Page 460 - Introduction to Continuum Mechanics

444 Integral Formulation of General Principles

The tension at the corner is given by

We note that the motion can also be obtained by considering the whole rope as a system. In
fact, the total linear momentum of the rope at any time t is

its rate of change is

and the total resultant force on the rope is

Thus, equating the force to the rate of change of momentum for the whole rope, we obtain

and

'2
Eliminating x from the above two equations, we arrive at Eq. (vi) again.

Example 7.6.2
Figure 7.4 shows a steady jet of water impinging onto a curved vane in a tangential direction.
Neglect the effect of weight and assume that the flow at the upstream region, section A, as
well as at the downstream region, section B is a parallel flow with a uniform speed v 0. Find
the resultant force (above that due to the atmospheric pressure) exerted on the vane by the
jet. The volume flow rate is Q.
Solution. Let us take as control volume that portion of the jet bounded by the planes at A
and B. Since the flow at A is assumed to be a parallel flow, therefore the stress vector on the
planed is normal to the plane with a magnitude equal to the atmospheric pressure which we
take to be zero. [We recall that in the absence of gravity, the pressure is a constant along any
direction which is perpendicular to the direction of a parallel flow (See Section 6.7)]. Thus,
the only forces acting on the material in the control volume is that from the vane to the jet.
Let F be the resultant of these forces. Since the flow is steady, the rate of increase of momentum