The displacement of a particle in $SHM$ is given by $x = 5 \sin(\pi t)$,where $x$ is in $cm$. What is the time taken by the particle to travel from the mean position to the position of maximum displacement (in $, s$)?

Give the relation between periodic time and angular frequency.

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $(t = 0)$ position of the particle,the radius of the circle,and the angular speed of the rotating particle. For simplicity,the sense of rotation may be fixed to be anticlockwise in every case: ($x$ is in $cm$ and $t$ is in $s$).
$(a)\; x = -2 \sin (3t + \pi/3)$
$(b)\; x = \cos (\pi/6 - t)$
$(c)\; x = 3 \sin (2\pi t + \pi/4)$
$(d)\; x = 2 \cos \pi t$

The $S.H.M.$ of a particle is given by the equation $x = 2 \sin \omega t + 4 \cos \omega t$. Its amplitude of oscillation is ........ units.

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

Equations $y = 2A \cos^2 \omega t$ and $y = A (\sin \omega t + \sqrt{3} \cos \omega t)$ represent the motion of two particles.