spherical pendulum using N2

[ee7klt]

Hi All,

I'm trying to solve the spherical pendulum using newton's laws, but am coming up with something different than that obtained using the lagrangian formalism. The two forces acting on the point mass are the tension $T$ and the gravitational force $mg$, which (in the case where it's moving in a circle of constant \theta) should conspire to produce the centripetal force necessary to keep it in circular motion. Thus,

r: $T - m g \cos \theta = m \ddot{r}$
\theta: $m g \sin \theta = m (r \ddot{\theta} + \dot{r} \dot{\theta}$
\phi : $r \sin \theta \ddot{\phi} + r \cos \theta \dot{theta} \dot{\phi} + \dot{r} sin \theta \dot{\phi} = 0$

In particular, this yields $T = mg \cos \theta$ with the constraint $r = l$ for the dynamic equilibrium case \theta = const. but I know the answer to be $T = -m l \sin^2 \theta {\dot{phi}}^2 - m g \cos \theta$.

Any ideas ?

[Xerxes314]

You just need to add the centripetal force mv²/r, and then convert v to phi.

Xerxes

[jbelew]

dynamic equilibrium, ouch!