Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli's equation? Explain.
No, it does not matter if one uses gauge pressure instead of absolute pressure in applying Bernoulli's equation.
Bernoulli's equation is given by P + ½ρv² + ρgh = constant along a streamline. When we apply this equation between two points, we equate the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume at both points.
Since gauge pressure (P_g) = P - P_a (where P_a is atmospheric pressure), the absolute pressure can be written as P = P_a + P_g. When we substitute this into Bernoulli's equation for two points along a streamline, the atmospheric pressure term P_a appears on both sides and cancels out.
Therefore, using gauge pressure instead of absolute pressure yields the same result, making it perfectly valid for practical applications.
Explanation
The textbook context defines gauge pressure as P - P_a = P_g, meaning absolute pressure P = P_a + P_g. Bernoulli's equation (P + ½ρv² + ρgh = constant) is applied between two points along a streamline. Since atmospheric pressure P_a is constant and appears on both sides of the equation when comparing two points, it cancels out. This is why pressure-measuring devices like tyre pressure gauges and sphygmomanometers measure gauge pressure, and these values can be directly used in Bernoulli's equation without conversion to absolute pressure.
Solution Steps
Step 1: Write Bernoulli's equation: P + ½ρv² + ρgh = constant
Step 2: Express absolute pressure in terms of gauge pressure: P = P_a + P_g
Step 3: Apply equation between two points - atmospheric pressure P_a cancels on both sides
Step 4: Conclude that gauge pressure can be used instead of absolute pressure