# Homework Help: Confirm the differential equation for a falling rope using newtons law

1. Oct 23, 2007

### sapiental

1. The problem statement, all variables and given/known data

A uniform chain of length L, measured in feet, is held vertically so that the lower end just touches the floor. The chain weighs 2lb/ft. The upper end that is held is released from rest at t = 0 and the chain falls straight down. Ignore air resistance, assume that the + direction is downward, and let x(t) denote the length of the chain on the floor at time t. Use the fact that the net force F in (18) is 2L to show that a differential equation for x(t) is:

(L - x) (d^2x / dt^2) - (dt/dx)^2 = Lg

2. Relevant equations

the problem mentions equation 18 in our book which is

F = d/dt (mv)

3. The attempt at a solution

I rewrote the original equation as

(L - x) (d^2x / dt^2) - Lg = (dt/dx)^2

by analyzing the system I'm guessing the weight = 2(L-x)

mass = 2(L-x)/g

since F = ma

2(L-x)/g * a = 2L

a = Lg/(L-x)

a = (d^2x / dt^2) or the second derivative of t with respect to x

then

(L-x)(Lg/(L-x) - Lg = (dx/dt)^2

0 = (dx/dt)

When I tried to reproduce the same d.e. for this system i get

(L - x) (d^2x / dt^2) + (dt/dx)^2 = Lg

*the + instead of the - before the (dt/dx)^2 is what throws me off.