Chapter 13, Problem 85P

You’ve inherited your great-grandmother’s mantle clock. The clock’s timekeeping is established by a pendulum consisting of a 15.0-cm-long rod and a disk 6.35 cm in diameter. The rod passes through a hole in the disk, and the disk is supported at its bottom by a decorative nut mounted on the bottom portion of the rod, which is threaded: see Fig. 13.38. As shown, the bottom of the disk is 1.92 cm above the bottom of the rod. There are 20.0 threads per cm, meaning that one full turn of the decorative nut moves the disk up or down by 1/20 cm. The clock is beautiful, but it isn’t accurate; you note that it’s losing 1.5 minutes per day. But you realize that the decorative nut is an adjustment mechanism, and you decide to adjust the clock’s timekeeping.

1. (a) Should you turn the nut to move the disk up or down?
2. (b) How many times should you turn the nut? Note: The disk is massive enough that you can safely neglect the mass of the rod and nut. But you can’t neglect the disk’s size compared with the rod length, so you don’t have a simple pendulum. Furthermore, note that both the effective length of the pendulum and its rotational inertia change as the disk moves up or down the shaft. You can either solve a quadratic or you can use calculus to get an approximate but nevertheless very accurate answer.

FIGURE 13.38 Problem 85