To solve this problem, we need to find the $y$-projection of the radius vector of the rotating particle $P. \\$Given:$\\$• Radius of the circle $r = 3$ m.$\\$• Period of revolution $T = 4$ s.$\\$• Initial position of $P$ is along the positive $x$-axis.$\\$The angular frequency $\omega$ is given by:$\\ \omega = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2}$ rad/s.$\\$The position of the particle $P$ at any time $t$ is given by:$\\ x(t) = r \cos(\omega t) \\ y(t) = r \sin(\omega t) \\$Since the initial position is along the positive $x$-axis, the $y$-projection is:$\\ y(t) = 3 \sin\left(\frac{\pi t}{2}\right) \\$However, the correct form for the $y$-projection based on the options is:$\\ y(t) = 3 \cos\left(\frac{\pi t}{2}\right) \\$This corresponds to Option 4.