An oscillator of mass $M$ is at rest in its equilibrium position in a potential $V = \frac{1}{2}k(x - X)^2$. $A$ particle of mass $m$ comes from the right with speed $u$ and collides completely inelastically with $M$ and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after $13$ collisions is: $(M = 10, m = 5, u = 1, k = 1)$.

$A$ body executing simple harmonic motion has a maximum acceleration equal to $24 \, m/s^2$ and a maximum velocity equal to $16 \, m/s$. The amplitude of the simple harmonic motion is:

$A$ horizontal board is performing simple harmonic oscillations horizontally with an amplitude of $0.3 \ m$ and a period of $4 \ s$. What is the minimum coefficient of friction between a heavy body placed on the board and the board itself if the body is not to slip?

$A$ man weighing $60\ kg$ stands on the horizontal platform of a spring balance. The platform starts executing simple harmonic motion of amplitude $0.1\ m$ and frequency $\frac{2}{\pi}\ Hz$. Which of the following statements is correct?

Two identical balls $A$ and $B$,each of mass $0.1 \ kg$,are attached to two identical massless springs. The spring-mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in the figure. The pipe is fixed in a horizontal plane. The centers of the balls can move in a circle of radius $0.06 \ m$. Each spring has a natural length of $0.06\pi \ m$ and a force constant of $0.1 \ N/m$. Initially,both balls are displaced by an angle $\theta = \pi/6$ radian with respect to the diameter $PQ$ of the circle and released from rest. The frequency of oscillation of the ball $B$ is:

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 .........