Introduction What is Judo? Judo is an MMA style sportthat involves physical and mental discipline, it involves various techniques tothrow your opponent or to pin them to the ground. Judo helps an individualdevelop strength, balance, speed, agility, and flexibility. In judo, one of the main techniques presentare the throwing techniques and grappling techniques. Although there are manytechniques for throwing, they all are encompassed under 3 distinct stages,”Kuzushi” which is the breaking of your opponent’s balance, “Tsukuri” which iswhen you prepare you opponent for the throw, and then the “Kake” which is theexecution of the throw. Three main throws that most judo competitors use arethe hip throws, leg throws, and arm throws since they are the easiest and mostbasic to learn.

Judo’s underlying concept involves enormous amounts of physicsand math, especially when the grappling and throwing aspects of the sport areinvolved. Asa former MMA fighter, I have always learned my techniques in fighting visuallyfrom my coach rather than on paper. Fighting against an opponent came mostlyinstinctively, I never gave a second thought to the math and physics behindeach punch or how to carry out the most effective throw on an individual. I wascurious to understand more about the theoretical aspects of Martial Arts andhow gathering knowledge about it could possibly make me a better and moreaccurate fighter. After learning the art for almost 6 years, I still had noclue how the techniques I use work theoretically, how effectively I can performthem for maximum efficiency, and the various aspects of physics and math thatplay behind each move. I am going to investigate 1 main throwwithin judo called hip-throw(harai-goshi) and 2 main physics principles involved in these throws that relateto the weight of an individual and their force. The center of mass and torque aremost prevalent in executing throws efficiently, understanding them will helpthe attacker gain an advantage against the opponent.

These variables varywithin the person and they affect how the throw can be carried out. Thisthrow is known to be a greater advantage to smaller and faster since it basicprinciple is to knock the opponents balance before can do anything. In thisthrow the opponents face each other, the attacker steps forward with theirright foot to the middle between the feet of their opponent. The attacker pullsthe opponent downward and toward their right. The opponent would be stable againsta pull directly toward the attacker, but because of the position of the feet,an instability is created. Then, holding the opponent’s arm, the attackerrotates their hip and throws the opponent to the ground. Although this doesn’t looklike it’s possible visually, it works theoretically using physics and basicmath. The opponent’s center of mass is stable as long as it is over the supportarea of his feet.

A normal person’s center of mass is between their spine andnavel, because of this, many throws target that area to manipulate a person’scenter of mass to execute a throw with minimal effort. The reason why thisthrow is most beneficial to lighter individuals is because if a light-weightedperson were to fight a heavier person, the center of mass would be closer tothe heavier person, giving the lighter individual an advantage since the torquebecomes greater making up for the amount of strength of the lighter individual.To figure out where the center of mass between two individuals is going to be,you have to use an equation where m = the weight of the individualin kilograms, x= position of the individual, and X=position of center of mass.

Forexample, if a 25 kg individual is going to throw a 75 kg individual and theyare 60 cm apart, you would find the center of mass by plugging in the numbersaccordingly to the variables. Onceyou get the answer, the center of mass will be 45cm from the starting pointwhich was 0. In this case, the center of mass will be closer to the 75 kgindividual creating a smaller torque giving the light-weighted person anadvantage in the throw.

In this specific throw, the two opponents are around 1foot apart from each other, and then depending on the weights of theindividuals, the center of mass will change. In the problem above, Isubstituted x1 for 0 and the other for 70. 25cm 45cm 75cm(0) (60)Ifwe were to switch around the values for the x’s so that x1 is 70 andthe other x is 0, we would get another number.

Nonetheless, the center of masswould still be in the same place as it was in the above example. + (Positive)25cm 52.5cm 75cm(0) – (Neg) (70) Imodeled a basic diagram to visualize the equation in the previous problems,following the same properties as a number line, going right is positive andgoing left is negative. If we were to do the same equation but switching thevalues of the x’s, we would get the same center of mass but a different number. + (Positive)25cm 15cm 75cm(60) – (Neg) (0) Inthe equation above, x1 became -70 since the initial position iscoming from the right heading to the left end of the number line, going in thedirection of negative. Although we got a different number, it is still in thesame place since now we are working from the right to the left ends of the linerather than the regular positive route, left to right ends.

To double checkthat answer and its position, just subtract 45 from 60 and you will get 15 cm,thus proving this method of solving the equation. During a throw, if an individual’scenter of mass moves more towards their feet, gravity will create a torque fromits pull on the individual’s center of mass. The equation to calculate torqueis T=torque F=linear force r=distancefrom axis of rotation to where force isapplied Inthis case, the force bringing a rotation and the lever arm between the pivotpoint and the force would be multiplied to find the torque on the unstableindividual since the weight vector is perpendicular to the lever arm. However,when you don’t know the vector that is perpendicular “r” in the diagram above,you would use the sine of the angle given as an easier way to find the torque.But in this scenario since we already know the weight vector, the equation canbe simplified to whereTperp.= the vector perpendicular to r.

To put this into a scenario thatcorresponds with this specific throw, if an individual were to put 60N of forceonto the hip getting ready to throw their opponent, what would be the torque ifthe force is 11 cm away and the direction of the force created a 45 degreeangle?Tosolve this equation, you would use,, and in the diagram, F2 beinga force parallel to r, and F3 being the force perpendicular to r.The hole in the diagram represents the axis of rotation and the rectangle beingthe hip of the opponent. Fperp. encompasses the force in newtontimes the sine of the angle. Although this doesn’t accurately display the shapeof the hip and its axis of rotation, this diagram was the closest I could thinkof for a model to this sort of equation since it conceptually would work thesame.

Inthis problem, since we have to find the force that is exerted perpendicular, wewould have to draw a triangle to find out the leg of the triangle that we need,F3. Based on alternate interior angles, the left angle in thetriangle is also 45 degrees, and since the identical angle is directly oppositefrom F3, it becomes 60 N. ) Nowwe plug this answer into the equation to find torque and multiply it by r inmeters. So,after solving this equation, the torque of the opponent would be 4.667 Nm when60 N of force is applied at a 45 degree angle to the opponent’s hip. When the opponent is unstable, gravitywill pull the opponent down based on the opponent’s weight, this can beconceptually thought as a vector going down from the individual’s center ofmass, the weight vector. Inthis picture, the lever arm is the horizontal line running from the individual’spivot point and the end of the weight vector. The weight of the opponent andthe distance of the level arm multiplied gives the torque.

However, when the opponentis stable standing up the lever arm for his weight vector is zero which makesthe torque is zero. If the person’s center of mass moves forward to where theirfeet are, the lever arm won’t be zero and the torque will cause the person’srotation. If the person is leaning more, the lever arm will increase resultingin a greater torque. For example, if the weight vector of two individuals areboth 20 N and one individual has a lever arm of .5 m and the other .

7m, theindividual with a lever arm of .7m would have a greater torque than the other. Thisparticular throw is fairly simple mathematically when trying to find the centerof mass and the torque, but it gets a little more complicated with other throwsthat don’t target near the center of mass. Doing the research for this topic notonly has helped me gain more insight into the mathematical works of the art butit also helped inspire me to learn Judo and MMA once again. Researching indepth on how physics affects the throw and its quality affected by center ofmass and torque will make me more aware when practicing and learning newtechniques.

My research wasn’t just limited to judo, I gained an interest inwhat other forms of MMA were about and their underlying concepts. Even though Iwon’t be calculating equations in my head whenever I practice, learning more indepth theoretically will aid me when I need to perform a technique accurately,using my newly found knowledge of center of mass and torque.