Classical Dynamics of Particles and Systems · Problem 9-17

Motion in a Noninertial Reference Frame — Marion & Thornton

Physics — Mechanics · falling-chain-numerical

Problem

Integrate Equation 9.17 in Example 9.2 numerically and make a plot of the speed versus the time using dimensionless parameters,

\dot{x}/\sqrt{2gb}

vs.

t/\sqrt{2b/g}

where

\sqrt{2b/g}

is the free fall time,

t_{\text{free fall}}

. Find the time it takes for the free end to reach the bottom. Define natural units by

\tau = t\sqrt{g/2b}

\alpha \equiv x/2b

and integrate

d\tau/d\alpha

from

\alpha = \varepsilon

(some small number greater than 0) to

\alpha = 1/2

. One can't integrate numerically from

\alpha = 0

because of a singularity in

d\tau/d\alpha

. The expression

d\tau/d\alpha

\frac{d\tau}{d\alpha} = \sqrt{\frac{1 - 2\alpha}{2\alpha(1 - \alpha)}}