Q.The period of conical pendulum in terms of its length (l), semi-vertical angle (θ) and acceleration due to gravity

A conical pendulum is a mass (the bob) suspended by a string from a fixed support point, made to revolve so that the bob traces a horizontal circle at constant speed while the string itself sweeps out the surface of a cone of constant half-angle θ measured from the vertical. Only two forces act on the bob: its weight $mg$ (vertically downward) and the string tension $T$ (along the string, toward the support). Resolving the tension into vertical and horizontal components: the vertical component $T\cos\theta$ must balance gravity exactly ($T\cos\theta=mg$, since the bob stays at constant height), and the horizontal component $T\sin\theta$ supplies the centripetal force needed for the circular motion ($T\sin\theta=\dfrac{mv^2}{r}$). …