Which statement concerning Bernoulli's principle is true?

Question

BlackTom AI · Accepted Answer

Bernoulli’s equation for steady, incompressible, non‑viscous flow along a streamline is the central law: P + 1/2 ρv^2 + ρgh = constant. This expresses conservation of mechanical energy per unit volume (pressure energy + kinetic energy + potential energy).

1) Relevant knowledge points and concepts
- P is fluid pressure, ρ is density, v is flow speed, g is gravity, h is elevation.
- Assumptions: steady flow, incompressible fluid, negligible viscosity, and evaluation along a single streamline.
- Consequence: changes in kinetic energy (½ρv^2) are balanced by opposite changes in pressure (P) or potential energy (ρgh).

2) Step‑by‑step reasoning
Start from Bernoulli’s equation:
P + 1/2 ρv^2 + ρgh = C (constant along a streamline).
Compare two points 1 and 2 on the same streamline:
P1 + 1/2 ρv1^2 + ρgh1 = P2 + 1/2 ρv2^2 + ρgh2.

3) Intermediate algebraic steps (horizontal flow example)
If h1 = h2 (horizontal): P1 + 1/2 ρv1^2 = P2 + 1/2 ρv2^2.
Rearrange: P2 − P1 = 1/2 ρ(v1^2 − v2^2) = −1/2 ρ(v2^2 − v1^2).
Thus if v2 > v1, then (v2^2 − v1^2) > 0, so P2 − P1 < 0 → pressure decreases where velocity increases.

4) Why the correct answer is correct
An increase in flow speed raises the kinetic energy term 1/2ρv^2; because the sum must stay constant, the pressure term P must decrease (unless compensated by a change in elevation). Therefore "The pressure of a fluid decreases at points where the velocity of the fluid increases" directly follows from Bernoulli’s relation.

5) Why the other options are incorrect
- "It applies only to gases and vaporized liquids." Incorrect: Bernoulli’s principle commonly applies to liquids and gases when the assumptions hold; it is not limited to gases.
- "The pressure of a fluid increases at points where the velocity of the fluid increases." Incorrect because sign is reversed; increased velocity corresponds to decreased pressure under Bernoulli’s equation.

6) Conclusion
Hence the correct statement is: "The pressure of a fluid decreases at points where the velocity of the fluid increases." This matches the provided answer.

Which statement concerning Bernoulli's principle is true?单项选择题

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