Water flows downwards through a vertical tapering pipe. The cross-sectional area of the top end is 4 cm^2 and that of the bottom end is 2 cm^2 . The vertical distance between the two ends is 15 cm . If the pressure of

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Quantrex Academy · Accepted Answer

Correct answer: A. 400 cm^3 s^ -1

Let the top end be section 1 and the bottom end be section 2.

Given:

A_1 = 4 \text{ cm}^2

A_2 = 2 \text{ cm}^2

h = 15 \text{ cm}

g = 10 \text{ m s}^{-2} = 1000 \text{ cm s}^{-2}

P_1 = P_2

Applying Bernoulli's equation between the top and bottom ends:

P_1 + \rho g h + \dfrac{1}{2} \rho v_1^2 = P_2 + 0 + \dfrac{1}{2} \rho v_2^2

Since

P_1 = P_2

, the pressure terms cancel out:

\rho g h + \dfrac{1}{2} \rho v_1^2 = \dfrac{1}{2} \rho v_2^2

2 g h = v_2^2 - v_1^2

From the equation of continuity, the volume flow rate

Q = A_1 v_1 = A_2 v_2

.

Thus,

v_1 = \dfrac{Q}{4}

and

v_2 = \dfrac{Q}{2}

.

Substitute these into the simplified Bernoulli's equation:

2 \times 1000 \times 15 = \left(\dfrac{Q}{2}\right)^2 - \left(\dfrac{Q}{4}\right)^2

30000 = \dfrac{Q^2}{4} - \dfrac{Q^2}{16}

30000 = \dfrac{4Q^2 - Q^2}{16}

30000 = \dfrac{3Q^2}{16}

Q^2 = \dfrac{30000 \times 16}{3} = 10000 \times 16 = 160000

Q = \sqrt{160000} = 400 \text{ cm}^3 \text{ s}^{-1}

Answer:

400 \text{ cm}^3 \text{ s}^{-1}

Water flows downwards through a vertical tapering pipe. The...

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