How to calculate total raised mass and body submass

Sometimes, in fitness exercises, the exercising individual would have to raise a part of their body (body submass) as an exercise. Eventually the body submass raised can be associated with an extra free weight to add intensity to the exercise. A typical problem is to log this type of exercises properly.

Exercises raising purely a free external weight can be logged as such and do not need any precaution. On the contrary, for exercises involving a part of the mass of the body (the so called body submass) one should always log the TOTAL WEIGHT for proper and reusable data records.

The knowledge of total pulled weight (not body weight, only the part of it that acts as a weight) is typically crucial for a controlled decrease or increase of intensity based on percentage. For example, it is now a well known fact that the most efficient progression is obtained for 5-10% increases in exercise weight.

A common mistake is to apply say a 5-10% change in only the extra free weight. It necessarily does not produce the targeted effect because it would induce a different percentage of change at the scale of the total raised weight.

The solution for this problem is elementary mathematically speaking but I am surprised I found in nowhere online at this point of time.

The small bits of maths to get this problem solved

The modeling of the problem is based on the WEIGHT (w) vs. MAX_REPS (r) graph. If you never heard about this : Two-set 1-RM calculator. Basically, experimental studies taken over by Matt. Brizcky agrees that, under certain hypothesis, a good approximation of the connection between WEIGHT pulled and the MAX amount of repetitions one can perform with that weight, is a straight line :

Relation between WEIGHT and MAX REPS

Relation between WEIGHT and MAX REPS

This equation presents 3 unknowns, the total raised weight w_tot, and the parameters a and b which the literature presents as being unique for each person and each exercise.

We therefore need 3 equations :

In this set of equations, w_tot refers to the total weight moved during the exercise. It consists of a submass of the body weight (w) and an eventual free weight (w_fw), and a and b described earlier.

Solving this set, you get w which is the submass of your body that is always involved. Add up whatever extra free weight you like to use and the total (w_tot) is the weight you can log for this exercise. Every change of intensity should be done on w_tot and never on the free extra weight alone.

Practical Procedure

In practice, the protocol is the following:

  1. Take 3 extra free weights. (zero is an acceptable value). These will be w_fw1, w_fw2, and w_fw3.
  2. For each weight, perform repetition until fatigue, and write down the respective maximum amount of reps performed. These values will respectively be r_1, r_2 and r_3.
  3. Use a solver to solve the equation (online calculator in project)
  4. Always use w_tot (= w + w_fw) to log your progress.
  5. Accessorily, you can use a and b to plot your w vs. r straight line to link weight and reps to failure.

Examples of use

Typical critical examples of use are presented below:

Example 1 :

You perform fix bar pull-ups (solicitates the back) with no extra weight and you manage to perform 12 reps (fatigue). Your program is an alternate program telling to switch from 70% to 95% of your 1RM. What is your 1RM ? (What would you input in a 1RM calculator ? zero ? because you pulled zero extra weight ? how about your own weight ?)

Example 2 :

You perform fix bar pull-ups (sollicitates the back) with no extra weight and you manage to perform 12 reps (fatigue). You therefore want to add weight to add intensity and get back to a maximum potential of 8 reps (this usually corresponds to an increase of about 10% in total raised weight). What weight should you add ?

Recommendations and limits on the use of this method

The values of a and b change with your progress. Similarly, if you progress it makes sense that your body mass changes, therefore potentially inducing a change the weight you raise. Although this method does not need to be used every time you log data, remember to renew the data every time obvious progress is observed.

This simplified method also is better for exercises involving a translation. Exercises involving a rotation induce momentum. Weight involved in a momentum cannot be added just like that: the contribution of its distance to the center of rotation must be considered.

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One thought on “How to calculate total raised mass and body submass

  1. Ben mon vieux ! Tu m’as l’air plus passionné par le sport que par ta recherche et tes cours à NUS.
    Mais je te comprends bien… 😉


    YB: Pour les cours a NUS on est d’accord, ils sont tristement de bien pietre qualite compare aux cours en ecoles d’inge, eux-meme pas forcement irreprochables… Ma recherche me passionne beaucoup, je suis juste tenu par le secret, tu dois bien connaitre ce probleme.

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