Explanation

Moment of inertia of a uniform circular disc about its central axis is:

$$ I = \frac{1}{2}MR^2 $$

Since both discs are made of the same material, their density is same.

Mass of a disc:

$$ M = \rho \times \text{Volume} = \rho \pi R^2 T $$

Substitute mass in the moment of inertia expression:

$$ I = \frac{1}{2} (\rho \pi R^2 T) R^2 $$

$$ I = \frac{1}{2} \rho \pi R^4 T $$

Given that moments of inertia of both discs are equal:

$$ R_1^4 T_1 = R_2^4 T_2 $$

$$ \frac{T_1}{T_2} = \left(\frac{R_2}{R_1}\right)^4 $$

Given:

$$ \frac{R_1}{R_2} = 2 \Rightarrow \frac{R_2}{R_1} = \frac{1}{2} $$

$$ \frac{T_1}{T_2} = \left(\frac{1}{2}\right)^4 = \frac{1}{16} $$

Comparing with:

$$ \frac{T_1}{T_2} = \frac{1}{\alpha} $$

$$ \alpha = \boxed{16} $$

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