Correct Answer to Question 7

To find the pressure P in the constricted part of the tube, we will use Bernoulli's equation which says that the quantity

P + gh + (1/2)V2 remains constant.

(P = pressure, = density, h = vertical height, V = local velocity). To find PB, we must first find the velocity VB in the constriction. Since the fluid is inconpressible, the volume of a sample of the fluid must remain constant. Consider the sample at A between the broken lines. When the sample reaches B, its length must increase. The diameter at B is one-half that at A. Thus, the sample length at B is four times that at A. The sample lengths are proportional to velocities. So the velocity VB in the constriction is four times VA:

VB = 4 * VA = 4 * 0.5 = 2 m/sec.

We are now ready to use Bernoulli's equation. Since the tube is horizontal, the height hA at A (measured from any reference level) must be the same as hB at B. So the equation

PA + ghA + (1/2)VA2 = PB + ghB + (1/2)VB2

reduces to

PA + (1/2)VA2 = PB + (1/2)VB2

or

PB = PA + (1/2)(VA2 - VB2).

The density is the same at both places since the fluid is incompressible. We substitute PA = 2*104 Pa, = 1*103 Kg/m3, VA = 0.5 m/sec and VB = 2 m/sec with the result of PB = 1.81*104 Pa.

Formula Sheet
Question 7 ... Question 8 ... Up to Unit 5 Table of Contents