Water flows in a streamline motion through a horizontal pipe of circular cross-section as shown in the figure. The pressure difference of water between P and Q is 15 Nm⁻². The area of cross-section at P and Q are 40 cm² and 20 cm², respectively. The rate of flow of water through the pipe, in cm³ s⁻¹, is: [Take density of water = 1000 kg m⁻³ ] (a) 400 (b) 100 (c) 200 (d) 300

Answer: (a). Continuity: vQ = 40/20vP = 2vP Bernoulli: PP - PQ = 1/2ρ(vQ² - vP²) = 3/2ρ vP² 15 = 1500 vP² ⇒ vP = 0.1 m/s = 10 cm/s Q = AP vP = 40×10 = 400 cm³/s So the answer is (a) 400.

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Bernoulli's Equation

Water flows in a streamline motion through a horizontal pipe of circular cross-section as shown in the figure. The pressure difference of water between $P$ and $Q$ is $15\text{ Nm}^{-2}$ . The area of cross-section at $P$ and $Q$ are $40\text{ cm}^2$ and $20\text{ cm}^2$ , respectively. The rate of flow of water through the pipe, in $\text{cm}^3\text{ s}^{-1}$ , is:

[Take density of water $= 1000\text{ kg m}^{-3}$ ]

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Answer: (a)

Continuity: $v_Q = \dfrac{40}{20}v_P = 2v_P$
Bernoulli: $P_P - P_Q = \frac{1}{2}\rho(v_Q^2 - v_P^2) = \frac{3}{2}\rho v_P^2$
$15 = 1500\,v_P^2 \Rightarrow v_P = 0.1\text{ m/s} = 10\text{ cm/s}$
$Q = A_P v_P = 40\times10 = 400\text{ cm}^3/\text{s}$
So the answer is (a) 400.

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