The length of a pendulum is measured as 20.0 cm. The time interval for 100 oscillations is measured as 90 s with a stop watch of 1s resolution. Find the approximate percentage change

Asked by Anil | 25th May, 2017, 02:35: PM

Expert Answer:

begin mathsize 12px style Time space period space of space simple space pendulum space is space straight T space equals space 2 straight pi space square root of straight l over straight g end root or space straight T squared space equals space 4 straight pi squared space straight l over straight g rightwards arrow space straight g space equals space fraction numerator 4 straight pi squared straight l over denominator straight T squared end fraction Taking space log space on space both space the space sides comma space we space get log space straight g space equals space log space left parenthesis 4 straight pi squared straight l right parenthesis space minus space log space straight T squared log space straight g space equals space log 4 space plus space logπ squared space plus space logl space minus space 2 log space straight T Differentiating space the space above space comma space we space get fraction numerator increment straight g over denominator straight g end fraction space equals space fraction numerator increment straight l over denominator straight l end fraction space minus space 2 fraction numerator increment straight T over denominator straight T end fraction For space maximum space relative space error fraction numerator increment straight g over denominator straight g end fraction space equals space fraction numerator increment straight l over denominator straight l end fraction space plus space 2 fraction numerator increment straight T over denominator straight T end fraction increment straight l space equals space 0.1 space cm space and space increment straight T space equals space 1 space straight s therefore space fraction numerator increment straight g over denominator straight g end fraction space equals space fraction numerator 0.1 over denominator 20 end fraction space plus space 2 open parentheses 1 over 90 close parentheses therefore space fraction numerator increment straight g over denominator straight g end fraction space equals space 0.32 space Percentage space error space in space straight g space is space fraction numerator increment straight g over denominator straight g end fraction space cross times space 100 space equals space fraction numerator increment straight l over denominator straight l end fraction space cross times space 100 space plus open parentheses space 2 fraction numerator increment straight T over denominator straight T end fraction close parentheses space cross times space 100 space space space space space space space space space space space space space space space space space space space space equals space 3 percent sign end style

Answered by Yashvanti Jain | 25th May, 2017, 06:20: PM

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