Halliday, Resnick & Walker — Rotation: Problem Problem_86

Bone screw insertion. An increasingly common method of surgically stabilizing a broken bone is by inserting a screw into the bone with an automated surgical screwdriver. As the screw enters the bone, the medical team monitors the torque applied to the screw. The purpose is to drive the screw inward until the screw head meets the bone and then to rotate the screw a bit more to tighten the screw threads against the bone threads that the screw has cut along its path. The danger is to tighten the screw too much because then the screw threads destroy (strip) the bone threads. Figure 10.36 shows an idealized plot of torque magnitude \(\tau\) versus angle of rotation \(\theta\), all the way to the failure stage. Initially, as more of the screw enters the bone, the required torque increases until it reaches a short plateau at \(\tau_{\text{plateau}} = 0.10 \text{ N}\cdot\text{m}\), which occurs as the head makes contact. Then the torque sharply increases as the screw tightens. The surgical team would like to stop at or near the peak at \(\tau_{\text{peak}} = 1.7 \text{ N}\cdot\text{m}\) and avoid passing into the failure region. They might be able to predict the peak from the plateau and the work done on the screw by the screwdriver. (a) What multiple of the plateau torque gives the peak torque? (b) How much work is done from the left end of the plot to the peak?