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,

T = 2 \pi \sqrt{l\over g}

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 = √l₁ : √l₂  ……. [Equation 2]

So we get,

√l₁ : √l₂ = 2 : 1

Squaring both sides we get,

l₁ : l₂ = 4 : 1

Hence, ratio of lengths = 4 : 1