The conical pendulum

A conical pendulum is an object hanging from a string that rotates, describing a horizontal circle: the string sweeps out the surface of a cone, hence the name. It is the archetypal example in which the centripetal force is provided by a component of the tension.

The string, of length $\ell$ and inclined at an angle $\theta$ to the vertical, supports the mass as it turns. The radius of the circle described is

$r = \ell\sin\theta$

The tension $\vec{t}$ is directed along the string, and it resolves into a horizontal component $t\sin\theta$ (towards the centre, which acts as the centripetal force) and a vertical component $t\cos\theta$ (upwards, which supports the weight). Newton’s second law then gives two equations:

$\begin{cases} t\sin\theta = m\omega^2 r = m\omega^2\ell\sin\theta \\ t\cos\theta = mg \end{cases}$

From the first equation, $\sin\theta$ cancels, leaving $t = m\omega^2\ell$. Substituting into the second:

$m\omega^2\ell\cos\theta = mg \quad\Rightarrow\quad \cos\theta = \frac{g}{\omega^2\ell}$

Key formula — Conical pendulum

$\cos\theta = \frac{g}{\omega^2\ell}$

The relation is rich in physical consequences. The faster the rotation (larger $\omega$), the smaller $\cos\theta$, i.e. the more the string “opens out”, approaching the horizontal. But $\cos\theta$ can never vanish: the angle $\theta = 90^\circ$ (horizontal string) would require $\omega \to \infty$ and is therefore unreachable — see the limiting case in Critical conical pendulum. The numerical applications are in Conical pendulum and Speed of a conical pendulum.

Links

Topics: Dynamics Concepts: Centripetal force · Tension Skills: Applying Newton’s laws · Symbolic set-up Objects: Conical pendulum

Related exercises: Speed of a conical pendulum · Centripetal force on a string · Critical conical pendulum