The mathematics of old people

“A little thought shows that the old person can equally be modeled as a system of weights and pulleys.”

[Finnish version: click here]

I recently sprained the ligaments in my knee — cosmically insignificant, but led to an interesting vector analysis, and gave insights into how and why old people exist the way they do.

Walking on a steady surface was easy after just a couple of weeks. However, it took me almost two months to be able to freely walk up and down stairs. At the same time, I started to notice that old people have problems with stairs. Why is this?

All that’s required is some high school physics. An old person can be modeled in a variety of ways. I believe the model in Figure 1 is the simplest. Assume that the old person is a lump attached to two fulcrums (left side). Assume the thigh and leg to be equally long (differences don’t materially change the results).

A little thought shows that the old person can equally be modeled as a system of weights and pulleys, as shown on right side of Figure 1. The old person is a lump hanging from a rope that passes over a pulley (the hip). The rope then passes through another pulley (the knee) to a third pulley (the ankle, which is assumed fixed). Another identical lump hangs from this end of the rope. When the system is static, there is a force F on the knee pulley, which needs to be calculated.

Figure 1. Modeling an old person in two different ways

Calculating F requires some vector analysis and trigonometry (Figure 2). The vertical forces are in opposite directions, Fv=mg*(cosa-cosb) The horizontal forces are added, so Fp=mg*(sina+sinb).

Figure 2. Vector analysis

The magnitude of the force vector is F= sqrt(Fp²+Fv²), which after a few steps gives  F=2*m*g*(1- cos(a+b)). The force is smallest (zero) when a+b=0. This happens for example when standing straight or lying flattened. It is largest (4*m*g) when a+b=180 degrees.

The vector sums are easily drawn graphically. Figure 3 shows three examples. (The forces have been normalized to one, so that a vector length of 1 corresponds to a force 2*m*g). If the feet move around by 30 degrees when walking, the maximum force on the knee is about 0.5, or m*g. Thus the knee needs to support the whole weight of the old person while walking.

Figure 3. Vector analysis

When climbing up a stair, the force abruptly rises up to almost 1.5, or 3*m*g. The force is thus larger than the old person’s weight. This seems countrintuitive, until one remembers that the old person is lifting herself up through a fulcrum of her thigh’s length.

Rising from a deep squat can require a force of up the 4*m*g, but at that point the model may be too simplified.

In other words: in rising up a stair, the old person’s knee needs to support a force that is more than three times her own weight. That’s formidable, for anyone.

The next time you see an old person walking up a staircase, remember F=2*m*g*(1- cos(a+b)). And stop to see if she needs help. Please.