A bead slides along a straight rod that rotates about a fixed pivot at constant…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

A bead slides along a straight rod that rotates about a fixed pivot at constant angular velocity

\omega = 4.0\,\mathrm{rad\,s^{-1}}

in a horizontal plane. At the same time the bead slides outward along the rod at a constant radial speed

u = 0.50\,\mathrm{m\,s^{-1}}

, starting from

r_0 = 0.10\,\mathrm{m}

from the pivot at

t=0

, so that

r(t) = r_0 + ut

and

\theta(t) = \omega t

. The bead is treated as a point and the rod as rigid. ((a)) Write the bead's velocity at

t=2.0\,\mathrm s

in terms of

\hat{\mathbf r}

and

\hat{\boldsymbol\theta}

, and find its speed. ((b)) Write its acceleration at a general time

t

in terms of

\hat{\mathbf r}

and

\hat{\boldsymbol\theta}

, and say which of the four terms vanish. ((c)) Just after the start, is the acceleration mostly along the rod or mostly across it? Find the distance from the pivot, and the time, at which the two parts are equal in size, and say which part dominates at

t = 2.0\,\mathrm s

Browse the problem library