We will apply Bernoulli's Equation(1/2) 1 v12 + P1 +
to the two regions. Since they are at the same height, the two y-terms are the same and we have only
(1/2) The density is the same in both regions so we have
(1/2) As we did in the previous Q&A Example, we can change the subscripts to "n" and "W" for the "narrow" and "Wide" regions; then Bernoulli's Equation is
(1/2) and we know everything in this equation except Pn,
Pn = (1/2) Pn = PW + (1/2)
Pn = PW + (1/2)
Pn = 15 mmHg + (1/2) [ 1000 kg/m3] [ (2.5 m/s)2 - (6.2 m/s)2 ]
Pn = 15 mmHg + (1/2) [ 1000 kg/m3] [ (6.25 m2/s2) - (38.44 m2/s2) ]
Pn = 15 mmHg + (1/2) [ 1000 kg/m3] [ - 32.19 m2/s2 ]
Pn = 15 mmHg - (1/2) (1000 kg/m3)(32.19 m2/s2)
Before we go on, notice that the pressure is going to be less in the narrow region. And that is just as we would/should expect since the velocity is greater there!
Pn = 15 mmHg - (1/2) (32,190 (kg m/s2) / m2 Pn = 15 mmHg - 16,095 N / m2
Pn = 15 mmHg - 16,095 Pa
Remember, you can't add apples and oranges! What about terms in mmHg and in Pa. Those both measure pressure but we have to convert one of them. Since we started with a pressure of 15 mmHg in the wide region, let's stay with units of mmHg and convert the term with units of Pa,
16.095 kPa = 16,095 Pa [ 1 mmHg / 133 Pa ] = 121 mmHg Pn = 15 mmHg - 121 mmHg
Pn = - 106 mmHg
What does this negative pressure mean? Both PW and Pn are gauge pressures. When we state
Pw = 15 mmHg that means the pressure in the wide region is 15 mmHg greater than atmospheric pressure. Now that we have calculated
Pn = - 106 mmHg that means the pressure in the narrow regions is 106 mmHg less than atmospheric pressure. That means the U-tube manometers would look something like this --
or
(c) 2000, Doug Davis; all rights reserved.
