Let u consider the following diagram in which a block of mass M is being supported by a uniform rope of mass 1 kg and length 10 meter. A pulse is created at the bottom of the rope and it reaches the top. In column I, the value of M in kg is given and in column II, time (in sec) after which the pulse reaches the top is given. Match the items given in column I to that in column II.
Column -I Column-II
A) 1 p) $2 (2 - \sqrt{3})$
B) 2 q) $2 (\sqrt{5} - 2)$
C) 3 r) $2 (\sqrt{3} - \sqrt{2})$
D) 4 s) $2 (\sqrt{2} - 1)$

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$A - s, B - p, C - r, D - q$

$A - s, B - r, C - q, D - p$

$A - s, B - r, C - p, D - q$

$A - r, B - s, C - p, D - q$

answer is A.

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Detailed Solution

Tension in the string at point ‘A’ is
$T = M g + (\frac{m}{L} x) g = (M + \frac{1}{10} x) g$

$v = \sqrt{\frac{T}{μ}} \Rightarrow \frac{d x}{d t} = \sqrt{\frac{(M + \frac{x}{10}) g}{1 / 10}} = \sqrt{(10 M + x) g}$

$\int d t = \int_{L = 0}^{L = 10} \frac{d x}{\sqrt{(10 M + x) g}}$ $t = \frac{2}{\sqrt{10}} [{(10 M + 10)}^{1 / 2}] = 2 [{(M + 1)}^{1 / 2} - M^{1 / 2}]$

a) $M = 1 k g, t = 2 [\sqrt{2} - 1]$
b) $M = 2 k g, t = 2 [\sqrt{3} - \sqrt{2}]$
c) $M = 3 k g, t = 2 [2 - \sqrt{3}]$
d) $M = 4 k g, t = 2 [\sqrt{5} - 2]$