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A massless inextensible string is wrapped over a cylinder of radius 20 cm and mass 2kg. It is connected to a hanging block of mass 1kg. If the surface were smooth, the cylinder would reach a distance $ℓ$ in time $t_{1}$ . If the surface were sufficiently rough, the cylinder would reach the distance $ℓ$ in time $t_{2}$ .
Then $\frac{t_{1}}{t_{2}} =$

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$\sqrt{\frac{10}{7}}$

$\frac{1}{\sqrt{2}}$

$\sqrt{\frac{7}{10}}$

answer is C.

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Detailed Solution

Smooth

$10 - T = 1 a_{1} = a + R α$
$TR = \frac{1}{2} \times 2 R^{2} α = Rα$
T = 2a $\Rightarrow 10 - T = \frac{T}{2} + T$
10 = 2.5T $\Rightarrow T = 4$
$a = 2 m / s^{2} \Rightarrow ℓ = \frac{1}{2} \times 2 t_{1}^{2} t_{1} = \sqrt{ℓ}$
Case (ii)

$10 - T' = 1 a_{1}^{'} = a' + R α' + 2 R α'$

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