Modeling the Human Arm and Measuring Forces
We were interested in testing how much force a bicep requires to hold a weight at given angles.
Which angle was the most strenuous? The least? To measure and analyze these forces we
needed to build a model. To represent the upper and lower arm, we used two properly
proportioned pieces of wood. To represent the elbow, we used a metal pivot. We connected a
string to each piece of wood in the right place to represent the bicep muscle. To measure the
force we tightly connected a force transducer across the string joining the upper arm and
forearm. Finally, we attached the entire system to a wooden baseboard and clamped the board
onto a blackboard for ease of use.
Before we could experiment, we needed a theoretical analysis
Analysis
The first step in our analysis consisted of finding an expression for beta a name we gave
to the sum of the angle of rotation (theta) and the angle between the string and the forearm
(alpha). By applying the law of cosines we were able to solve for the length of the hypotenuse at
any theta the arm was rotated to. Then using our knowledge of all three sides for all thetas we
were able to apply the law of cosines a second time to find alpha. Then it was a simple matter of
adding theta to solve for beta. However, this analysis was largely disregarded when we adopted a
more simple approximation of beta = ((90 – theta)/1.2) + theta.
Having found beta we summed all forces in the x-direction (Fc - Fb*cos(beta)), the ydirection (Fa + (weight of arm) + (load weight) – Fb*sin(beta)) and the total torque ( Fc(.051)sin(theta)
+
Fa(.051)cos(theta)
–
2.35(.132)cos(theta)-1.96(.437)cos(theta))
and
respectively set all of them equal to zero. Then, using the system of equations we were able to
solve for Fa, Fb , and Fc.
Now that we have analyzed the experiment, we can take measurements to compare. To take a
measurement, we followed multiple steps:
Step 1: We applied the load to the end of the arm. The load in this case is a 200g mass.
Step 2: Using a protractor, we measured our angle for the arm.
Step 3: We pulled the string until it was tight enough to make the wood stay in position, then we
clamped the arm
Step 4: We recorded the measurement from the force transducer.
We repeated this process a total of five times for every angle from 0 o – 90o (excluding 90) in
increments of 15o. This adds up to a total of 25 measurements. For each angle, we calculated
the mean and standard deviation of the multiple trials. Our results are as follows:
Data
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Mean
SD
Theta=0 Theta=15 Theta=30 Theta=45 Theta=60 Theta=75
29.6
30.4
31.4
31.8
31.9
33.5
29.8
30.8
31.3
31.9
32.8
33.2
29.5
30.5
31.3
31.6
32.1
32.9
29.9
29.7
30.9
31.8
32.8
33.3
29.7
30.3
31.2
31.4
32.3
33.7
29.7
30.34
31.22
31.7
32.38
33.32
0.158114 0.403733 0.192354
0.2 0.408656 0.303315