Question

Four massless springs whose force constants are 2k, 2k, k and 2k respectively are attached to a mass M kept on a frictionless plane as shown in the figure. If the mass M is displaced in the horizontal direction, then the frequency of the system is:

A) ${\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{k}{4M}}}$
B) ${\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{4k}{M}}}$
C) ${\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{k}{7M}}}$
D) ${\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{7k}{7M}}}$

Answer 1

Hint: Formula for frequency is:
${\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{1}{LC}}}$ (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:
$$\begin{gathered} \Rightarrow K_1={\displaystyle \frac{1}{2k}}+{\displaystyle \frac{1}{2k}} \\ \Rightarrow K_1={\displaystyle \frac{2k\times 2k}{2k+2k}} \end{gathered}$$ (Taking LCM)
$\Rightarrow K_1={\displaystyle \frac{2k}{2k}}=1k$
Now, we will calculate for the springs in parallel:
$$\begin{gathered} \Rightarrow K_2={\displaystyle \frac{1}{{\displaystyle \frac{1}{2k}}}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{k}}}} \\ \Rightarrow K_2=2k+k=3k \end{gathered}$$ (in parallel connection we have to take the reciprocal of the spring constants)
Total value of K comes out to be:
$$\begin{gathered} \Rightarrow K=K_1+K_2 \\ \Rightarrow K=k+3k=4k \end{gathered}$$
From the equation of frequency:
$$\begin{gathered} \Rightarrow f={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{1}{M{\displaystyle \frac{1}{K}}}}} \\ \Rightarrow f={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{K}{M}}} \end{gathered}$$ (We have substituted the value of LC as per formula of frequency)
$\Rightarrow f={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{4k}{M}}}$ (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.