Two simple pendulums of lengths 100 cm and 101 cm are set into oscillation at the same time. After what time does one pendulum gain one complete oscillation over the other?

Two simple pendulums of lengths 100 cm and 101 cm are set into oscilla

1.	Two simple pendulums of lengths 100 cm and 101 cm are set into oscillation at the same time. After what time does one pendulum gain one complete oscillation over the other?
Answer» Solution :Length of the first pendulum is comparatively less, and hence, its time period is also less, thus the first pendulum oscillates faster. By the time the second-pendulum executes n oscillations, SUPPOSE the first one completes (n + 1) oscillations. Hence, if `T_(1)andT_(2)` are time periods of the first and the second pendulums, `(n+1)T_(1)=nT_(2)or,T_(2)/T_(1)=(n+1)/n=1+1/n""...(1)` Again, `T_(2)/T_(1)=sqrt(L_(2)/L_(1))=sqrt(101/100)=(1+1/100)^(1//2)` `=1+1/2xx1/100=1+1/200""...(2)` From equations (1) and (2) `1/n=1/200or,n=200,i.e.,n+1=201` Thus, the REQUIRED time = time of 201 COMPLETE oscillations of the first pendulum = `201xxT_(1)` `=201xx2pisqrt(L_(1)/g)` `=201xx2xxpisqrt(100/980)~~403s=6 min43s`.