The time period of two pendulums is in the ratio of 2:1. What will be the corresponding ratio of their lengths?

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Solution

Step 1: Given data and assumptions
Let the two pendulums be A and B.
The ratio of the time period of two pendulums $= 2 : 1$
Step 2: Relationship between time period and length
Time period of a pendulum is given by, $T = 2 π \sqrt{\frac{L}{g}} w h e r e, T = t i m e p e r i o d o f a p e n d u l u m L = e f f e c t i v e l e n g t h$
We know that time period is directly proportional to the square root of the length.
$T \propto \sqrt{L} w h e r e, T = t i m e p e r i o d o f a p e n d u l u m L = e f f e c t i v e l e n g t h$
Hence it can be written as,
$T^{2} = L w h e r e T = t i m e p e r i o d o f a p e n d u l u m L = e f f e c t i v e l e n g t h$
Step 3:Calculating corresponding ratio of their length
$\sqrt{\frac{L_{1}}{L_{2}}} = \frac{T_{1}}{T_{2}} B y S q u a r i n g o n b o t h s i d e s . \frac{L_{1}}{L_{2}} = \frac{{(2)}^{2}}{1^{2}} = \frac{4}{1} \frac{L_{1}}{L_{2}} = 4 : 1$
Hence, the ratio of their length is 4:1.

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