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. Let mp = 0.7 kg and r = 0.1 m
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. If the pendulum disk is initially at a distance L = 0.85 m from the pin at O, how much would the period of the pendulum change if the adjustment nut with a lead of 0.5 mm was rotated four complete rotations closer to the disk? In addition, how much time would the clock gain or lose in a 24 h period if this were done?
The period of the pendulum change is ____ s. (Round the final answer to six decimal places.)
The clock would gain ____ min every 24 h. (Round the final answer to four decimal places.)
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