$A$ mass $m_1$ performs $S.H.M.$ with amplitude $A$ while connected to a horizontal spring. While mass $m_1$ is passing through the mean position, another mass $m_2$ $(m_2 < m_1)$ is placed on it so that both masses move together with amplitude $A_1$. The ratio of $A/A_1$ is

$A$ small mass $m$ is suspended at the end of a wire having negligible mass,length $L$,and cross-sectional area $A$. The frequency of oscillation for the $S.H.M.$ along the vertical line is ($Y =$ Young's modulus of the wire).

$A$ spring-block system in horizontal oscillation has a time period $T$. If the spring is cut into four equal parts and the block is re-connected with one of the parts,what will be the new time period of oscillation?

What is the condition for a body suspended at the end of a spring to undergo simple harmonic oscillation?

$A$ mass attached to a spring is free to oscillate,with angular velocity $\omega$,in a horizontal plane without friction or damping. It is pulled to a distance $x_{0}$ and pushed towards the centre with a velocity $v_{0}$ at time $t=0$. Determine the amplitude of the resulting oscillations in terms of the parameters $\omega, x_{0}$ and $v_{0}$. [Hint: Start with the equation $x=A \cos (\omega t+\theta)$ and note that the initial velocity is negative.]

$A$ stone of mass $m$ is tied to an elastic string of negligible mass and spring constant $k$. The unstretched length of the string is $L$. The other end of the string is fixed to a nail at a point $P$. Initially,the stone is held at the same level as point $P$ and is dropped vertically.
$(a)$ Find the distance $y$ from the top when the mass comes to rest for an instant,for the first time.
$(b)$ What is the maximum velocity attained by the stone in this drop?
$(c)$ What shall be the nature of the motion after the stone has reached its lowest point?