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**Hint:**Formula for frequency is:

$\dfrac{1}{{2\pi }}\sqrt {\dfrac{1}{{LC}}} $ (L is the inductor, C is the capacitor)

As per electrical and mechanical analogy conversion, in force current analogy M is the capacitor (C) and k is the reciprocal of the inductor (1/L).

Let’s find the value of k using series and parallel connections (using 1/L = k, in series connections are added with their direct connections and the parallel connection have reciprocal addition).

**Complete step by step solution:**

As we are provided with an inductor and capacitor in the system then we will add the reciprocal of the inductor for the series connection.

$K = {K_1} + {K_2}$

First, we will do the calculation for series connection:

$

\Rightarrow {K_1} = \dfrac{1}{{2k}} + \dfrac{1}{{2k}} \\

\Rightarrow {K_1} = \dfrac{{2k \times 2k}}{{2k + 2k}}

$ (Taking LCM)

$ \Rightarrow {K_1} = \dfrac{{2k}}{{2k}} = 1k$

Now, we will calculate for the springs in parallel:

$

\Rightarrow {K_2} = \dfrac{1}{{\dfrac{1}{{2k}}}} + \dfrac{1}{{\dfrac{1}{k}}} \\

\Rightarrow {K_2} = 2k + k = 3k$ (in parallel connection we have to take the reciprocal of the spring constants)

Total value of K comes out to be:

$

\Rightarrow K = {K_1} + {K_2} \\

\Rightarrow K = k + 3k = 4k $

From the equation of frequency:

$

\Rightarrow f = \dfrac{1}{{2\pi }}\sqrt {\dfrac{1}{{M\dfrac{1}{K}}}} \\

\Rightarrow f = \dfrac{1}{{2\pi }}\sqrt {\dfrac{K}{M}} $ (We have substituted the value of LC as per formula of frequency)

$ \Rightarrow f = \dfrac{1}{{2\pi }}\sqrt {\dfrac{{4k}}{M}} $ (We have substituted the values M and K).

**Hence, Option B is correct.**

**Note:**In the question above we have used electrical to mechanical equivalent system of force current, where current is acting as the force in a mechanical system, mass as capacitor, frictional coefficient as reciprocal of R resistor, spring constant as reciprocal of L inductor, displacement as magnetic flux and velocity as voltage.

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