$A$ particle executes simple harmonic motion according to the equation $x(t) = A \sin^2(\alpha t)$. If the time period of the $SHM$ is $0.2 \ s$, then the value of $\alpha$ (in units of $rad/s$) is (in $\pi$)

In simple harmonic motion,the acceleration of a particle is given by $a = -bx$. What is the time period of the motion?

Which of the following functions of time represent $(a)$ periodic and $(b)$ non-periodic motion? Give the period for each case of periodic motion $[\omega$ is any positive constant].
$(i)$ $\sin \omega t + \cos \omega t$
$(ii)$ $\sin \omega t + \cos 2\omega t + \sin 4\omega t$
$(iii)$ $e^{-\omega t}$
$(iv)$ $\log(\omega t)$

$A$ particle free to move along the $x$-axis has potential energy given by $U(x) = k[1 - \exp(-x^2)]$ for $-\infty \le x \le +\infty$,where $k$ is a positive constant of appropriate dimensions. Then:

$A$ particle is executing simple harmonic motion with an amplitude of $2 \,m$. The difference in the magnitudes of its maximum acceleration and maximum velocity is $4$. The time-period of its oscillation and its velocity when it is $1 \,m$ away from the mean position are respectively:

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