Two pendulums of lengths 200 cm and 220.50 cm start oscillating in phase. After how many oscillations will they be in same phase for the first time ?

Explanation

The two pendulums will be in the same phase for the first time when larger pendulum complestes n oscillations in time t and smaller pendulum completes `(n+1)` oscillatinos in time t.
For larger pendulum, time period
`T=(t)/(n)=2pisqrt((l)/(g))`
or `t=2pinsqrt((l)/(g))` ...(i)
For smaller pendulum, time period,
`T^(')=(t)/((n+1))=2pisqrt((l^('))/(g))`
or `t=2pi(n+1)sqrt((l^('))/(g))` ...(ii)
From (i) and (ii), we have
`2pinsqrt((l)/(g))=2pi(n+1)sqrt((l^('))/(g))`
or `(n+1)/(n)=sqrt((l)/(l^(')))=sqrt((220.50)/(200))=(1+(20.50)/(200))^(1/2)`
or `1+(1)/(n)=1+(1)/(2)xx(20.50)/(200)=1+0.05125`
or `(1)/(n)=0.05125~~0.05`
or `n=(1)/(0.05)=20`