# Two simple pendulums of time period 2 sec and 2.1 sec are made to vibrate simultaneously. They are in phase initially. After how many vibrations they are in the same phase?

Q. Two simple pendulums of time period 2 sec and 2.1 sec are made to vibrate simultaneously. They are in phase initially. After how many vibrations they are in the same phase?

• 21
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• 35

Solutions:

First simple pendulum has a time period of 2 sec.

Second pendulum has a time period of 2.1 sec.

That means: After each complete oscilation, the second pendulum is ahead by (2.1 – 2.0) = 0.1 sec.

if we show this in a list:

after 1 vibration, second pendulum leads by 0.1 s

after 2 vibrations, second pendulum leads by 2×0.1s = 0.2 s

after 3 vibrations, second pendulum leads by 3×0.1s = 0.3 s

………………………….

after 10 vibrations, second pendulum leads by 10×0.1s = 1s

after 20 vibrations, second pendulum leads by 20×0.1s = 2s

after 21 vibrations, second pendulum leads by 21×0.1s = 2.1s (This is a time period of second pendulum.)

So, both pendulum comes at mean position after 21 oscillations which means they are at the same phase.

Ans: 21

OR,

Difference of time period of two pendulum is (2.1 – 2.0)s = 0.1 s

Let’s say, both pendulum after completing complete oscillations, they come to same phase.

So,

Ans: 21