Question

Correct Bernoulli's equation is (symbols have their usual meaning):

• A. \(P + mgh + \frac{1}{2} mv^2 = \text{constant}\)
• B. \(P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\)
• C. \(P + \rho gh + \rho v^2 = \text{constant}\)
• D. \(P + \frac{1}{2} \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\)

Answer 1

The Correct Option is B

To solve this problem, we need to identify the correct form of Bernoulli's equation. Bernoulli's equation is a principle in fluid dynamics that describes the conservation of energy in a flowing fluid. It is applicable to incompressible, non-viscous fluids. The equation relates the pressure energy, kinetic energy per unit volume, and potential energy per unit volume of a fluid flowing along a streamline.

The general form of Bernoulli's equation is given as:

\(P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}\)

• \(P\) is the pressure energy per unit volume.
• \(\frac{1}{2} \rho v^2\) is the kinetic energy per unit volume, where \(v\) is the velocity of the fluid and \(\rho\) is the density of the fluid.
• \(\rho gh\) is the potential energy per unit volume, where \(g\) is the acceleration due to gravity and \(h\) is the height above a reference point.

Now let's analyze the given options:

1. \(P + mgh + \frac{1}{2} mv^2 = \text{constant}\):
   • This option uses \(mgh\) and P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}:
     • This option correctly reflects the known form of Bernoulli's equation, matching our derived equation above.
   • \(P + \rho gh + \rho v^2 = \text{constant}\):
     • This option incorrectly represents the kinetic energy term; kinetic energy should have the form \(\frac{1}{2} \rho v^2\), rather than \(\rho v^2\).
   • \(P + \frac{1}{2} \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\):
     • This option incorrectly modifies the potential energy term. Potential energy per unit volume should be \(\rho gh\).

Therefore, the correct form of Bernoulli's equation is represented by option 2: \(P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\).

Answer 2

Bernoulli’s equation for fluid flow is:

\[ P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}. \]

Here:
P is the pressure,
\(\rho\) is the density of the fluid,
g is the acceleration due to gravity,
h is the height,
v is the velocity.

Final Answer: \[ P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}. \]