The pendulums of lengths 100 cm and 110.25 cm start oscillating in phase. After how many oscillation will they again be in same phase ?

Explanation

In order to be in same phase again, large pendulum completes v oscillation and small pendulum completes (v + 1) oscillations.
So, far larger pendulum with l = 90 cm.
`v = (1)/(2pi) sqrt((g)/(l)) =(1)/(2pi) sqrt((g)/(90)) " "…(i)`
For smaller pendulum,
` v + 1 = (1)/(2pi) sqrt((g)/(70)) " "...(ii)`
Dividing (ii) by (i)
`therefore (v + 1)/(v) = sqrt((90)/(70))`
` = ((70 + 20)/(70))^(1//2) = (1 + (2)/(7))^(1//2)`
`1 + (1)/(v) = 1.142`
or `(1)/(v) = 0.142 or v = 7.04 = 7`
So, the two pendulums will be in same phase when the large pendulum completes 7.0 oscillations or smaller pendulum completes 8.0 oscillations.