At $t=\frac{2T}{3}$ a particle is at $\frac{-A}{2}$ and moving towards mean position. Find the equation of SHM. Also find the speed at $\frac{T}{4}$

The motion of a mass on a spring, with spring constant $K$ is as shown in figure.

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

One end of a spring of force constant $k$ is fixed to a vertical wall and the other to a block of mass $m$ resting on a smooth horizontal surface. There is another wall at a distance $x_0$, from the block. The spring is then compressed by $2x_0$ and released. The time taken by the block to strike the other wall is

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is $100 \mathrm{g}$. The time period of the motion of the particle will be (approximately)

(take $g=10 {\mathrm{ms}}^{-2}$, radius of earth $=6400 \mathrm{km}$)

The position co-ordinates of a particle moving in a $3D$ coordinate system is given by

$x=a\cos \mathrm{\omega }\mathrm{t}$

$y=a\sin \omega t$

and $z=a\omega t$

The speed of the particle is: