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## Q. 8.9

A WEIGHTED FOREARM BIO

GOAL Apply the equilibrium conditions to the human body.

PROBLEM A 50.0-N (11-lb) bowling ball is held in a person’s hand with the forearm horizontal, as in Figure 8.16a. The biceps muscle is attached $0.030 0 \mathrm{~m}$ from the joint, and the ball is $0.350 \mathrm{~m}$ from the joint. Find the upward force $\overrightarrow{\mathbf{F}}$ exerted by the biceps on the forearm (the ulna) and the downward force $\overrightarrow{\mathbf{R}}$ exerted by the humerus on the forearm, acting at the joint. Neglect the weight of the forearm and slight deviation from the vertical of the biceps.

STRATEGY The forces acting on the forearm are equivalent to those acting on a bar of length $0.350 \mathrm{~m}$, as shown in Figure 8.16b. Choose the usual $x$ – and $y$-coordinates as shown and the axis at $O$ on the left end. (This completes Steps 1 and 2.) Use the conditions of equilibrium to generate equations for the unknowns, and solve.

## Verified Solution

Apply the second condition for equilibrium (Step 3) and solve for the upward force $F$ :

$\sum \tau_{i}=\tau_{R}+\tau_{F}+\tau_{\mathrm{BB}}=0$

$R(0)+F(0.030 0 \mathrm{~m})-(50.0 \mathrm{~N})(0.350 \mathrm{~m})=0$

$F=583 \mathrm{~N}(131 \mathrm{lb})$

Apply the first condition for equilibrium (Step 4) and solve (Step 5) for the downward force $R$ :

\begin{aligned}\sum F_{y} &=F-R-50.0 \mathrm{~N}=0 \\R &=F-50.0 \mathrm{~N}=583 \mathrm{~N}-50 \mathrm{~N}=533 \mathrm{~N}(120 \mathrm{lb})\end{aligned}

REMARKS The magnitude of the force supplied by the biceps must be about ten times as large as the bowling ball it supporting!