Double 90 vs single 45 degree muscle contractions

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Introduction

If you train the muscles that move a joint through two perpendicular directions of strength training only vs a single direction of strength training at a diagonal between those two directions only.  Would you be stronger when moving in that diagonal direction if you trained in that diagonal direction only?  My suspicion is the maximum amount of force you could resist in that diagonal direction would be greater with diagonal strength training but only by a limited amount.  And that the maximum force would be multiplied by the square root of 2 or less when training at the diagonal only vs training in those two perpendicular directions only. 

Model 1 A finds no difference

Model 1 B has the diagonal only training have twice the force at a diagonal as training at 2 perpendiculars only

Model 2 finds no difference

Model 3 has the diagonal only multiply the maximum force by up to the square root of 2 times that of training at 2 perpendiculars only 

How muscles are aligned in the body

If you drew a straight line between the average location of the insertion points and the average location of the points of origin of any muscle when someone is in anatomical position there would be a wide variety of angles instead of a nice easy to think about set of angles that are perpendicular or parallel to each other

Limited number of directions for orthogonal coordinate systems

Translation is to move in a straight line

The movement of an rigid object in three dimensions can be described as a sum  of three perpendicular rotations and their opposites and three perpendicular translations and their opposites

Potential application to save time in strength training from a limited number of directions

It would save time if the principle of orthogonal coordinate systems could limit the number of exercises someone needs to do to be strong moving a joint in any direction.

Question to determine if the potential application could work

A question to figure out if this principle can be used would be how much force and movement length as well as related parameters to those measurements of impulse, power and work would be lost if two muscles that move a joint in directions that are 90 degrees apart from each other substituted for one muscle that moves a joint at a 45 degree angle from both of them

Model 1A

The premise of this model is you have so much mass to distribute among muscles and you can either split that mass into two separate muscles or put it all into one muscle.  How would the force be different one the diagonal between those two options? 

The pair of muscles will be called muscle 1 and muscle 2

The single muscle will be called muscle 3

The muscles form a rectangular solid shape in resting position (which is not true for muscles but will simplify things)

The two muscles have the same resting mass and resting volume as the one muscle

The volume of each muscle is it's resting length times it's resting area

The force each muscle produces is equal to a constant times it's resting area 

The force a muscle can produce does not change as it's length changes (which is not true but will simplify things)

The one muscle, muscle 3 has an origin at ( 0, 0 ) and an insertion at ( L, L )

The two muscles have a origin at ( L, 0) for muscle 1 and ( 0, L ) for muscle 2 and an insertion at ( L, L )

The origin of the muscles will not move as they contract but the object they insert into at the insertion coordinate will move and change coordinates as the muscles contract

The muscles are going to pull a object through the process of translation from the location ( L, L ) to the limit as deltaL approaches positive zero of ( L-deltaL, L-deltaL ) 

Note : this would be problematic because the two muscles would no longer pull the object at perpendicular angles to each other any more as the object moves closer to ( 0.5L, 0.5L ) from the original point of ( L, L ), so instead it will be assumed that the muscles pull the object an infinitesimal length so that pulling the object at two perpendicular angles can be compared with pulling the object at 45 degrees with a single muscle.  No matter how hard the muscles two contract they should not be able to move the object closer to ( 0, 0 ) than ( 0.5L, 0.5L ) if they both pull at the object they insert into with an equal force.

Length = L, for double muscle pair of muscle 1 and muscle 2

Length = ( 2 ^ 0.5 ) * L  for single muscle of muscle 3

Volume for muscle 3 = Volume for muscle 1 + Volume for muscle 2

Volume of muscle 3 = 2V

Volume of muscle 1 = Volume of muscle 2 = V

Area of muscle 1 = Area of Muscle 2 = V / L

Area of muscle 3 = 2V / [ (2 ^ 0.5) * L ] = ( 2 ^ 0.5 ) * V / L

K is a proportionality constant

K = Force per area

Magnitude of force of muscle 1 = Magnitude of force of muscle 2 = K * V / L

Magnitude of force of muscle 3 = K * ( 2 ^ 0.5 ) * V / L

Force 1 + Force 2 = ( - K * V / L, - K * V / L )

Magnitude of ( Force 1 + Force 2 ) = K * ( 2 ^ 0.5 ) * V / L

Magnitude of Force 3 / Magnitude of ( Force 1 + Force 2 ) = 1

Model 1B

Same as model 1 A except origin of Muscle 3 is ( 0.5L, 0.5L )

