A rigid, massless rod of length pivots at one end; the pivot is driven horizontally…

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 rigid, massless rod of length

\ell = 0.40\,\mathrm{m}

pivots at one end; the pivot is driven horizontally so that its position is

X(t) = X_0\sin(\Omega t)

with

X_0 = 0.050\,\mathrm{m}

and

\Omega = 2.0\,\mathrm{rad\,s^{-1}}

. A bob is fixed at the rod's far end, and the rod can swing all the way around the pivot; its angle from the downward vertical is

\theta

. At

t=0

the rod is at

\theta = 25^\circ

with

\dot\theta = 1.5\,\mathrm{rad\,s^{-1}}

. ((a)) Write the bob's position

\mathbf r(\theta,t) = \big(X(t)+\ell\sin\theta,\,-\ell\cos\theta\big)

and its real velocity

\dot{\mathbf r}

t=0

, in Cartesian components, to two significant figures. ((b)) Write the bob's virtual displacement

\delta\mathbf r

t=0

(components), and say which part of

\dot{\mathbf r}

has no counterpart in

\delta\mathbf r

. ((c)) State the dimension and shape of this system's configuration space.