Introduction involves enormous amounts of physics andIntroduction involves enormous amounts of physics and


         What is Judo? Judo is an MMA style sport
that involves physical and mental discipline, it involves various techniques to
throw your opponent or to pin them to the ground. Judo helps an individual
develop strength, balance, speed, agility, and flexibility.  In judo, one of the main techniques present
are the throwing techniques and grappling techniques. Although there are many
techniques for throwing, they all are encompassed under 3 distinct stages,
“Kuzushi” which is the breaking of your opponent’s balance, “Tsukuri” which is
when you prepare you opponent for the throw, and then the “Kake” which is the
execution of the throw. Three main throws that most judo competitors use are
the hip throws, leg throws, and arm throws since they are the easiest and most
basic to learn. Judo’s underlying concept involves enormous amounts of physics
and math, especially when the grappling and throwing aspects of the sport are

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a former MMA fighter, I have always learned my techniques in fighting visually
from my coach rather than on paper. Fighting against an opponent came mostly
instinctively, I never gave a second thought to the math and physics behind
each punch or how to carry out the most effective throw on an individual. I was
curious to understand more about the theoretical aspects of Martial Arts and
how gathering knowledge about it could possibly make me a better and more
accurate fighter. After learning the art for almost 6 years, I still had no
clue how the techniques I use work theoretically, how effectively I can perform
them for maximum efficiency, and the various aspects of physics and math that
play behind each move.


         I am going to investigate 1 main throw
within judo called hip-throw
(harai-goshi) and 2 main physics principles involved in these throws that relate
to the weight of an individual and their force. The center of mass and torque are
most prevalent in executing throws efficiently, understanding them will help
the attacker gain an advantage against the opponent. These variables vary
within the person and they affect how the throw can be carried out.


throw is known to be a greater advantage to smaller and faster since it basic
principle is to knock the opponents balance before can do anything. In this
throw the opponents face each other, the attacker steps forward with their
right foot to the middle between the feet of their opponent. The attacker pulls
the opponent downward and toward their right. The opponent would be stable against
a pull directly toward the attacker, but because of the position of the feet,
an instability is created. Then, holding the opponent’s arm, the attacker
rotates their hip and throws the opponent to the ground. Although this doesn’t look
like it’s possible visually, it works theoretically using physics and basic
math. The opponent’s center of mass is stable as long as it is over the support
area of his feet. A normal person’s center of mass is between their spine and
navel, because of this, many throws target that area to manipulate a person’s
center of mass to execute a throw with minimal effort. The reason why this
throw is most beneficial to lighter individuals is because if a light-weighted
person were to fight a heavier person, the center of mass would be closer to
the heavier person, giving the lighter individual an advantage since the torque
becomes 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 individual
in kilograms, x= position of the individual, and X=position of center of mass.


example, if a 25 kg individual is going to throw a 75 kg individual and they
are 60 cm apart, you would find the center of mass by plugging in the numbers
accordingly to the variables.






you get the answer, the center of mass will be 45cm from the starting point
which was 0. In this case, the center of mass will be closer to the 75 kg
individual creating a smaller torque giving the light-weighted person an
advantage in the throw. In this specific throw, the two opponents are around 1
foot apart from each other, and then depending on the weights of the
individuals, the center of mass will change. In the problem above, I
substituted x1 for 0 and the other for 70.


25cm                                                                                            45cm                                    75cm

(0)                                                                                                           (60)

we were to switch around the values for the x’s so that x1 is 70 and
the other x is 0, we would get another number. Nonetheless, the center of mass
would still be in the same place as it was in the above example.


+ (Positive)

25cm                                                                                            52.5cm                                 75cm

(0)                                                                      – (Neg)                                   (70)


modeled a basic diagram to visualize the equation in the previous problems,
following the same properties as a number line, going right is positive and
going left is negative. If we were to do the same equation but switching the
values of the x’s, we would get the same center of mass but a different number.






+ (Positive)

25cm                                                                                            15cm                                    75cm

(60)                                                                    – (Neg)                                   (0)



the equation above, x1 became -70 since the initial position is
coming from the right heading to the left end of the number line, going in the
direction of negative. Although we got a different number, it is still in the
same place since now we are working from the right to the left ends of the line
rather than the regular positive route, left to right ends. To double check
that 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’s
center of mass moves more towards their feet, gravity will create a torque from
its pull on the individual’s center of mass. The equation to calculate torque



linear force

from axis of rotation to   where force is


this case, the force bringing a rotation and the lever arm between the pivot
point and the force would be multiplied to find the torque on the unstable
individual 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 can
be simplified to

Tperp.= the vector perpendicular to r.

         To put this into a scenario that
corresponds with this specific throw, if an individual were to put 60N of force
onto the hip getting ready to throw their opponent, what would be the torque if
the force is 11 cm away and the direction of the force created a 45 degree

solve this equation, you would use,, and in the diagram, F2 being
a 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 being
the hip of the opponent. Fperp. encompasses the force in newton
times the sine of the angle. Although this doesn’t accurately display the shape
of the hip and its axis of rotation, this diagram was the closest I could think
of for a model to this sort of equation since it conceptually would work the


this problem, since we have to find the force that is exerted perpendicular, we
would 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 the
triangle is also 45 degrees, and since the identical angle is directly opposite
from F3, it becomes 60 N.



we plug this answer into the equation to find torque and multiply it by r in





after solving this equation, the torque of the opponent would be 4.667 Nm when
60 N of force is applied at a 45 degree angle to the opponent’s hip.

         When the opponent is unstable, gravity
will pull the opponent down based on the opponent’s weight, this can be
conceptually thought as a vector going down from the individual’s center of
mass, the weight vector.


this picture, the lever arm is the horizontal line running from the individual’s
pivot point and the end of the weight vector. The weight of the opponent and
the distance of the level arm multiplied gives the torque. However, when the opponent
is stable standing up the lever arm for his weight vector is zero which makes
the torque is zero. If the person’s center of mass moves forward to where their
feet are, the lever arm won’t be zero and the torque will cause the person’s
rotation. If the person is leaning more, the lever arm will increase resulting
in a greater torque. For example, if the weight vector of two individuals are
both 20 N and one individual has a lever arm of .5 m and the other .7m, the
individual with a lever arm of .7m would have a greater torque than the other. This
particular throw is fairly simple mathematically when trying to find the center
of mass and the torque, but it gets a little more complicated with other throws
that don’t target near the center of mass.


         Doing the research for this topic not
only has helped me gain more insight into the mathematical works of the art but
it also helped inspire me to learn Judo and MMA once again. Researching in
depth on how physics affects the throw and its quality affected by center of
mass and torque will make me more aware when practicing and learning new
techniques. My research wasn’t just limited to judo, I gained an interest in
what other forms of MMA were about and their underlying concepts. Even though I
won’t be calculating equations in my head whenever I practice, learning more in
depth theoretically will aid me when I need to perform a technique accurately,
using my newly found knowledge of center of mass and torque.