A rectangular narrow U-tube has equal arm lengths and base length, each equal to ℓ . The vertical arms are filled with mercury up to ℓ /2 and then one end is sealed. By heating the enclosed gas all th

Question

A rectangular narrow U-tube has equal arm lengths and base length, each equal to ℓ . The vertical arms are filled with mercury up to ℓ /2 and then one end is sealed. By heating the enclosed gas all the mercury is expelled. If atmospheric pressure is P 0 ​ , the density of mercury is ρ and cross-sectional area is S, then [Neglect thermal expansion of glass and mercury]

TestHub · Accepted Answer

Total mercury length pushed out against the atmosphere: $ℓ + ℓ + \frac{ℓ}{2} = \frac{5 ℓ}{2}$

$W_{a t m} = P_{0} \cdot S \cdot \frac{5 ℓ}{2} = \frac{5 ℓ}{2} P_{0} S$ - so (A) correct, (C) incorrect

Initial PE (two columns of height $ℓ /2$ , CM at $ℓ /4$ ): $U_{i} = 2 (ρS \frac{ℓ}{2}) g \frac{ℓ}{4} = \frac{ρS g ℓ ^{2}}{4}$

Final PE (all mercury lifted out of the tube): $U_{f} = 2 ρS g ℓ^{2}$

$W_{g r a v} = U_{f} - U_{i} = \frac{7}{4} Sρ g ℓ^{2}$ - so (B) correct, (D) incorrect

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