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

Hint:

Bernoulli’s theorem is fundamental for ideal fluid systems. Always ensure that terms represent energy per unit volume (pressure, potential energy, and kinetic energy) and use consistent units throughout.

Answer 1

The Correct Option is B

Bernoulli’s theorem describes the conservation of energy in a flowing fluid and can be derived by analyzing the work done by pressure forces, gravity, and changes in velocity along a streamline. It is mathematically expressed as: \[ P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}, \] where: \( P \): Pressure energy per unit volume, \( \rho \): Density of the fluid, \( g \): Gravitational acceleration, \( h \): Height above a reference level, \( v \): Velocity of the fluid. To analyze the given options: Option (a): The use of \( m \) (mass) instead of \( \rho \) (density) is incorrect because Bernoulli’s equation is based on energy per unit volume, not mass. Option (c): Modifying the potential energy term as \( mg \) instead of \( \rho gh \) is dimensionally inconsistent. Option (d): Changing the kinetic energy term from \( \frac{1}{2} \rho v^2 \) to \( \frac{1}{2} m v^2 \) is incorrect for fluid dynamics, as it does not conform to energy per unit volume. Only option (b) correctly preserves the balance of energy terms per unit volume: \[ P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}. \] Final Answer: \[ \boxed{P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}} \]