$A$ mass $m$ attached to a spring oscillates every $2 \, s$. If the mass is increased by $2 \, kg$,then the time period increases by $1 \, s$. The initial mass is ..... $kg$.

$A$ block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: $(i)$ when the block is at $x_0$; and $(ii)$ when the block is at $x = x_0 + A$. In both cases, a particle with mass $m$ is placed on the mass $M$. Which of the following statements are correct?

In the situation as shown in the figure,the time period of vertical oscillation of the block for small displacements will be:

The motion of a mass on a spring,with spring constant $K$ is as shown in the figure. The equation of motion is given by $x(t) = A \sin \omega t + B \cos \omega t$ with $\omega = \sqrt{\frac{K}{m}}$. Suppose that at time $t = 0$,the position of the mass is $x(0)$ and velocity is $v(0)$,then its displacement can also be represented as $x(t) = C \cos (\omega t - \phi)$,where $C$ and $\phi$ are:

The springs shown in the figure are identical,each having a spring constant $K$. When mass $A = 4\, kg$ is attached,the elongation of the spring is $1\, cm$. If mass $B = 6\, kg$ is attached to the system of two springs in series as shown,the total elongation produced is ..... $cm$.

Two springs of force constants $300 \, N/m$ (Spring $A$) and $400 \, N/m$ (Spring $B$) are joined together in series. The combination is compressed by $8.75 \, cm$. The ratio of energy stored in $A$ and $B$ is $E_A / E_B$. Then $E_A / E_B$ is equal to: