Correct Bernoulli's equation is (symbols have their usual meaning):
- \(P + mgh + \frac{1}{2} mv^2 = \text{constant}\)
- \(P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\)
- \(P + \rho gh + \rho v^2 = \text{constant}\)
- \(P + \frac{1}{2} \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\)
The Correct Option is B
Approach Solution - 1
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:
- \(P + mgh + \frac{1}{2} mv^2 = \text{constant}\):
- 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 \(\rho gh\).
- This option uses \(mgh\) and P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}:
Therefore, the correct form of Bernoulli's equation is represented by option 2: \(P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\).
Approach Solution -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: \[ P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}. \]
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