A Classless System

Digger

If you want 200 kg blocks moved about… then Stuart is your man.

Some time ago I tried to convince Chris Beardsley that the 3 class lever system of levers was a disaster, for some reason he remained unconvinced.

I first came across this conundrum many decades ago whilst wondering around the local library. I picked up a book on exercise and found an example of a calf diagrammed as a second class lever. I thought it particularly clever that the body would do this because it meant that the calves handled less than bodyweight, but I was disturbed that by turning the book upside down that the calves were now a first class lever and that the calves now had to handle more than bodyweight. At school my physics teacher explained that yes the system was correct and did change when I turned the book upside down… he was clearly insane.

Why the texts (and Interweb) are always wrong

Look in virtually any A-Level text book or any fitness book (or look online) and you will see the following diagram using the calf raise as an example of a 2nd class lever and comparing it to a wheelbarrow:

Fig 1

According to the biology texts this an example of a 2nd class lever. If the bodyweight of the person is 600 Newtons this is taken to be the load on the ankle. The pivot is taken to be the ball of the foot. Thus the force exerted by the muscle is calculated in the above example to be 400 N, i.e. muscle tension is calculated to be less than bodyweight.

Yet something about it doesn’t quite make sense because if you turn it over so that the person performs a calf press with a weigh equal to his bodyweight of 600 Newtons (e.g. on a vertical leg press machine) it becomes a 1st class lever – so the forces come out different when obviously they shouldn’t, i.e. muscle tension is now calculated to be 1200 N which is twice the bodyweight of 600 N:

Fig 2

As a 1st class lever the pivot is taken to be the ankle, the load of 600 N is on the ball of the foot thus the calf force is calculated to be 1200 N.

The reason for this discrepency in calf forces between the 2nd and 1st class levers is that the load on the ankle joint is not bodyweight – the load on the ankle is instead bodyweight + calf muscle tension.

It is much better to avoid using the 3 class system of levers and so bear in mind Newton’s 3rd Law:

Fig 3

The 3rd Law states that for every action there must be an equal and opposite reaction. Thus when it comes to levers all opposing forces must balance AND all moments must balance. I have turned this diagram on its side and used cars pushing against a rigid pole to illustrate the absurdity of thinking of a ‘fixed’, ‘forceless’ pivot. Note that this style of diagram also works if more points are added as in a 4, 5, 6 or more points lever.

Thus a 3 point lever should obey the following:

Opposing Forces

F1 + F2 = F3

Opposing Moments

F1l1 = F2l2

F3l1 = F2l3

F3l2 = F1l3

Also of course l1 + l2 = l3.

The awful 3 class system of levers arbitrarily places where the pivot is and by convention ignores this as an opposing force… so it is best to avoid the 3 class system of levers and instead make sure that all opposing forces balance and all moments balance, i.e. use the classless system in Fig 3.

So the diagram of the calf should look like this:

Fig 4

The load on the ball of the foot is obviously the weight of the person (600 N) and if the pivot points are at 20 cm and 10 cm, this means that the calf force is 1200 N. Thus the total force on the ankle joint is 1800 N – this is because the calf muscle attaches to the tibia thus compressing it into the ankle with 1200 N of force which added to the 600 N bodyweight totals 1800 N. Thus all opposing forces balance and all torques balance.

Conclusion

The only way the calf, in the traditional 3 class system, would operate as a 2nd class lever in wheelbarrow mode is if the calf muscle at it’s upper end were attached to an external object. A wheelbarrow is a 2nd class lever because the person lifting the wheelbarrow is external to the wheelbarrow. A calf is 1st class lever because the muscle is attached to its own lifting system, it is not attached to an external system.

The fact is that any 3 point lever can be considered as either a 1st, 2nd or 3rd class lever and hence the 3 class system of levers is useless – all levers are in fact classless; Figs 3 and 4 should always be used.