A mass $M$ is attached to a spring which oscillates with a period of $2 \mathrm{s}$. If the mass is increased by $2 \mathrm{kg}$, the period increased by $1 \mathrm{s}$. Find the initial mass $M$, assuming that Hooke’s law is obeyed.

Two masses $m_1$ and $m_2$ are suspended together by a massless spring of spring constant $k$. When the masses are in equilibrium, $m_1$ is removed without disturbing the system. Find the angular frequency of oscillation of $m_2$.

Find its frequency, when a cylindrical block of wood of mass $m$, radius $r$ & density $p$ is floating in water with its axis vertical. It is depressed a little and then released. If the motion of the block is simply harmonic.

A particle moves in $x-y$ plane according to the equation $\overrightarrow{r}=\left(\hat{i}+2\hat{j}\right) A \cos \omega t$ . The motion of the particle is 

$\left(\mathrm{a}\right)$on a straight line 

$\left(\mathrm{b}\right)$ on an ellipse

$\left(\mathrm{c}\right)$ periodic 

$\left(\mathrm{d}\right)$ simple harmonic.

A particle executing Simple Harmonic Motion with an amplitude $A$. The displacement of the particle when its potential energy is half of its total energy is

When a particle executing S H M is $2 \mathrm{cm}$ Away from the mean position its kinetic energy is double the potential energy. At what distance from the mean position will the P.E. be double the K.E.?