The time periods of two simple pendulums at a place are in the ratio 2:1. What will be the ratio of their lengths ?

As we know that,

$T = 2 π \sqrt{\dfrac{l}{g}} \\[0.5em]$

Time period is directly proportional to the square root of the length of the pendulum.

In the case when,
T₁ : T₂ = 2 : 1     [Equation 1]

we know that,

T₁ : T₂ = $\sqrt{l$1} : \sqrt{l2}     [Equation 2]

So we get,

$\sqrt{l$1} : \sqrt{l2} = 2 : 1 \\[0.5em] \Rightarrow \dfrac{\sqrt{l1}}{\sqrt{l2}} = \dfrac{2}{1}

Squaring both sides we get,

$\dfrac{l$1}{l2} = \dfrac{2^2}{1^2} \\[0.5em] \Rightarrow \dfrac{l1}{l2} = \dfrac{4}{1} \\[0.5em] \Rightarrow l1 : l2 = 4 : 1

Hence, ratio of lengths = 4 : 1