For a simple harmonic motion given by $x = 0.01 \sin 100\pi (t + 0.05)$,what is the time period in $\text{sec}$?

$A$ 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:

Identify the function which represents a periodic motion.

The periodic time of $SHM$ is also measured by $T = 2 \pi \sqrt{\frac{\text{displacement}}{\text{acceleration}}}$. Can we say that the time period depends on the displacement?

Which of the following functions of time represent $(a)$ simple harmonic motion and $(b)$ periodic but not simple harmonic? Give the period for each case.
$(1)$ $\sin \omega t - \cos \omega t$
$(2)$ $\sin^2 \omega t$

The equation of $SHM$ of a particle is given as $2 \frac{d^2x}{dt^2} + 32x = 0$,where $x$ is the displacement from the mean position of rest. The period of its oscillation (in seconds) is