Suppose we want to do some weight training in the gym to help us with sport and be healthy. We might need to get a few things worked out first.
- How do we know how much weight to lift for a given exercise?
- How do we know how many times (“repetitions”) we should do?
- How do we know how many sets of exercises to do ?
- How do we know how long to rest in between sets ?
- How many times should we go to the gym a week ?
- How long should we rest between trips to the gym ?
There is no ready made mathematical answer to all of these questions.
We need to hire a professional trainer to work out a starting point for us, and that will be money well spent.
However we can also do some internet research and a bit of mathematics, to find out some relevant information ourselves.
While it is good to do some math to determine the training weights that you can use, you should always check with a personal trainer or gym staff, before commencing your program.
The contents of this post are a mathematical review of some websites for weight training, and are definitely not weight training advice for someone to set up their own training program.
Here at Passy World we do regular weights training, but we had our program set up by professional staff at a gym. We recommend that you always do the same.
Let’s look at some of the mathematics of weight training.
Here is a short YouTube video which mentions some of the mathematics involved with weight training.
As demonstrated in the video, the key to each exercise is the “repetition” which is the full motion we do to complete the exercise once.
“Brian Mac – Sports Coach” is a brilliant website about weight training, and the source of most of the material which follows in this post.
We highly recommend visiting the above link for a great overview of muscle strength development and sports training.
On the “Brian Mac” site, ranges of values for recommended repetitions are given, depending on the type of training that the person needs to do.
This is summarized in the following table.
The types of training indicated in the above table, as per the “Brian Mac” site are as follows:
The aim is to develop muscles that are able to to produce repeated contractions under conditions of fatigue. This requires high repetitions (15+) with light loading (30-50% of 1RM). Appropriate for field sports, rowing and martial arts.
The aim is to develop fast powerful movements. This requires medium number of repetitions (6-10) with medium to heavy loading (70-80% of 1RM). Appropriate for power based events e.g. sprinting, jumping (long jump), throwing (Javelin).
The aim is to enable maximum loads to be lifted. This requires low number of repetitions (1-5) with heavy loads (80-100% of 1RM). Appropriate for Power Lifting, Olympic Lifting, Shot Putt.
In the table shown previously, all weight to be lifted is expressed as a percentage of the “1RM” weight value which is set at 100%.
The 1RM weight is the maximum weight we can lift for a specific exercise, where the weight is so heavy that we are only able to do one repetition of that weight. We are not strong enough to do a second lift of that weight.
As we train and get stronger, this 1RM value may become larger, and so it needs to be recalculated from time to time.
But how do we determine the 1RM weight for a particular exercise?
The “Brian Mac” site has a very nice online calculator that looks like this:
Three estimates are provided by this calculator:
- Novice – an adult with less than 1 year of regular weight training
- Experienced – an adult with 1 to 2 years of regular weight training
- Advanced – an adult with more than 2 years of regular weight training
Once we have determined the weight we should lift for 1RM (One Repetitition Maximum) at 100% training weight from the calculator, we can then work out what weight we would need to do for different amounts of repititions, according to the training program we are following: Strength, Power, or Endurance.
For example, Let’s say we obtained 120kg for Military Bench Press as our 1RM from the calculator. If we want to do Endurance Training, then we need to consult our training table.
The table shows that if we want to do 14 reps per exercise for Endurance, then we need to have our barbell weight set at 65% of 1RM.
So our required weight is calculated as 65% of 120kg = 0.65 x 120 = 78kg.
For Endurance training we need to Military Bench Press 78kg of weight.
But what if we want to determine the 1RM for an exercise that is not in the Brian Mac calculator, such as Bicep Curls.
What we need to do is work out the maximum weight we could manage for several reps, until such point as our muscles are fatigued, and we cannot lift the weight any more.
For example, to do a 1 rep max test for the bicep curls, we might lift a 40kg barbell and discover that we can only lift the weight 6 times until our muscles are completely fatigued.
From our previous table we know that 6 repetitions represents 85% of the 1RM weight value.
85% of 1RM is 40kg (Divide both sides by 85)
1% = 40 divided by 85
1% = 0.470588 (Now Multiply both sides by 100)
100% = 47 kg
So our calculated 1RM value for us doing 1 rep of bicep curls is 47kg.
To check the mathematics we have done, we can use an Online 1RM Calculator.
There is a very good calculator for determining the 1RM value, (as well as the training percents), at the following link:
Interestingly, when we use this calculator for our 6 reps of 40kg Bicep Curls, the answer given is 46kg and not 47kg.
There are several slightly different 1RM math formulas, made by different researchers. This is discussed on Wikipedia, and the common formulas are by Brzycki, Beachle, Epley, Lander, Lombardi, and others.
The Online Calculator appears to be using the Brzycki formula:
1RM = Weight / (1.0278 – (0.0278 x Reps))
which is the most commonly used formula by most Sports Scientists and Personal Trainers.
The % Table we have been using in this blog is only an approximation, with rounded off percents. It is therefore not suitable for calculating accurate 1RPM values.
We need to use the Brzycki formula to work out our 1RM correctly. This is simple to do, and involves some basic Algebra substitution.
For our Bicep Curls we have:
Weight = 40
Reps = 6
1RM = Weight / (1.0278 – (0.0278 x Reps))
= 40 / (1.0278 – (0.0278 x 6))
= 40 / 0.861
Now that we know the 1RM for our Bicep Curls is 46kg:
If we want to do “Power Training” then we could work at
12 reps using 70% of 46 = 12 reps using a 32kg weight.
And if we wanted to do “Endurance Training” for our bicep curls, we could work a set of 14 reps using 65% of 46 = 14 reps using a 30kg weight.
You may be wondering what exactly are bicep curls exercises ?
“Expert Village” on YouTube have some really good YouTube videos on how to do different weight training exercises.
So if you forgot some of the things your personal trainer showed you at the Gym, then find a video on YouTube to watch.
Here is a great example video by Expert Village on bicep curls.
Finally, let’s take a look at some of the Mathematics involved in Olympic Weight Lifting.
Competition Weight Lifting is all about overcoming the force of gravity in the most efficient way possible. This involves applying some specific mathematical geometry and alignment between the weight being lifted and the the person lifting the weight.
Here is an excellent science documentary video, which explains the mathematics and geometry very clearly.
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