- #1

Saitama

- 4,243

- 93

## Homework Statement

A rope of mass M and length ##l## lies on a friction less table, with a short portion, ##l_0## hanging through a hole. Initially the rope is at rest.

a. Find a general solution for x(t), the length of rope through the hole.

(Ans: ##x=Ae^{\gamma t}+Be^{-\gamma t}##, where ##\gamma^2=g/l##)

b. Evaluate the constants A and B so that the initial conditions are satisfied.

## Homework Equations

## The Attempt at a Solution

The forces acting on the rope are weight and tension (T) due to the part of rope on the table. If x is the length of rope hanging, l-x is the length of rope on the table. Let ##\lambda## be the mass per unit length of rope.

Newton's second law for hanging part,

$$\lambda xg-T=\lambda xa$$

Newton's second law for rope on table,

$$T=\lambda (l-x)a$$

From the two equations,

$$a=\frac{gx}{l+2x}$$

I can substitute a=d^2x/dt^2 but Wolfram Alpha gives no solution for this.

Any help is appreciated. Thanks!