Identify the correct definition.

$A$ particle performs $SHM$ on the $x-$axis with a time period of $0.5 \, s,$ such that its velocity is zero at $x = -3 \, cm$ and at $x = 9 \, cm$. It was located at $x = 0$ and moving in the negative $x-$direction at $t = 0$. The equation of $SHM$ of the particle is:

$A$ particle is moving with constant angular velocity along the circumference of a circle. Which of the following statements is true?

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
$(a)$ The rotation of Earth about its axis.
$(b)$ Motion of an oscillating mercury column in a $U$-tube.
$(c)$ Motion of a ball bearing inside a smooth curved bowl,when released from a point slightly above the lowermost point.
$(d)$ General vibrations of a polyatomic molecule about its equilibrium position.

Obtain the expression of displacement from the force law for simple harmonic motion.

Explain periodic function.