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71. Problem 17.26 (RHK) Consider a container of fluid subject to a vertical upward acceleration a. (a) We have to show that variation of gauge pressure with depth in the fluid is given by
where h is the depth and
(c) What is the state of affair in free fall? |
is
the density. (b) We have to show also that if the fluid as a whole
undergoes a vertical downward acceleration a, the gauge pressure at depth
h is given by
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Solution: Click For PDF Version
We want to find variation of pressure with height in a fluid
that as a whole is undergoing upward acceleration a. Density of the fluid
is
Let us consider an element of the fluid at a depth h
from the top of cross-sectional area A and width
dh. Let the variation
of hydrostatic pressure with depth be given by the function
Downward forces acting on this fluid element will be its weight w and the force due to pressure at depth h, and the upward force on it will be due to the pressure at depth h+dh. As this fluid element is undergoing upward acceleration a, the equation of motion is
Retaining terms up to first order in
Integrating this differential equation, we find that
As
(b) Carrying out a similar calculation for the situation when the fluid as a whole undergoes a vertical downward acceleration a, we will find that the gauge pressure at a depth h will vary as
(c) If the fluid is under free fall, that is
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.
The free-body diagram of the fluid element will be as shown below.
,
we get the following equation for the function
,
is the
pressure at the top layer of the fluid, the gauge pressure in the fluid varies
as
.
,
the gauge pressure in the fluid will be zero.