$A$ mass attached to a spring performs $S.H.M.$ whose displacement is $x = 3 \times 10^{-3} \cos(2\pi t) \text{ m}$. The time taken to obtain maximum speed for the first time is (in $s$):

$A$ block of mass $1 \ kg$ is fastened to a spring of spring constant $100 \ N \ m^{-1}$. The block is pulled to a distance $x = 10 \ cm$ from its equilibrium position $(x = 0 \ cm)$ on a frictionless surface,from rest at $t = 0$. The kinetic energy and the potential energy of the block when it is $5 \ cm$ away from the mean position are:

If a vertical mass-spring system is taken to the moon,what will happen to its time period?

The graph shown was obtained from experimental measurements of the period of oscillations $T$ for different masses $M$ placed in the scale pan on the lower end of the spring balance. The most likely reason for the line not passing through the origin is that the

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?

$A$ spring of length $l$ and force constant $k$ is attached to a mass $m$ and oscillates with frequency $f_1$. If the spring is cut into two equal parts and one part is attached to the same mass $m$ to oscillate,its frequency becomes $f_2$. Find the relation between $f_1$ and $f_2$.