Q.What is a conical pendulum? Show that its time period is given by $2\pi\sqrt{\dfrac{l\cos\theta}{g}}$, where l is the length of the string, θ is the angle that the string makes with the vertical and g is the acceleration due to gravity.

[!TLDR] A conical pendulum's bob revolves in a horizontal circle while its string sweeps a cone; balancing gravity against the vertical component of tension, and centripetal force against the horizonta …

A conical pendulum is a mass $m$ hung from a fixed point O by a string of length $l$ which, instead of swinging in a vertical plane, revolves so that the bob traces a horizontal circle of radius $r$ at constant speed while the string sweeps the surface of a cone of half-angle $\theta$ from the vertical. Two forces act on the bob: weight $mg$ downward and tension $T$ along the string. Resolving $T$ vertically and horizontally: $T\cos\theta = mg$ (no net vertical motion) and $T\sin\theta = \dfrac{mv^2}{r}$ (the horizontal component supplies the centripetal force). Dividing, $\tan\theta = \dfrac{v^2}{rg}$. Using $v=\dfrac{2\pi r}{T_{period}}$ and the cone geometry $r=l\sin\theta$: $\tan\theta = \dfrac{4\pi^2 r}{T_{period}^2 g} …

Resolve the string tension into vertical (balancing gravity) and horizontal (supplying centripetal force) components, then combine with the geome …

Confusing the period of revolution T with the tension symbol T used in the same derivation — keep them distinct (e.g. …