Chapter 8, Problem 69RPP

BIO Improper lifting and the back A careful study of human anatomy allows medical researchers to use the conditions of equilibrium to estimate the internal forces that body parts exert on each other while a person lifts in a bent position (see Figure 8.19). Suppose an 800-N (180-lb) person lifts a 220-N (50-lb) barbell in a bent position. The situation can be represented with a mechanical model (Figure 8.26a). The cable (the back muscle) exerts a tension force ${\stackrel{\to }{T}}_{\text{M on B}}$ on the backbone and the support at the bottom of the beam (the disk in the lower back) exerts a compression force ${\stackrel{\to }{F}}_{\text{D on B}}$ on the backbone. The backbone in turn exerts the same magnitude force on the 2.5-cm- diameter fluid-filled disks in the lower backbone. Such disk compression can cause serious back problems. A force diagram of this situation is shown in Figure 8.26b. The magnitude of the gravitational force ${\stackrel{\to }{F}}_{\text{E on}}{}_{\text{B}}$ that Earth exerts on the center of mass of the upper stomach-chest region is 300 N. Earth exerts a 380-N force on the head, arms, and 220-N barbell held in the hands. Using the conditions of equilibrium, we estimate that the back muscle exerts a 3400-N (760-lb) force ${\stackrel{\to }{T}}_{\text{M on}}{}_{\text{B}}$ on the backbone and that the disk in the lower back exerts a 3700-N (830-lb) force ${\stackrel{\to }{F}}_{\text{D on B}}$ on the backbone. This is like supporting a grand piano on the 2.5-cm-diameter disk.

Which expression below best describes the torque caused by force $F_3=F_{\text{E on }}{}_{\text{B}}$, the force that Earth exerts on the upper body at its center of mass for the backbone of length L?

a. 0

b. $F_3\left(2L/3\right)\text{sin }12°$

c. $F_3\left(L/2\right)\text{cos}30°$

d. $-F_3\left(2L/3\right)\text{sin }12°$

e. $—F(L/2)\text{cos }30°$