# Question about Formulating an Equation

1. Feb 14, 2012

### chomeschooler

Hello.

I am in need of assistance formulating an equation for a certain project of mine.

I am building a model knee joint(picture included)

I would like to solve the following:

How much tension is exerted upon the patellar tendon when one engages in a squatting position at various angles?

This concept of tension exerted upon the patellar tendon can be tested by building a simple mechanical model of a knee joint. The model will incorporate the femur, patella, tibia/fibula, quadriceps femoris, and the gastrocnemius. A spring will be used to represent the tendon of quadriceps femoris. Springs have a certain property known as Hooke’s Law, named after the physicist Robert Hooke. This law states that the “restoring force (F) produced by the spring is proportional to the distance by which the spring has been lengthened (x).” In equation form, this looks like: F= -kx
In other words, the equation asserts that: “the force (F) of the spring is equal to the spring constant (k, a measure of stiffness of the spring) times the distance (x) that the spring has been stretched. The minus sign says that the force is exerted in the opposite direction of the stretching.” In other words, when stretching the spring out, the spring force is pulling back in the other direction.
Making use of Hooke’s Law (F=-kx) allows one to measure the relative levels of force necessary to retain a position of the mechanical knee model at varying angles.

So here's some of the angles/ length of spring I have discovered)

The constant of the spring is 4 cm

100 degrees, spring= 4 cm and 1mm
70 degrees, spring= 4 cm and 3 mm
55 degrees, spring = 4 cm and 3 mm
25 degrees, spring = 5 cm and 2 mm

As you can see, as the angle becomes more acute, the relative tension on the spring increases.

Here's where I could use your help:
Can you help me formulate an equation to describe this relationship to show the length of the spring as a function of angle of the knee joint?

Thanks!

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