Water and coconut oil are poured into the two limbs of a U-tube as shown in the figure. The water–oil interface is assumed to lie vertically along the centre of the tube (ρ_w = density of water, ρₒ = density of coconut oil). Consider the following statements about this situation. (A) The pressure at point P = the pressure at point Q (B) h₁ρ_w = h₂ρₒ (C) h₃ρ_w = h₄ρₒ Of the above statements,

(5) all of (A), (B), (C) are true.

2020 A/L Physics Past Paper · Question 20 (English Medium)

Water and coconut oil in a U-tube — which statements are true?

Water and coconut oil are poured into the two limbs of a U-tube as shown in the figure. The water–oil interface is assumed to lie vertically along the centre of the tube ( $\rho_w$ = density of water, $\rho_o$ = density of coconut oil). Consider the following statements about this situation.

(A) The pressure at point $P$ = the pressure at point $Q$

(B) $h_1\rho_w = h_2\rho_o$

Of the above statements,

(1) only (A) is true.
(2) only (B) is true.
(3) only (A) and (B) are true.
(4) only (B) and (C) are true.
(5) all of (A), (B), (C) are true.

Answer and explanation

Answer: (3) only (A) and (B) are true.

What the question asks

We must find which of the three statements are true.

Basis: in a liquid at rest, the pressure at depth $h$ is $p_0 + h\rho g$ ; within one liquid, the pressure is equal at the same level.

Relevant theory

Consider a point A at a depth h below the surface of a liquid exposed to the atmosphere.

$p = p_0 + h\rho g$

The pressure at the point B which is at the same level, is also the same.

Core idea: the pressure of a liquid column is $h\rho g$ .

At both open surfaces the pressure is $p_0$ .

The rule " $same\,level \Rightarrow same\,pressure$ " only holds within one continuous liquid at rest.

Reading the options

The options are combinations of true statements: {A}, {B}, {A, B}, {B, C}, {A, B, C}.

We judge all three, then match.

Reasoning and calculation

$The\,pressure\,at\,point\,P = the\,pressure\,at\,point\,Q$

$P$ and $Q$ are the two points touching each other on either side of the interface.

Since the interface is at rest, the pressures on both sides of it are equal.

Otherwise the interface would move.

$h_1\rho_w = h_2\rho_o$

Pressure of the water column $h_1$ :

$p_P = p_0 + h_1\rho_w g$

Pressure of the oil column $h_2$ :

$p_Q = p_0 + h_2\rho_o g$

From (A) these are equal:

$h_3\rho_w = h_4\rho_o$

$h_3$ , $h_4$ are measured up to one shared dashed level — at height $z$ above $P$ .

$h_3 = h_1 - z$ , $h_4 = h_2 - z$

So:

$h_3\rho_w - h_4\rho_o = (h_1\rho_w - h_2\rho_o) - z(\rho_w - \rho_o) = -z(\rho_w - \rho_o) \neq 0$

since $\rho_w > \rho_o$ .

That level runs through two different liquids, so "same level, same pressure" does not apply.

Why the false statement (C) is tempting: below the level, there is $z$ height of water on the water side.

On the oil side, there is $z$ height of oil.

These give different weights, so the pressures at that level differ — lower on the water side.

With numbers ( $\rho_w = 1000$ , $\rho_o = 920$ , $h_1 = 0.20$ m, $z = 0.10$ m, $g = 10$ ):

$h_3\rho_w g = 1000$ Pa

$h_4\rho_o g \approx 1080$ Pa

Why the other options fail

(B) is also true.

(A) is also true.

Conclusion

(A), (B) are true; (C) is not — the answer is (3).

Before using "same level, same pressure", check that both points are connected through one continuous liquid. In a U-tube with two liquids, the interface level is the only safe level to compare across.

Exam tip

தமிழ் · සිංහල