$A$ spring having a spring constant $K$ is loaded with a mass $m$. The spring is cut into two equal parts and one of these is loaded again with the same mass. The new spring constant is

In the figure,$S_1$ and $S_2$ are identical springs. The oscillation frequency of the mass $m$ is $f$. If one spring is removed,the frequency will become

Two springs of constant $k_1$ and $k_2$ are joined in series. The effective spring constant of the combination is given by

In the following questions,match Column-$I$ with Column-$II$ and choose the correct options.

Two identical springs of spring constant $k$ are attached to a block of mass $m$ and to fixed supports as shown in the figure. Show that when the mass is displaced from its equilibrium position on either side,it executes simple harmonic motion. Find the period of oscillations.

$A$ highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidly onto another cubical block $B$ of the same dimensions and of low modulus of rigidity $\eta$ such that the lower face of $A$ completely covers the upper face of $B$. The lower face of $B$ is rigidly held on a horizontal surface. $A$ small force $F$ is applied perpendicular to one of the side faces of $A$. After the force is withdrawn,block $A$ executes small oscillations. The time period of which is given by: