A mass $m$ is attached to two springs of same force constant $K$, as shown in following four arrangements. If $T_1, T_2, T_3$ and $T_4$ respectively be the time periods of oscillation in the following arrangements, in which case time period is maximum?

A weightless spring which has a force constant oscillates with frequency $n$ when a mass $m$ is suspended from it. The spring is cut into two equal halves and a mass $2m $ is suspended from it. The frequency of oscillation will now become

A force of $6.4\  N$ stretches a vertical spring by $0.1\ m$. The mass that must be suspended  from the spring so that it oscillates with a time period of $\pi/4\  second$ is .... $kg$

Two bodies $M$ and $N $ of equal masses are suspended from two separate massless springs of force constants $k_1$ and $k_2$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude $M$ to that of $N$ is

A spring having with a spring constant $1200\; N m ^{-1}$ is mounted on a hortzontal table as shown in 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 mass when the spring is unstreched as $x=0,$ and the direction from left to right as the positive direction of $x$ -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 the inittal phase?

Two springs of force constants $300\, N / m$ (Spring $A$) and $400$ $N / m$ (Spring $B$ ) are joined together in series. The combination is compressed by $8.75\, cm .$ The ratio of energy stored in $A$ and $B$ is $\frac{E_{A}}{E_{B}} .$ Then $\frac{E_{A}}{E_{B}}$ is equal to