The angular frequency of small oscillations of the system shown in the figure is

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Answer 1

Let block of mass $m$ is pulled down by $X$ due to this spring is stretched by $X_{1}$ .
From string constraint $2 X_{1} = X$ $\Rightarrow X_{1} = \frac{X}{2}$
Total energy of the system= Kinetic energy of block + spring energy,
$\Rightarrow \frac{1}{2} m \dot{X}^{2} + \frac{1}{2} K X_{1}^{2} = C o n s t a n t$
Put the $X_{1} = \frac{X}{2}$
$\Rightarrow \frac{1}{2} m \dot{X}^{2} + \frac{1}{8} K X^{2} = C o n s t a n t$

Differentiating the equation,
$\Rightarrow m \dot{X} \ddot{X} + \frac{1}{4} K X \dot{X} = 0$ $\Rightarrow m \ddot{X} + \frac{1}{4} K X = 0$ $\Rightarrow \ddot{X} = - \frac{K}{4 m} X$ $\Rightarrow \ddot{X} = - ω^{2} X$ ,
$\Rightarrow Angular frequency ω = \frac{K}{4 m}$

Answer 2

Let block of mass

m

is pulled down by

X

due to this spring is stretched by

X_{1}

.

From string constraint

2 X_{1} = X

\Rightarrow X_{1} = \frac{X}{2}

Total energy of the system= Kinetic energy of block + spring energy,

\Rightarrow \frac{1}{2} m \dot{X}^{2} + \frac{1}{2} K X_{1}^{2} = C o n s t a n t

Put the

X_{1} = \frac{X}{2}

\Rightarrow \frac{1}{2} m \dot{X}^{2} + \frac{1}{8} K X^{2} = C o n s t a n t

Differentiating the equation,

\Rightarrow m \dot{X} \ddot{X} + \frac{1}{4} K X \dot{X} = 0

\Rightarrow m \ddot{X} + \frac{1}{4} K X = 0

\Rightarrow \ddot{X} = - \frac{K}{4 m} X

\Rightarrow \ddot{X} = - ω^{2} X

,

\Rightarrow Angular frequency ω = \frac{K}{4 m}