman


This article has been edited for clarity. This article and the following one in this series, “The Application Of The Physics Of Forces in Aikido,” originally appeared the Scientific American in the July 1980 issue under to column, “The Amateur Scientist.” It is reproduced with the consent of Scientific American and the author, Jearl Walker. We would also like to thank Michael Goodman for the many illustrations, that also appeared in the original article, as well as Oscar Ratti for his contribution of the first illustration in this article.

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About the Author:

Jearl Walker studied tae-kwon-do in high school but had to give it up to survive college. After earning a PhD in physics, he began to use karate in his physics classes at Cleveland State University, where he is a professor.
He published on the physics of karate strikes in 1975 in the American
Journal of Physics. The article reprinted here was one of his monthly
"Amateur Scientist" articles from his 13 years of writing for Scientific American. It is reprinted with permission,

To find more articles of interest, search on one of these keywords:

Judo, physics of judo, judo throws, judo matwork, leverage, hip throw, major outer reaping throw, o-soto-gari, osotogari, sweeping ankle throw, Okuri-ashi-barai, cross armlock, juji-gatame, hold downs

Read more articles by Jearl Walker

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