The Mathematics of Torque

 

How heavy are Clubbells, really? How hard is it to swing a heavier Clubbell®?  How much work is it to swing it faster?  How quickly can I expect to progress in weight?  What is the increase in perceived effort as I move through the different weight divisions? This article will provide answers to these questions that make both logical and intuitive sense.

First we consider a relatively well known aspect of circular motion, the applied moment to moment force causing an object to rotate, called torque.   This is just one part of the total work involved in swinging a Clubbell®, but it is easy to understand and nicely illustrates the answers to our questions.  Also, the concepts we need will give us a deeper appreciation for the nuances of work involved in swinging any weight.

There are a couple of ways we might proceed.  We could use Newton 's Laws of Motion, applying vector analysis to the various forces involved, a method more suitable to the mathematically masochistic.  Or we could look at the work-energy relationship involved in swinging, using a few easily understood and intuitive concepts to draw some useful conclusions.                     

Let's avoid complex formulas and take a simpler approach...examining circular motion from an energy expenditure point of view (work), and how this might apply to the various Clubbell® weights.

To answer our questions and aid our understanding, we need a few concepts:

• Force is the product of an object's mass and its acceleration. 
• Work (perceived effort) is force acting through a distance over time.
• Energy is the capacity to do work.
• Power is the rate at which work is accomplished.
• Torque is force causing rotational motion (Clubbell® weight times radius of swing)

Here, energy can be thought of as potential (inactive) and work as kinetic (active).  Then power generation becomes the rate at which we can conduct or channel our energy (convert potential to kinetic) as we swing the club.

From physics, we also know that for an object of constant mass and a circle of constant radius, the net force required to move the object in a circle is directly proportional to the square of the object's speed.   If the speed is doubled, the force is quadrupled.  In other words, to swing a Clubbell® twice as fast takes four times the work.

So the energy (expended as work) required to generate power involves torque applied over a distance (arc or full circle) for a given time (duration of swing * total reps).  Things affecting power generation are the mass of the Clubbell®, the acceleration (both keeping it going and stopping it on a dime), the distance the club moves through space (shorter or longer club, arc or full circle), the duration of the motion, and finally the speed.

We can think of (some of) the work required to swing a Clubbell® as torque applied along the arc of the swing at a given rate (speed) for a given duration (reps).  This is rather simplistic and ignores many other forces, such as centripetal force (grip strength) required just to hang on to the Clubbell®.  But the concepts are pretty intuitive.  We all know how the number and speed of reps influences our perceived effort, and our bodies know that there is a lot more going on than just torquewhen we work out.

Looking at the following chart (here's the multiplication), we can see how the amount of work (torque) goes up quickly as the size of the Clubbell® increases:

Weight	Length	Torque
5 lbs	20 inches	100
10 lbs	25 inches	250
15 lbs	25 inches	375
20 lbs	25 inches	500
25 lbs	27 inches	675
45 lbs	27 inches	1215

The difference in weight between the smallest and largest Clubbell® is only forty pounds.  But the amount of work necessary to use it is more than ten times as much.  (Units are irrelevant here for torque, since we're just looking at relative percentages).   And this isn't counting grip strength!  In addition, the explosive start and the rapid stopping on a dime compress the time component, increasing the acceleration/deceleration, so the work component jumps up (force equals mass times acceleration).

If someone did somehow manage to double the speed and also jump in size from 5 to 45, it would take an energy expenditure that goes from 100 to 4860, an increase of 48 times.  Applying this same reasoning, we can figure out the relationships between any two Clubbell®s.  For example, just a five pound jump from the 5 to the 10 pound Clubbell® increases the workload by one hundred and fifty percent, assuming no increase in rotational speed.

It is important to remember that torque is just one aspect of the work involved in Clubbell® training.  The movements are multi-planar and non-uniform, and when combination routines are considered, it would take a super-computer to figure out all the forces involved and the energy expended.

It is clearly important to accurately assess one's strength and capacity for work before jumping to the next size Clubbell®.  Circular Strength Training places real demands on the body that increase rapidly with just small increases in weight. Adding weight multiplies work.

Michael Gannon, CST Alpha