$A$ particle moves such that its acceleration $a$ is given by $a = -bx$,where $x$ is the displacement from the equilibrium position and $b$ is a constant. The period of oscillation is

The motion of a particle varies with time according to the relation $y = a(\sin \omega \,t + \cos \omega \,t)$,then

Write the phase difference between displacement and velocity,displacement and acceleration,and velocity and acceleration for a particle executing Simple Harmonic Motion $(SHM)$.

The displacement of a particle in an oscillatory motion at a time $t$ is given by $x = 8 \sin \frac{\pi t}{4} \text{ cm}$. Calculate its displacement in the time interval $t = 0 \text{ s}$ to $t = 2 \text{ s}$. (in $\text{ cm}$)

The mass of a particle is $1\,kg$ and it is moving along the $x-$axis. The period of its small oscillation is $\frac{\pi}{2}$. Its potential energy may be:

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].
$(i)$ $\sin \omega t + \cos \omega t$
$(ii)$ $\sin \omega t + \cos 2\omega t + \sin 4\omega t$
$(iii)$ $e^{-\omega t}$
$(iv)$ $\log(\omega t)$