Two bodies $P$ and $Q$ of equal mass are suspended from two separate massless springs of force constants $k_1$ and $k_2$ respectively. If the maximum velocities of them are equal during their motion,the ratio of the amplitude of $P$ to $Q$ is:

Two identical particles each of mass $m$ are interconnected by a light spring of stiffness $k$. The time period for small oscillations is equal to:

$A$ spring with a spring constant $1200 \; N m^{-1}$ is mounted on a horizontal table as shown in the figure. $A$ mass of $3 \; kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \; cm$ and released. Let us take the position of the mass when the spring is unstretched as $x = 0$,and the direction from left to right as the positive direction of the $x$-axis. Give $x$ as a function of time $t$ for the oscillating mass if at the moment we start the stopwatch $(t = 0)$,the mass is:
$(a)$ at the mean position,
$(b)$ at the maximum stretched position,and
$(c)$ at the maximum compressed position.
In what way do these functions for $SHM$ differ from each other: in frequency,in amplitude,or in the initial phase?

$A$ particle is suspended from a vertical spring which is executing $S.H.M.$ of frequency $5 \ Hz$. The spring is unstretched at the highest point of oscillation. What is the maximum speed of the particle? (Take $g = 10 \ m/s^2$)

$A$ heavy brass sphere is hung from a light spring and is set in vertical small oscillations with a period $T$. The sphere is now immersed in a non-viscous liquid with a density $1/10$th the density of the sphere. If the system is now set in vertical $S.H.M.$,its period will be

The potential energy of a particle of mass $0.1\,kg,$ moving along the $x-$axis,is given by $U = 5x(x-4)\,J,$ where $x$ is in metres. It can be concluded that