A metal cylinder of radius R is placed on a wooden plank BD. The plank is kept horizontal suspended with the help of two identical string AB and CD each of length L. The temperature coefficient of linear expansion of the cylinder and the strings are $α_{1}$ and $α_{2}$ respectively. Angle $θ$ shown in the figure is $30^{0}$ . It was found that with change in temperature the center of the cylinder did not move. Then the ratio $\frac{α_{1}}{α_{2}}$ is ? If it is know that L = 4R. Assume that change in value of $θ$ is negligible for small temperature changes.

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answer is 8.

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Detailed Solution

Let change in temperature be $Δ T$

Length of a string changes by $Δ L = L α_{2} Δ T$
The wooden plank descends by $Δ y = \frac{Δ L}{\sin θ}$
$Δ y = 2 Δ L [∵ \sin θ = \frac{1}{2}]$
Change in radius of the ball: $Δ R = R α_{1} Δ T$
The center of the ball will not move if $Δ y = Δ R$
$\Rightarrow 2 L α_{2} Δ T = R α_{1} Δ T \Rightarrow 8 R α_{2} = R α_{1} \Rightarrow \frac{α_{1}}{α_{2}} = \frac{8}{1}$