Text and example by Michael Frind.
(June 29, 2005)
Here is a very basic example of how forces on joints in the human body can be calculated. (This calculation really does qualify as a "back-of-the-envelope" calculation. As you can see from the scan below, I did it on a 3M Post-It Note. With all the time I spend working at the computer these days, my handwriting has deteriorated to the point of "chicken-scratch", but my scribblings are still legible.)
Because all the forces in this example act vertically, we do not need to worry about sines and cosines, and so the calculations are very easy. (The sine of 90 degrees is 1, and the cosine of 0 degrees is 1.) This example was deliberately kept as simple as possible in order to show the principle behind how forces act on knee joints.
The example here assumes a joint-to-end-of-limb distance of 1 metre (yes, I know this is unrealistically long, but I wanted to keep the numbers and units as uncomplicated as possible), and a joint-centre-to-tendon-insertion-point distance of 0.1 metres. (In a real person, these two distances would probably be roughly 0.5 metres and 0.05 metres, hence 50 centimetres and 5 centimetres, but the overall 10:1 ratio would remain the same. Try the calculation shown below for yourself, but instead use 0.5-metre and 0.05-metre moment arms. You will find that the final results will be the same. This is so because the ratio of the tendon-to-joint moment arm to the weight-to-joint moment arm remains 10:1. If you change this ratio of moment-arm lengths, you will find that the forces at the joint and tendon will vary accordingly.)
This sample problem is a staple in upper-year high-school physics courses and first-year university physics courses. In engineering (civil, structural, mechanical) programs, it is covered in a course dedicated to statics. (In statics, no motion occurs. If movement is actually occuring, then the problem falls into the category of dynamics. In dynamics, instead of the sum of forces being zero, the sum of forces equals mass multiplied by acceleration, in accordance with Newton's Second Law of Motion.)
Figure 1: Back-of-the-envelope calculation of forces on a human joint.
So, we can see that if the weight being carried (e.g. by the person's hand) is 400 Newtons, then the tensile force in the biceps tendon is 4000 Newtons, and the compressive force on the bearing surfaces of the elbow joint is 3600 Newtons. (This simplified example assumes that the forearm is weightless.) Clearly, if the elbow joints harbours some type of injury, for example articular-cartilage damage, then serious concerns might arise.
Note that if motion is involved, dynamic loadings come into play. Dynamic loadings can be many-fold greater than their static counterparts. As a simple example, consider how much a bed mattress deflects when you stand still on it. This would be an example of a static loading.
But if you jump up and down on the mattress, you are loading it dynamically. The dynamic loadings can be so great that the mattrees springs might incur damage. The same concern of dynamic loadings applies to a bathroom scale: if you jump up and down on the scale, the readings will be outside of the range of what the unit can measure. Because the scale is not designed to measure such high loadings, the mechanism would incur damage. This gives you some idea of the enormous dynamic loadings exerted on the knee jump during running and jumping activities.
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Detailed information on knee anatomy, physiology and injuries can be found by reading the full-text articles provided in this Knee Library. (It is also possible to find abstracts-only articles on websites such as National Library of Medicine and PubMed. If you come across abstracts of articles for which you want the full-text article, e-mail Michael Frind and I will dig them up for you.) Writings on knee-related topics by Michael Frind can also be found by searching Interim Archive of Bob's Board, the the Bob's Board (Kneeboard) discussion forum, and by using said forum's integrated search feature (be sure to set the search-engine time frame for more than the default three days).
