Question

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

• A. $\sqrt{\left(K/2m\right)}$
• B. $\sqrt{\left(2K/m\right)}$
• C. $\sqrt{\left(K/4m\right)}$
• D. $\sqrt{\left(4K/m\right)}$

Answer 1

The correct option is C $\sqrt{\left(K/4m\right)}$

Let block of mass $m$ is pulled down by $X$ due to this spring is stretched by $X_1$.

From string constraint $2X_1=X$ $\Rightarrow X_1=\frac{X}{2}$

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

$\Rightarrow \frac{1}{2}m{\dot{X}}^2+\frac{1}{2}K{X}_1^2=Constant$

Put the $X_1=\frac{X}{2}$

$\Rightarrow \frac{1}{2}m{\dot{X}}^2+\frac{1}{8}K{X}^2=Constant$

Differentiating the equation,

$\Rightarrow m\dot{X}\ddot{X}+\frac{1}{4}KX\dot{X}=0$ $\Rightarrow m\ddot{X}+\frac{1}{4}KX=0$ $\Rightarrow \ddot{X}=-\frac{K}{4m}X$ $\Rightarrow \ddot{X}=-{\omega }^{2}X$,

$\Rightarrow \text{Angular frequency}\omega =\sqrt{\frac{K}{4m}}$