## Chapter 8

## Q. 8.9

## 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.

## Step-by-Step

## 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!