$A$ block is resting on a piston which executes simple harmonic motion with a period $2.0 \, s$. The maximum velocity of the piston,at an amplitude just sufficient for the block to separate from the piston,is .......... $m \, s^{-1}$.

Show that the oscillations due to a spring are simple harmonic oscillations and obtain the expression for the period of oscillation.

Two identical springs are connected to a mass $m$ as shown in the figure ($k$ = spring constant). If the time period of the configuration in $(a)$ is $2 \,s$, what is the time period of the configuration in $(b)$?

Two bodies $A$ and $B$ of equal mass are suspended from two separate massless springs of spring constants $K_1$ and $K_2$ respectively. The two bodies oscillate vertically such that their maximum velocities are equal. The ratio of the amplitude of $B$ to that of $A$ is

Three masses $500 \ g$,$300 \ g$,and $100 \ g$ are suspended at the end of a spring as shown in the figure and are in equilibrium. When the $500 \ g$ mass is removed,the system oscillates with a period of $2 \ s$. When the $300 \ g$ mass is also removed,it will oscillate with a period of (in $s$)

$A$ particle of mass $m$ is attached to three identical springs $A, B$ and $C$ each of force constant $k$ as shown in the figure. If the particle of mass $m$ is pushed slightly against the spring $A$ and released,then the time period of oscillations is