First slide

Motion of block on Smooth incline

Question

Two identical smooth planks A and B are kept on ground making angles $30^{o} and 45^{o}$ respectively with horizontal. Two smooth blocks are simultaneously released from the top of planks A and B. If $t_{A} and t_{B}$ are times taken by A and B respectively, then $t_{A} : t_{B}$ is

Easy

A

$2 \sqrt{2} : 1$
B

$\sqrt{3} : \sqrt{2}$
C

${(2)}^{\frac{1}{4}} : 1$
D

${(2)}^{\frac{1}{2}} : 1$

Solution

Acceleration of block along the plank $a = g \sin θ$
$∴ l = length of plank = \frac{1}{2} a t^{2} \Rightarrow t = \sqrt{\frac{2 l}{g \sin θ}}$
$∴ t_{A} : t_{B} = \sqrt{\frac{2 l}{g \sin 30}} : \sqrt{\frac{2 l}{g \sin 45^{o}}} = \sqrt{2} : {(\sqrt{2})}^{\frac{1}{2}} = {(2)}^{\frac{1}{4}} : 1$

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