On a smooth inclined plane,a body of mass $M$ is attached between two springs. The other ends of the springs are fixed to firm supports. If each spring has a force constant $K$,the period of oscillation of the body (assuming the springs are massless) is

$A$ 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?

$A$ flat horizontal board moves up and down in $SHM$ with an amplitude $\alpha$. What is the shortest permissible time period of the vibration such that an object placed on the board does not lose contact with the board?

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

$A$ particle of mass $m$ is attached to three identical massless springs of spring constant $k$ as shown in the figure. The time period of vertical oscillation of the particle is

$A$ spring has a certain mass suspended from it and its period of vertical oscillations is $T_1$. The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations is now $T_2$. The ratio of $T_2 / T_1$ is