$A$ mass $m = 1.0\,kg$ is placed on a flat pan attached to a vertical spring fixed on the ground. The mass of the spring and the pan is negligible. When pressed slightly and released,the mass executes simple harmonic motion. The spring constant is $k = 500\,N/m$. What is the amplitude $A$ of the motion,so that the mass $m$ tends to get detached from the pan? (Take $g = 10\,m/s^2$).

$A$ block is fastened to a horizontal spring. The block is pulled to a distance $x = 10 \, cm$ from its equilibrium position (at $x = 0$) on a frictionless surface from rest. The total energy of the block at $x = 5 \, cm$ is $0.25 \, J$. The spring constant of the spring is $......... \, N \, m^{-1}$.

$A$ mass $M$,attached to a horizontal spring,executes $S.H.M.$ with amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\frac{A_1}{A_2}$ is

$A$ mass $M$ is suspended from a light spring. An additional mass $M_1$ added extends the spring further by a distance $x$. Now the combined mass will oscillate on the spring with period $T=$

$A$ $10 \,kg$ metal block is attached to a spring of spring constant $1000 \,N \,m^{-1}$. The block is displaced from the equilibrium position by $10 \,cm$ and released. The maximum acceleration of the block is: (in $\,m/s^2$)

$A$ stone of mass $m$ is tied to an elastic string of negligible mass and spring constant $k$. The unstretched length of the string is $L$. The other end of the string is fixed to a nail at a point $P$. Initially,the stone is held at the same level as point $P$ and is dropped vertically.
$(a)$ Find the distance $y$ from the top when the mass comes to rest for an instant,for the first time.
$(b)$ What is the maximum velocity attained by the stone in this drop?
$(c)$ What shall be the nature of the motion after the stone has reached its lowest point?