Page 459 - Introduction to Continuum Mechanics

Principle of Linear Momentum 443

and the net momentum outflux is

Thus, if T denotes the tension at the corner point of the rope at time f, we have

i.e.,

as expected.

On the other hand, by considering the control volume (F c) 2 (see Fig. 7.3), we have, the
momentum in the downward direction is (m /l)xx and the momentum influx in the same
direction is [(m /l)x]x. Thus,

i.e.,

From Eqs. (ii) and (iv), we have

i.e.,

The general solution of Eq. (vi) is

Thus, if the rope starts from rest with an initial overhang of x 0, we have

so that C\ = €2 = x 0 /2 and the solution is