Physics quiz question: ideal fluid flow through horizontal pipe

Physics 8A (currently Physics 208A) Quiz 10, spring semester 2007
Cuesta College, San Luis Obispo, CA

Cf. Young and Freeman, University Physics, 11/e, Exercise 14.40

Water enters a horizontal section of pipe at a rate of 1.80 m³/s at point [1], where the radius is 0.300 m, and the pressure is 2.50×10⁵ Pa. Assume ideal fluid flow.

At a point [2] further along the pipe, the pressure is less than at point [1]. The radius r₂ of the pipe at point [2] is __________ compared to the radius r₁ point [1].
(A) smaller.
(B) the same.
(C) larger.
(D) (Not enough information given.)

Correct answer (highlight to unhide): (A)

Writing out Bernoulli's equation:

0 = ∆P + ρ·g·∆y + (1/2)·ρ·∆(v²),

the decrease in pressure as the water flows from [1]→[2] means that the first term ∆P is negative, and the second term ρ·g·∆y is zero because there is no change in elevation y. For the equality to hold, the third term (1/2)·ρ·∆(v²) must then be positive, such that:

0 < (1/2)·ρ·∆(v²),

0 < ∆(v²),

0 < v₂² - v₁²,

v₁² < v₂²,

v₁ < v₂.

From the continuity equation (conservation of fluid flow):

A₁·v₁ = A₂·v₂,

since v₁ < v₂, then A₁ > A₂, and thus r₁ > r₂ (the pipe is smaller at point [2] than at point [1]).

Student responses:
(A) : 9 students
(B) : 0 students
(C) : 5 students
(D) : 0 students