Critical conical pendulum

Problem

A conical pendulum of length $\ell = 1\;\mathrm{m}$ rotates at angle $\theta$ from the vertical. What happens to the tension $T$ and to the angular velocity $\omega$ as $\theta \to 90^\circ$? Why is the limiting angle never reached with a real string?

The string tension (along the string) and the weight (vertical) are the only forces: their horizontal resultant is the centripetal force.

Solution

Equations of motion. The mass traces a horizontal circle of radius $r = \ell\sin\theta$. The only forces are the tension $\vv T$ (along the string) and the weight $m\vv g$. We resolve them along the vertical and the horizontal.

Vertical (equilibrium, no vertical acceleration): $T\cos\theta = mg$ Horizontal (the horizontal component of $T$ is the centripetal force): $T\sin\theta = m\,\omega^2 r = m\,\omega^2 \ell\sin\theta$ From the second equation, cancelling $\sin\theta$: $T = m\,\omega^2 \ell$. Equating with the first ($T = mg/\cos\theta$): $m\,\omega^2 \ell = \frac{mg}{\cos\theta} \quad\Longrightarrow\quad \omega = \sqrt{\frac{g}{\ell\cos\theta}}$

Limit $\theta \to 90^\circ$. As $\theta \to 90^\circ$, $\cos\theta \to 0$, so: $T = \frac{mg}{\cos\theta} \to \infty, \qquad \omega = \sqrt{\frac{g}{\ell\cos\theta}} \to \infty$ Both the tension and the angular velocity diverge: an infinite tension and an infinitely fast rotation would be needed to bring the string perfectly horizontal. Physically, the reason is that the string, being taut, cannot have zero vertical component and still support the weight: the vertical component $T\cos\theta$ must always equal $mg$, and to do so as $\cos\theta \to 0$ requires $T \to \infty$.

Why it is unreachable. A real string has a finite breaking load: well before $\theta$ reaches $90^\circ$, the tension exceeds the string’s strength and it snaps. The perfectly horizontal configuration is therefore an ideal limit, physically unreachable.

$\ev{T = \frac{mg}{\cos\theta} \to \infty, \quad \omega = \sqrt{\frac{g}{\ell\cos\theta}} \to \infty \quad \text{as } \theta \to 90^\circ}$

Links

Topics: Dinamica Concepts: Tensione · Forza centripeta · Moto circolare uniforme Skills: Analisi di casi limite e fantafisica Objects: Pendolo conico