Grandfather clocks keep time by advancing the hands a set amount per oscillation of the pe...

Grandfather clocks keep time by advancing the hands a set amount per oscillation of the pendulum. Therefore, the pendulum needs to have a very accurate period for the clock to keep time accurately. As a fine adjustment of the pendulum’s period, many grandfather clocks have an adjustment nut on a bolt at the bottom of the pendulum disk. Screwing this nut inward or outward changes the mass distribution of the pendulum by moving the pendulum disk closer to or farther from the axis of rotation at O. Model the pendulum as a uniform disk of radius r and mass mp at the end of a rod of negligible mass and length L − r, and assume that the oscillations of θ are small. Let mp = 0.7 kg and r = 0.1 m.

The clock is running slow so that it is losing 2 minutes every 24 hours (i.e., the clock takes 1442 minutes to complete a 1440-minute day). If the pendulum disk is at L = 0.85 m, how many turns of the adjustment nut would be needed, and in what direction, to correct the pendulum’s period if the screw lead* is 0.5 mm?

Figure P9.22