Hypothetical model for measuring Caloric expenditure as a function of repetition speed or frequency

Disclaimer : This information is not guaranteed to be accurate and I am not liable if you make any decisions or take any actions, in terms of medical treatment, health decisions, exercise, behavior or anything else based on the information presented.  I am not telling you any decisions to make even if I use terms like "you" but am using the word "you" as part of a writing style to simplify writing.  Any suggestions for what "you" should do are not for you personally to do but what someone might do as part of a exercise or nutrition program which might help some people's health and make other people's health worse.  You should not do any activity that will make your health worse even if "you" should do it according to the program described.  If the information is wrong and you believe it is true, act on it and it causes you problems, I am not responsible because I have warned you the information is not guaranteed to be accurate.

Disclaimer : I have read, heard or saw a lot of material about exercise and some of these formulas might be found in other material but I can not remember the source if you know sources for any or all these formulas from outside this article please put them in the comments so I can source them.  I do not know if I umiquely discovered any of these formulas or if all of them can be found elsewhere.  Each of these individual formulas may or may not be true.

Hypothetical model for measuring Caloric expenditure as a function of repetition speed or frequency

Basis for formula.

Assumption 1 : If you hold a isometric pose it will use a constant amount of Calories per time.  If you hold it M as long it will expend M times as many calories

Assumption 2 : If a muscle resisted a force M times heavier with the same type of resistance in terms of concentric, eccentric or isometric it would expend M times as much calories per time.

Assumption 3 : If a muscle displaced a force a distance M times as far with the same type of resistance in terms of concentric or eccentric it would take M times as many calories 

Assumption 4  : The amount of force a muscle resists will not be the external force from an object but the torque from the object times a constant except when the torque is zero.  The amount of the force a muscles resists will be the external force from an object times a constant when the torque is zero, this force is to prevent bones from being pulled apart, although muscles can not push to prevent joints from beimg pushed together because muscles only pull.  These two things can be added together to get a approximation although it will be a slight overestimate.  An exception should be made where they are not added together when the force from the external object is always changing directions to produce maximum torque such as some weight machines as opposed to free weights and resistance bands.

Assumption 5 : During repeated concentric and eccentric contractions you can add two different components together to estimate Caloric expenditure, the first component being related to the distance of a path traveled by the object and the second component being related to the total amount of time these contractions have been occuring.  

Assumption 5A : The first component is based on  based on the equation for mechanical work.  Leverage must be considered, it is not simply the force of the object resisted times the distance traveled because there is a different torque in each position.  It will be the integral from the starting time to the ending time of the mechanical power as a function of time.  The mechanical power as a function of time will be the absolute value of the linear or translational velocity the object is being moved at in the direction of the force from the object providing resistance times the absolute value of the force from the object being moved times a constant plus the absolute value of the torque from the object being moved times a constant. 

Assumption 5 B : The second component is based on the assumption that holding a isometric contraction for a longer amount of time uses more calories even though these contractions are not isometric but concentric and eccentric.  The second component is assumed to be the amount of time spent in these contractions times a weighted average of the percent time a muscle spends in each position times the isometric power it would spend in each position to hold the resisted object stationary which is different in each position based on leverage.  The first component is also effected by leverage.

Let N = Number of repetitions

Let S = Number of repetitions per time

Path Length traveled = N * constant

Path Length traveled = S * time * constant

S is directly proportional to Speed either angular or linear

Energy = K1*Time + K3*Path Length traveled

Energy = K1*Time + K2*N

Energy = K1*Time+K2*S*Time

Energy/Time = K1 + K2 * (N / Time)

Energy/time = K1 + K2*S

K1 > 0 and K2 > 0 and K3 > 0

Int[y, g(y)] = the anti derivative of g(y) with respect to y

K1 = (Int[Theta = Ending Angle, Isometric Power(theta)] - Int[Theta = Starting Angle, Isometric Power(theta)]) / (Ending Angle - Starting Angle)

Isometric Power(theta) = Energy expenditure per time if the amount of force resisted (such as the weight of a weight) in the concentric repetition was instead held isometrically at a angle of theta

Isometric Power(theta) = Torque(theta) * constant + Force(theta) * constant

If you are isometrically pulling a weight to keep it from dropping with force in a direction parellel or anti parellel to gravity with a lever arm parellel to gravity there would be zero torque but still isometric energy expenditure

S is directly proportional to both linear speed and angular speed of concentric contractions.  Linear speed is directly proportional to angular speed.  Linear speed is equal to the radius times the angular speed in radians per time for a circular path of a constant radius.  An assumption of a path along a circular arc or a circular segment rotating around a single joint shall be made.

Angular Speed is assumed to be constant for the concentric portion of the repetition and also a constant for the eccentric portion of the repetition although not necessarily the same constant for both the concentric and eccentric portion. 

For a constant number of repetitions the lower the repetitions per time the more energy is expended

For a constant number of repetitions the more time is spent doing each repetition the more energy is expended

For a constant amount of time the more repetitions per time the more energy is expended

For a constant amount of time the more repetitions are done the more energy is expended

The more repetitions per time are done the more energy is expended per time

The faster the angular or linear speed the repetitions are done the more energy is expended per time

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