Biophysics Problem 22

For the example on the tension in the deltoid muscle of the arm, determine how the expression for the tension is changed if a weight $W$ is placed in the hand.

Diagram

This example is on page 102 of your text. Refer back to the diagram and imagine an additional force W acting vertically downwards at $B.$

In the text example, $R$ passes through $O$ and exerts no moment about it. The first equation in the example indicates that the torques of $T$ and $W$ about $O$ sum algebraically to zero.

Calculations

We have:

$(OA) \sin \Delta$ is the perpendicular distance from $O$ to the line of action $T$
$(OC) \sin \theta$ is the perpendicular distance from $O$ to the line of action $W$

And from the new force:

$(OB) \sin \theta$
is the perpendicular distance from $O$ to the line of action of the $\text{New Force}\; W_1$

So with this additional force, the first equation becomes:

$T(OA) \sin \Delta - W(OC) \sin \theta - W_1 (OB) \sin \theta = 0$

And since $OA = AC = 1/5\; OB,$ we get:

$T = \frac{(2W +5W_1) \sin \theta}{\sin \Delta}$