$A$ block of mass $(10 \alpha) \text{ g}$,where $\alpha$ is a constant,is moving with velocity $3 \text{ m/s}$ to the right. It collides inelastically with a block on the right of mass $10 \text{ g}$ and sticks to it. The right block is connected to three springs as shown in the figure. The spring constant of each spring is $k = 2 \text{ N/m}$. If the amplitude of the resulting simple harmonic motion is $A = \frac{1}{2\sqrt{2}} \text{ m}$,then the value of $\alpha$ is:

$A$ block is placed on a horizontal plank. The plank is performing $SHM$ along a vertical line with an amplitude of $40 \, cm$. The block just loses contact with the plank when the plank is momentarily at rest. Then:

You are holding a shallow circular container of radius $R$,filled with water to a height $h$ $(h \ll R)$. When you walk with speed $v$,it is seen that water starts spilling over. This happens due to the resonance of the periodic impulse given to the container (due to walking) with the oscillation of the water in the container. If the time period of water oscillating in the container is inversely proportional to $\sqrt{h}$,then $v$ is proportional to

Fill in the blanks:
$1.$ If a tunnel is dug along the diameter of the Earth and a body is dropped freely in it,the motion of this body is .........,assuming there is no frictional force of the medium acting on it.
$2.$ When a body executes simple harmonic motion and makes $\frac{1}{2\pi}$ oscillations,its phase increases by ......... $rad$.
$3.$ For a body executing simple harmonic motion,the phase increases by ......... every second.
$4.$ The ratio of force constants of two springs is $1:2$ and the ratio of their mechanical energies is $2:9$. The ratio of the amplitudes of the two bodies suspended at the ends of the springs is .........

The figure shows a graph between velocity $(v)$ and displacement $(x)$ from the mean position of a particle performing $SHM$.

$A$ clock $S$ is based on the oscillations of a spring,and a clock $P$ is based on pendulum motion. Both clocks run at the same rate on Earth. On a planet having the same density as Earth but twice the radius,then: