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