A bob hangs from a fixed pivot on a light string of length and is free to swing in…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch Constraints, generalised coordinates and Lagrange's equations twice over

Problem

A bob hangs from a fixed pivot on a light string of length

\ell_{\rm s}=0.60\,\mathrm{m}

and is free to swing in any direction (a spherical pendulum). Its position is fixed by the polar angle

\theta

, measured from the downward vertical, and the azimuthal angle

\phi

, measured about that vertical, with

\phi=0

chosen along a fixed horizontal direction

\hat{\mathbf x}

. A steady horizontal draught exerts a constant force

F=2.0\,\mathrm{N}

on the bob, directed along

\hat{\mathbf x}

. ((a)) State the dimension and shape of the configuration space of this pendulum. ((b)) Find the generalised force

\mathcal Q_\theta

at the instant

\theta=60^\circ

\phi=0

, and state its units.