We will apply Bernoulli's Equation( ^{1}/_{2})_{1}v_{1}^{2}+ P_{1}+_{1}g y_{1}= (^{1}/_{2})_{2}v_{2}^{2}+ P_{2}+_{2}g y_{2 }to the two regions. Since they are at the same height, the two y-terms are the same and we have only

( ^{1}/_{2})_{1}v_{1}^{2}+ P_{1}= (^{1}/_{2})_{2}v_{2}^{2}+ P_{2}The density is the same in both regions so we have

( ^{1}/_{2}) v_{1}^{2}+ P_{1}= (^{1}/_{2}) v_{2}^{2}+ P_{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}) v_{n}^{2}+ P_{n}= (^{1}/_{2}) v_{W}^{2}+ P_{W}and we know everything in this equation except P

_{n},P _{n}= (^{1}/_{2}) v_{W}^{2}+ P_{W}- (^{1}/_{2}) v_{n}^{2}P

_{n}= P_{W}+ (^{1}/_{2}) v_{W}^{2}- (^{1}/_{2}) v_{n}^{2}P

_{n}= P_{W}+ (^{1}/_{2}) [ v_{W}^{2}- v_{n}^{2}]P

_{n}= 15 mmHg + (^{1}/_{2}) [ 1000 kg/m^{3}] [ (2.5 m/s)^{2}- (6.2 m/s)^{2}]P

_{n}= 15 mmHg + (^{1}/_{2}) [ 1000 kg/m^{3}] [ (6.25 m^{2}/s^{2}) - (38.44 m^{2}/s^{2}) ]P

_{n}= 15 mmHg + (^{1}/_{2}) [ 1000 kg/m^{3}] [ - 32.19 m^{2}/s^{2}]P

_{n}= 15 mmHg - (^{1}/_{2}) (1000 kg/m^{3})(32.19 m^{2}/s^{2})Before we go on, notice that the pressure is going to be

lessin the narrow region. And that is just as we would/should expect since the velocity isgreaterthere!P _{n}= 15 mmHg - (^{1}/_{2}) (32,190 (kg m/s^{2}) / m^{2}P

_{n}= 15 mmHg - 16,095 N / m^{2}P

_{n}= 15 mmHg - 16,095 PaRemember, you can't add

applesandoranges!What about terms in mmHg and in Pa. Those both measurepressurebut 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 mmHgP

_{n}= 15 mmHg - 121 mmHg

P_{n}= - 106 mmHgWhat does this

negativepressure mean? Both P_{W}and P_{n}aregauge pressures.When we stateP _{w}= 15 mmHgthat means the pressure in the wide region is 15 mmHg

greater thanatmospheric pressure. Now that we have calculatedP _{n}= - 106 mmHgthat means the pressure in the narrow regions is 106 mmHg

less thanatmospheric pressure. That means the U-tube manometers would look something like this --or

(c) 2000, Doug Davis; all rights reserved.