Halliday, Resnick & Walker — Equilibrium and Elasticity: Problem Problem_85

Figure 12.72a shows details of a finger in the crimp hold of the climber in Fig. 12.36. A tendon that runs from muscles in the forearm is attached to the far bone in the finger. Along the way, the tendon runs through several guiding sheaths called pulleys. The A2 pulley is attached to the first finger bone; the A4 pulley is attached to the second finger bone. To pull the finger toward the palm, the forearm muscles pull the tendon through the pulleys, much like strings on a marionette can be pulled to move parts of the marionette. Figure 12.72b is a simplified diagram of the second finger bone, which has length \(d\). The tendon’s pull \(\vec{F}_t\) on the bone acts at the point where the tendon enters the A4 pulley, at distance \(d/3\) along the bone. If the force components on each of the four crimped fingers in Fig. 12.36 are \(F_h = 13.4 \text{ N}\) and \(F_v = 162.4 \text{ N}\), what is the magnitude of \(\vec{F}_t\)? The result is probably tolerable, but if the climber hangs by only one or two fingers, the A2 and A4 pulleys can be ruptured, a common ailment among rock climbers.