Two small bodies of mass $2 \, kg$ each are attached to each other using a thread of length $10 \, cm$,and hang on a spring whose force constant is $200 \, N/m$,as shown in the figure. We burn the thread. What is the distance between the two bodies when the top body first arrives at its highest position? (Take $\pi^2 = 10$)

An object of mass $m$ is performing simple harmonic motion on a smooth horizontal surface as shown in the figure. Just as the oscillating object reaches its extreme position,another object of mass $2m$ is placed gently on the oscillating object,and it sticks to it. Consider the following statements:
$(a)$ Amplitude of oscillation remains unchanged.
$(b)$ Time period of oscillation remains unchanged.
$(c)$ Total mechanical energy of the system does not change.
$(d)$ The maximum speed of the oscillating object changes.
Which of the above statements are correct?

The time period of oscillation for the given combination will be:

If a body of mass $0.98\, kg$ is made to oscillate on a spring of force constant $4.84\, N/m$,the angular frequency of the body is ..... $rad/s$. (in $.22$)

In figure $(A)$,mass '$2m$' is fixed on mass '$m$' which is attached to two springs of spring constant $k$. In figure $(B)$,mass '$m$' is attached to two springs of spring constant '$k$' and '$2k$'. If mass '$m$' in $(A)$ and $(B)$ are displaced by distance '$x$' horizontally and then released,then the time periods $T_{1}$ and $T_{2}$ corresponding to $(A)$ and $(B)$ respectively follow the relation.

$A$ mass $M$ attached to a horizontal spring executes simple harmonic motion with amplitude $A_1$. When mass $M$ passes the mean position,a smaller mass $m$ is attached to it,and both of them together execute simple harmonic motion with amplitude $A_2$. Then the value of $\frac{A_1}{A_2}$ is