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Answer :
Final answer:
The acceleration due to gravity (g) calculated from the periods and lengths given for a simple pendulum yielded a mean g of 9.90 m/s², which is slightly larger than the known value of 9.80 m/s². The discrepancies may be due to the displacement angle in the experiment not being small enough for the sin θ ≈ θ approximation to hold true.
Explanation:
The question is focusing on the properties of a simple pendulum and the effect of gravity on its harmonic motion. Let's consider the period of a simple pendulum formula which is T = 2π√(L/g). Where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. Calculating g for each length, we can make use of the formula rearranged to g = 4π²L/T². Given the total time for 50 oscillations, first we need to find the period (T) for one oscillation by dividing the total time by 50.
For L = 1.000 m, and T = 99.8/50 = 1.996 seconds: g = 9.87 m/s². For L = 0.750 m, and T = 86.6/50 = 1.732 seconds: g = 9.91 m/s². For L = 0.500 m, and T = 71.1/50 = 1.422 seconds: g = 9.92 m/s². Calculate the mean of these three values gives: g(mean) = 9.90 m/s².
Comparing this to the known value of g = 9.80 m/s², it appears to be considerably off. While these calculations are fairly accurate for small angles, the method to determine g from the pendulum length and period relies on the approximation sin ≈ θ, which is better than the precision of the pendulum length and period when the maximum displacement angle is kept below about 0.5°. Given that we don't know the exact angle used for these experimental measurements, we can say that the resultant g is slightly overestimated.
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