) 3.451 x 104 cm l) None of these Problem 15. A pendulum clock, with the rod mad
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) 3.451 x 104 cm l) None of these Problem 15. A pendulum clock, with the rod made of brass, is designed to keep accurate time at 20°C. If we approximate the clock as a simple pendulum undergoing small angle dis- placements from equilibrium, then how much will the error be, in seconds per hour, if the clochk operates at 0°C? The coefficient of linear expansion for brass is = 19 × 10-6 1/C" Recall: The period of a pendulurn is given by T-2, where L is the length of the pendu- lum arm and g is the acceleration of gravity, which is unaffected by local temperature changes. (a) 1.3 seconds (c) 0.68 seconds (e) None of these (b) 6.2 scconds (d) 13.2 secondsExplanation / Answer
difference in time ,T' - T = 2 pi sqrt[ L / g] - 2 pi sqrt[ L' / g]
Relative error = (T' - T) / T
= (sqrt(L) - sqrt(L') / sqrt(L)
= 1 - srt(L' / L)
and L' = L (1 + alpha deltaT )
= L (1 + (19 x 10^-6)(0-20))
= L (1 - 3.8 x 10^-4 )
error = 1 - sqrt(1 - 3.8 x 10^-4))
= 3.8 x 10^-4
error per hour = (3600 s)(3.8x 10^-4)
= 1.3 sec
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