A bead starts from rest at a point and slides without friction along a wire

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

Action and Symmetry, Ch The action, the first variation and the boundary term

Problem

A bead starts from rest at a point

O

and slides without friction along a wire, under uniform gravity

g=9.8\,\mathrm{m\,s^{-2}}

, to a vertical wall a horizontal distance

D=1.2\,\mathrm{m}

from

O

. The wire is shaped to make the travel time stationary, and the height at which it meets the wall is left free. Measure

x

horizontally and

y

downward from

O

, so that the travel time is

\dfrac{1}{\sqrt{2g}}\displaystyle\int_0^D\sqrt{\dfrac{1+y'^2}{y}}\,\mathrm{d}x

. ((a)) State the condition that the free end imposes where the wire meets the wall, and say why. ((b)) Show that the curve

x=R(\phi-\sin\phi)

y=R(1-\cos\phi)

, with

R

a constant length and

\phi

a parameter, satisfies the Beltrami first integral

y(1+y'^2)=\text{constant}

of this integrand, and find the constant. Then use part (a) to find

R

and the depth

y_2

at which the wire meets the wall. ((c)) Using energy conservation for the bead's speed, show that the travel time along this curve is

\pi\sqrt{R/g}

, and evaluate it. ((d)) Find the shortest time for the bead to reach the wall along any straight ramp from

O

, and compare it with part (c).