Measurements and Experimentation — Question 7

As we know that,

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

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 = $\sqrt{\text l_1} : \sqrt{\text l_2}$     [Equation 2]

So we get,

$\sqrt{\text l_1} : \sqrt{\text l_2} = 2 : 1 \\[0.5em] \Rightarrow \dfrac{\sqrt{\text l_1}}{\sqrt{\text l_2}} = \dfrac{2}{1}$

Squaring both sides we get,

$\dfrac{\text l_1}{\text l_2} = \dfrac{2^2}{1^2} \\[0.5em] \Rightarrow \dfrac{\text l_1}{\text l_2} = \dfrac{4}{1} \\[0.5em] \Rightarrow \text l_1 : \text l_2 = 4 : 1$

Hence, ratio of lengths = 4 : 1