Pa- Pa (air) Zero pressure level Pm P-

Some manometers are constructed as in Fig. , where
one side is a large reservoir (diameter D ) and the other side
is a small tube of diameter d , open to the atmosphere. In
such a case, the height of manometer liquid on the reservoir
side does not change appreciably. This has the advantage
that only one height needs to be measured rather than two.
The manometer liquid has density ρ _m while the air has density
ρ _a . Ignore the effects of surface tension. When there is
no pressure difference across the manometer, the elevations
on both sides are the same, as indicated by the dashed
line. Height h is measured from the zero pressure level as
shown. ( a ) When a high pressure is applied to the left side,
the manometer liquid in the large reservoir goes down,
while that in the tube at the right goes up to conserve mass.
Write an exact expression for p _1gage , taking into account the
movement of the surface of the reservoir. Your equation
should give p _1gage as a function of h , ρ _m , and the physical
parameters in the problem, h , d , D , and gravity constant g.
( b ) Write an approximate expression for p _1gage , neglecting
the change in elevation of the surface of the reservoir liquid.

( c ) Suppose h = 0.26 m in a certain application. If p _a =
101,000 Pa and the manometer liquid has a density of 820
kg/m ³ , estimate the ratio D / d required to keep the error
of the approximation of part ( b ) within 1 percent of the
exact measurement of part ( a ). Repeat for an error within
0.1 percent.