Question

Two simple pendulum of lengths 1. 44m and 1m start swinging at the Same time . After how time they will be (1)out of phase and (2)in phase again? Take g=10

Answer 1

(1)
After 1st oscillation (or even before) itself they will be out of phase as the time required to complete 1st oscillation is different for both pendulum.T1∝√L1 &T2∝√L2
You can approximate it to time to complete one oscillation by long pendulum(1.44m)
As Time period(T)∝Effective length(√L)
It cannt be actually calculated
(2)
By the law of simple pendulum we know

Time period(T)∝Effective length(√L)

Let the time period of the 1st pendulum having effective length L1=1.44m be T1s

And the time period of the 2nd pendulum having effective length L2=1m be T2s

So by the law T2/T1=√L2/√L1.....(1)

Let the two pendulum starts oscillation in same phase and after a minimum time t sec they again become in the same phase. During this t sec the 1st pendulum oscillates N1times and the 2ndpendulum oscillates N2times

So t=N1×T1=N2×T2

⇒N2/N1=T1/T2.....(2)

Comparing (1) and (2)

N2/N1=T1/T2=√L1/√L2=√1.44/√1=√1.44=1.2
Simplyfying to get an whole no., N2/N1=6/5
Here 1st pendulum will oscillate 5 times and 2nd one will oscillate 6 times before they come to same phase again.
time=

T=2π√L/g
t=5*T1=6*T2
=5*2π√1.44/g=6*2π√1/g
=12.036s