First slide

Applications of SHM

Question

The time period of vertical oscillations of the system shown in Fig. is $(given k_{1} = k_{2} = k and k_{3} = 2 k)$

Moderate

A

$2 π \sqrt{\frac{m}{k}}$
B

$2 π \sqrt{\frac{3 m}{k}}$
C

$2 π \sqrt{\frac{m}{3 k}}$
D

$2 π \sqrt{\frac{2 m}{3 k}}$

Solution

The spring constant of the parallel combination of springs of spring constants k₁ and k₂ is
$k_{p} = k_{1} + k_{2} = k + k = 2 k$
This combination is in series with the spring of spring constant k₃. So the effective spring constant of the system is
$\begin{array}{l} \frac{1}{k_{eff}} = \frac{1}{k_{p}} + \frac{1}{k_{3}} = \frac{1}{2 k} + \frac{1}{2 k} = \frac{1}{k} \\ ∴ k_{eff} = k \end{array}$
and $T = 2 π \sqrt{\frac{m}{k_{eff}}} = 2 π \sqrt{\frac{m}{k}}$
So the correct choice is (a).

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