The Momentum Equation And Its Applications

The Momentum Equation And Its Applications

Def:

An equation stating that the impulse (force multiplied by time) applied to the body of water is equal to the momentum (mass multiplied by velocity) acquired by it. The concepts of momentum and impulse, along with energy, are basic to all dynamics.


First off, 
Newton's 2^nd Law can be written:

The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force.

Resultant Force : F = ma

To determine the rate of change of momentum for a fluid we will consider a streamtube as we did for the Bernoulli equation,
We start by assuming that we have steady flow which is non-uniform flowing in a stream tube.

A streamtube in three and two-dimensions

In time , a volume of the fluid moves from the inlet a distance ,so the volume entering the streamtube in the time  is

this has mass,

and momentum

Similarly, at the exit, we can obtain an expression for the momentum leaving the steamtube:

We can now calculate the force exerted by the fluid using Newton's 2^nd Law. The force is equal to the rate of change of momentum. So

We know from continuity that 

, and if we have a fluid of constant density,

then we can write

The rate at which momentum leaves face 1 is

The rate at which momentum enters face 2 is

Thus the rate at which momentum changes across the stream tube is

i.e.

For two or more dimensional flow

Figure: 2 dimensional flow in streamtube

For the force in x-direction

  

For the force in y-direction

Example

Solution

Find: the vertical component of the anchoring force required to hold the nozzle in the place.

Application of momentum equation

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