Two light springs of force constant $k_{1} {and k}_{2}$ and a block of mass m are in one line AB on a smooth. Horizontal table such that one end of each spring is fixed on rigid supports and the other end is free as shown in the figure. The distance CD between the free ends of the spring is 60 cm. If the block moves along AB with a velocity $120 cm/s$ in between the springs, calculate the period of oscillation of the block. $(k_{1} = 1.8 {N/m, k}_{2} = 3.2, m = 2 kg)$

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1.41 S

2.82 S

5.64 S

1.92 S

answer is B.

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Detailed Solution

$t_{1} = \frac{T_{1}}{2} = \frac{2 π}{2} \sqrt{\frac{m}{K_{1}}}$

$t_{1} = π \sqrt{\frac{0.2}{1.8}} = \frac{π}{3} \sec$

Similarly, time $t_{2}$ taken by block contact with spring between A & C

$t_{2} = \frac{T_{2}}{2} = \frac{2 π}{2} \sqrt{\frac{m}{K_{2}}}$

$t_{2} = π \sqrt{\frac{0.2}{3.2}} = \frac{π}{4} \sec$

$t_{3} = \frac{2 L}{υ} = \frac{2 \times 0.6}{1.2} = 1 \sec$

Total time $t = t_{1} + t_{2} + t_{3}$

$t = \frac{π}{3} + \frac{π}{4} + 1$

t = 2.83 sec

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