Column I and Column II contain oscillating systems. All pulleys, springs and threads shown in figures are massless (negligible mass) there is no friction anywhere. All blocks have same mass M and the supporting surfaces are at rest with respect to ground. Systems of Column I oscillate in vertical plane and that in column II in horizontal plane. Force constants of springs are shown in figure. Match the systems given in Column I with the systems in Column II. Which are having same time period of oscillation.
COLUMN-I
COLUMN-II
A)
P)
B)
Q)
C)
R)
D)
S)

T)

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$A - R, B - S T, C - Q S, D - P Q$

$A - R S, B - S Q, C - Q, D - P$

$A - R, B - S T, C - Q, D - P$

$A - R, B - S, C - P, D - Q$

answer is C.

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Detailed Solution

If block is displaced downward by x spring elongates by 2x.

$T = 2 T^{|}$
$T^{|} = k (2 x)$
T = 4kx = restoring force on block
$t = 2 π \sqrt{\frac{M}{4 k}}$
If block is displaced by x pulley is displaced by $y = \frac{x}{2}$

$T = 2 T = k y T = \frac{k}{2} y = \frac{k}{4} x t = 2 π \sqrt{\frac{4 M}{k}}$
If block displaced downward by x such that spring 1 is elongating y and spring 2 is elongated by z.

$x = 2 y + z T^{|} = k y = 2 T T = k z x = \frac{4}{k} T + \frac{T}{k} T = (\frac{k}{5}) x t = 2 π \sqrt{\frac{5 M}{k}}$
If $y_{1}$ is elongation is 1 and $y_{2}$ in elongation 2 then displacement of block x = $2 y_{2} + 2 y_{2}$

$T_{1} = 2 T = T_{2} T_{1} = k y_{1} = 2 T y_{1} = \frac{2 T}{k} T_{2} = k y_{2} \Rightarrow y_{2} = \frac{2 T}{k} x = \frac{4 T}{k} + \frac{4 T}{k} T = \frac{k}{8} x t = 2 π \sqrt{\frac{8 M}{k}}$