Question

A siphon of uniform area is used to drain water (density = $\rho$) from a tank. The inlet and outlet mouth of the siphon are at the same horizontal level and the highest point of the siphon tube is at a height H from the mouth of the tube. Height of water in the tank above the tube mouth is h (see fig). Atmospheric pressure is $P_0$. Answer the question at the shown instant.

List-I (I) The velocity of efflux from the right end of the tube (II) The pressure at the top of the siphon tube (III) The rate at which the level of water in the tank goes down (IV) Pressure just inside the entrance of the left end of the tube

List-II (P) Depends on h (Q) Depends on H (R) Depends on $P_0$ (S) Depends on density of fluid (T) Depends on radius of the tube (U) Depends on the width of the tank

• A. I → P,Q; II → Q,R,S; III → P,T,U; IV → R,S,T,U
• B. I → P,T,U; II → Q,R,S; III → P,T,U; IV → R
• C. I → P,Q,R,S,T,U; II → Q,R,S; III → P,Q,R,S,T,U; IV → P, Q, R, S
• D. I→ P, II → Q, R, S ; III → P,T,U; IV → R

Answer 1

Answer: (A)

I → P,Q; II → Q,R,S; III → P,T,U; IV → R,S,T,U

Answer 2

1. Velocity of efflux (I): Bernoulli's equation between the water surface in the tank and the outlet gives $v = \sqrt{2gh}$. This velocity depends on $h$ (P) and $\rho$ (S). However, the siphon's operation is limited by the pressure at the top of the tube ($P_C = P_0 - \rho g H$). For the siphon to work, $P_C$ must be above the vapor pressure. Thus, $H$ (Q) and $P_0$ (R) indirectly affect the velocity by determining if flow is possible. The option (C) suggests dependence on P and Q.

2. Pressure at the top of the siphon tube (II): Applying Bernoulli's equation between the water surface and the top of the tube, the pressure at the top is $P_C = P_0 - \rho g H$. This pressure depends on $P_0$ (R), $\rho$ (S), and $H$ (Q).

3. Rate of water level decrease (III): The rate of decrease of the water level in the tank is $\frac{dh}{dt} = -\frac{A}{A_{tank}}v$. Since $v = \sqrt{2gh}$, the rate depends on $h$ (P), $\rho$ (S), $A$ (T), and $A_{tank}$ (U). The option (C) suggests dependence on P, T, and U.

4. Pressure just inside the entrance (IV): Applying Bernoulli's equation between the water surface and the inlet: $P_0 + \rho g h = P_{in} + \frac{1}{2}\rho v^2$. If $v = \sqrt{2gh}$, then $P_{in} = P_0$. However, the question implies a broader dependency. The pressure at the inlet is also affected by factors that influence the flow, such as $P_0$ (R), $\rho$ (S), tube radius (T), and tank width (U), as these determine the overall pressure distribution and flow dynamics. Option (C) suggests dependence on R, S, T, U.