Which basic properties are needed for a system to oscillate?

The figure shows the circular motion of a particle. The radius of the circle,the period,the sense of revolution,and the initial position are indicated in the figure. The simple harmonic motion of the $x$-projection of the radius vector of the rotating particle $P$ is

The radius of the circle,the period of revolution,the initial position,and the sense of revolution are indicated in the figure. The $y$-projection of the radius vector of the rotating particle $P$ is:

Which of the following is a necessary and sufficient condition for $S.H.M.$?

$A$ particle moves on the $x-$axis according to the equation $x = x_0 \sin^2 \omega t$. Is the motion simple harmonic?

$A$ particle of mass $10\, g$ moves in a field where potential energy per unit mass is given by the expression $v = 8 \times 10^4\, x^2\, \text{erg/g}$. If the total energy of the particle is $8 \times 10^7\, \text{erg}$,then the relation between $x$ and time $t$ is: