$A$ mass $m_1$ connected to a horizontal spring performs $S.H.M.$ with amplitude $A$. While mass $m_1$ is passing through the mean position,another mass $m_2$ is placed on it so that both the masses move together with amplitude $A_1$. The ratio of $\frac{A_1}{A}$ is:

$A$ 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?

The motion of a mass on a spring,with spring constant $K$ is as shown in the figure. The equation of motion is given by $x(t) = A \sin \omega t + B \cos \omega t$ with $\omega = \sqrt{\frac{K}{m}}$. Suppose that at time $t = 0$,the position of the mass is $x(0)$ and velocity is $v(0)$,then its displacement can also be represented as $x(t) = C \cos (\omega t - \phi)$,where $C$ and $\phi$ are:

One end of a long metallic wire of length $L$,area of cross-section $A$ and Young's modulus $Y$ is tied to the ceiling. The other end is tied to a massless spring of force constant $k$ and a mass $m$ is hung from the free end of the spring. If $m$ is slightly pulled down and released,then its time period of oscillation is

The springs shown in the figure are identical,each having a spring constant $K$. When mass $A = 4\, kg$ is attached,the elongation of the spring is $1\, cm$. If mass $B = 6\, kg$ is attached to the system of two springs in series as shown,the total elongation produced is ..... $cm$.

Two particles $A$ and $B$ of masses $m$ and $2m$ are suspended from massless springs of force constants $K_1$ and $K_2$. During their oscillation, if their maximum velocities are equal, then the ratio of amplitudes of $A$ and $B$ is