A small block A of mass m is placed symmetrically over rectangular block B of mass 2 m and having dimensions shown in figure. Block B is placed over frictionless horizontal surface. Mass A is connected to extended walls of block B by springs of spring constant $k = 2 m g / b$ . Now block A is made to perform SHM freely over block B. If block B does not topple during subsequent motion, then choose correct options.

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Period of oscillation of blocks will be $2 π \sqrt{\frac{b}{6 g}}$

Maximum amplitude of oscillations of block A for no toppling of block B is $\frac{3 b}{10}$

Torque due to contact force from ground about centre of mass of block will be 6 mgx where x is the change in length of the spring from its natural length

Block B will topple for amplitudes less than $\frac{b}{5}$ .

answer is B, C.

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Detailed Solution

$T = 2 π \sqrt{\frac{μ}{K_{a t}}}$
$= 2 π \sqrt{\frac{2 m}{32 k}}$ but $K = \frac{2 m g}{b}$
$T = 2 π \sqrt{\frac{m}{3 (\frac{2 m g}{b})}}$
$T = 2 π \sqrt{\frac{b}{6 g}}$
for the body to topple
$2 K A (b) = 2 m g \frac{b}{2} + m g (\frac{b}{2} - A) 2 (\frac{2 m g}{b} A (b)) = 2 m g \frac{b}{2} + m g (\frac{b}{2} - A) 4 A = b + \frac{b}{2} - A A = \frac{5 b}{10}$