A rectangular narrow U-tube has equal arm lengths and base length, each equal to l = 250 mm. The vertical arms are filled with mercury up to 2 l and then one end is sealed (figure 3.114). By heating

Question

A rectangular narrow U-tube has equal arm lengths and base length, each equal to $l = 250$ mm. The vertical arms are filled with mercury up to $\frac{l}{2}$ and then one end is sealed (figure 3.114). By heating the enclosed gas all the mercury is expelled. Determine the work done by the gas if the atmospheric pressure is $p_{0} = 1 0^{5}$ Pa, the density of mercury is $ρ = 13.6 \times 1 0^{3}$ kg $m^{- 3}$ and the cross-sectional area is $S = 1$ $c m^{2}$ .

TestHub · Accepted Answer

Mercury is pushed out in three stages. (1) Sealed-arm level falls x from 0 to 2 l ​ while the open arm rises x : p = p 0 ​ + 2 ρ gx , W 1 ​ = S ( 2 p 0 ​ l ​ + 4 ρ g l 2 ​ ) . (2) Mercury in the base is pushed with the open arm full: p = p 0 ​ + ρ g l over a volume Sl , W 2 ​ = Sl ( p 0 ​ + ρ g l ) . (3) The column in the open arm is pushed out, p = p 0 ​ + ρ g ( l − y ) : W 3 ​ = S ( p 0 ​ l + 2 ρ g l 2 ​ ) . Total W = S ( 2.5 p 0 ​ l + 1.75 ρ g l 2 ) = 1 0 − 4 ( 62500 + 14580 ) = 7.71 J.

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