Length = L, for double muscle pair of muscle 1 and muscle 2

0.25 + 0.25 = 0.5

( ( 1 - 0.5 ) ^ 2 + ( 1 - 0.5 ) ^ 2 ) ^ 0.5 = 0.5 ^ 0.5

Length = ( 0.5 ^ 0.5 ) * L  for single muscle of muscle 3

Volume for muscle 3 = Volume for muscle 1 + Volume for muscle 2

Volume of muscle 3 = 2V

Volume of muscle 1 = Volume of muscle 2 = V

Area of muscle 1 = Area of Muscle 2 = V / L

Area of muscle 3 = 2V / [ (0.5  ^ 0.5) * L ] = ( 2 ^ 1.5 ) * V / L

K is a proportionality constant

K = Force per area

Magnitude of force of muscle 1 = Magnitude of force of muscle 2 = K * V / L

Magnitude of force of muscle 3 = K * ( 2 ^ 1.5 ) * V / L

Force 1 + Force 2 = ( - K * V / L, - K * V / L )

Magnitude of (Force 1 + Force 2) = K * ( 2 ^ 0.5 ) * V / L

Magnitude of Force 3 / Magnitude of ( Force 1 + Force 2 ) = 2  

Model 2

A object is moved from ( 1 meter, 1 meter) to ( 0, 0 )

resisting a constant force of ( 1 Newton, 1 Newton ) with a magnitude of ( 2 ^ 0.5 ) Newtons while traveling along the path no matter what direction it travels

Is the work different if it follows each of the following two paths 

Work = Force dot Displacement

Path 1

( 1 meter, 1 meter ) to ( 1 meter, 0 ) to ( 0, 0) 

or

 

( 1 meter, 1 meter ) to ( 0, 1 meter ) to ( 0, 0) 

|  Work | = ( 1 Newton, 1 Newton) dot ( 1 meter, 0 ) + ( 1 Newton, 1 Newton ) dot ( 0, 1 meter )

| Work | = 1 Newton Meter + 1 Newton Meter = 2 Newton Meters

Path 2

( 1 meter, 1 meter ) to ( 0, 0 )

| Work | = ( 1 Newton, 1 Newton) dot ( 1 meter , 1 meter ) = 2 Newton meters

 

| Work | = ( 2 ^ 0.5 ) Newtons * ( 2 ^ 0.5 ) meters = 2 Newton meters

Model 3

( [ 2 ^ 0.5 ] Newton, 45 degrees) = ( 1 Newton, 0 degrees ) + ( 1 Newton, 90 degrees ) 

Muscle 1 is strongest pulling at 0 degrees

Muscle 2 is strongest pulling at 90 degrees

Muscle 3 is strongest pulling at 45 degrees

If muscle 1 can produce the most force when moving a joint in directions along a plane tilted at 0 degrees polar coordinates

Maximum force at plane tilted at angle Theta = Maximum Force * cos ( Theta ) for muscle 1

Maximum force at plane tilted at 45 degrees = Maximum Force * cos ( 45 degrees ) for muscle 1

cos ( 45 degrees ) = 1 / ( 2 ^ 0.5 )

If muscle 2 can produce the most force when moving a joint in directions along a plane tilted at 90 degrees polar coordinates

Maximum force at plane tilted at angle Theta = Maximum Force * cos ( 90 degrees - Theta ) for muscle 2

Maximum force at plane tilted at 45 degrees = Maximum Force * cos ( 45 degrees ) for muscle 2

cos ( 45 degrees ) = 1 / ( 2 ^ 0.5 ) 

Maximum force at plane tilted at 45 degrees = Maximum Force / ( 2 ^ 0.5 ) for muscle 1 and for muscle 2

2 * Force ( X hours training) > Force ( 2 * X hours training) > Force ( X hours training ) 

Time training muscle 3 = Time training muscle 1 + Time training muscle 2

Time training muscle 1 = Time training muscle 2

Time training is how many hours someone puts into strength training

Max force Muscle 3 at 45 degrees < 2 * Max force muscle 1 at 0 degrees 

Max force Muscle 3 at 45 degrees < 2 * Max force muscle 2 at 90 degrees

Max force Muscle 3 at 45 degrees > Max force muscle 1 at 0 degrees

Max force Muscle 3 at 45 degrees > Max force muscle 2 at 90 degrees

2 ^ 0.5 = [ 1 / ( 2 ^ 0.5 ) ] + [ 1 / ( 2 ^ 0.5 ) ]

2 / ( 2 ^ 0.5 ) = 2 ^ 0.5

( 2 ^ 0.5 ) / 2 = 2 ^ - 0.5

2 ^ 0.5 < Max force muscle 3 at 45 degrees / ( Max force muscle 1 at 45 degrees + Max force muscle 2 at 45 degrees ) > 2 ^ - 0.5

Copyright Carl Janssen 2022 September 28