Module 2 lec 8
Momentum:
The general conversion law also applies to momentum, М, which for mass m moving with
a velocity u, is defined as
Law of conversation of momentum:
Following the usual law, eqn. 2.2, the net rate of transfer of momentum into a system equals
the rate of increase of the momentum of the system. There are two principal modes, by
which momentum can be transferred; by force and by convection.
1. A force is readily seen to be equivalent to a rate of transfer of momentum by
examining its dimensions:
A momentum balance can be applied to a mass m falling with instantaneous velocity
u under gravity in air that offers negligible resistance. Considering momentum as
positive downwards, the rate of transfer of momentum to the system is the
gravitational force mg, and is equated to the rate of increase of downwards
momentum of the mass, giving:
m is taken out of the derivative because the mass is always constant. The
acceleration is therefore:
Another example is provided by the steady flow of a fluid in a pipe of length L and
diameter D, as shown in the figure below. The upstream pressure exceeds the
downstream pressure
and thereby provides a driving force for flow form left to
right. The shear stress
exerted by the wall on the fluid tends to retard the
motion.
Applying a steady – state momentum balance gives:
In eqn. 2.38, the first term is the rate of addition of momentum to the system
resulting from the net pressure difference
, which acts on a circular area
. The second term is the rate of subtraction of momentum from the system
by the wall shear stress, which acts to the left on the cylinder area
. Since the
flow is steady, there is no change in momentum of the system with time, so the
derivative term is zero. Eqn. 2.38 becomes
2. The convective transfer of momentum by flow is more subtle, but can be
appreciated with reference to figure shown above. Water from a hose of crosssectional area A impinges with velocity u on the far side of a trolley of mass m with
frictionless wheels. The dotted box delineates a stationary system within which the
momentum Mv is increasing to the right, because the trolley clearly tends to
accelerate in that direction. The reason is that momentum is being transferred
across the surface BC into the system by the convective action of the jet. The rate of
transfer is mass flow rate times the velocity:
The acceleration of the trolley can now be found by applying momentum balances in
two different ways, depending on whether the control volume is stationary or
moving. In each case, the water leaves the nozzle of cross-sectional area A with
velocity u and the trolley has a velocity v. Both velocities are relative to the nozzle.
1. Control surface moving with trolley. As shown in the figure below, the control
surface delineating the system is moving to the right at the same velocity v as
the trolley. The observer perceives water entering the system across BC not with
velocity u but with a relative velocity (u – v), so that the rate m of convection of
mass into the control volume is
A momentum balance gives:
The mass of the system is not constant, but increasing at the rate given by:
It follows from the last three equations that the acceleration of the trolley to the
right is:
2. Control fixed volume: In the figure shown below, the control fixed volume is
now fixed in space, so that the trolley is moving within it. Also – and the part of
the jet of length L inside the control surface is lengthening at a rate dL / dt = v
and increasing its momentum. A mass balance gives
A momentum balance gives
By eliminating dM / dt between the above two eqns and rearranging, the acceleration
becomes: