Question

The lower end of a capillary tube of radius 1 mm is dipped vertically into mercury. (a) Find the depression of mercury column in the capillary. (b) If the length dipped inside is half the answer of part (a), find the angle made by the mercury surface at the end of the capillary with the vertical. Surface tension of mercury = 0.465 N m−1 and the contact angle of mercury with glass −135 °.

Answer 1

A::B::C::D

A::B::C::DSolution :

A::B::C::DSolution : r=1mm=10−3m,θ=1.35∘r=1mm=10-3m,θ=1.35∘

A::B::C::DSolution : r=1mm=10−3m,θ=1.35∘r=1mm=10-3m,θ=1.35∘ h=2Tcosθrρgh=2Tcosθrρg

A::B::C::DSolution : r=1mm=10−3m,θ=1.35∘r=1mm=10-3m,θ=1.35∘ h=2Tcosθrρgh=2Tcosθrρg =2×0.465×cos135∘10−3×13600×(9.8)=2×0.465×cos135∘10-3×13600×(9.8)

A::B::C::DSolution : r=1mm=10−3m,θ=1.35∘r=1mm=10-3m,θ=1.35∘ h=2Tcosθrρgh=2Tcosθrρg =2×0.465×cos135∘10−3×13600×(9.8)=2×0.465×cos135∘10-3×13600×(9.8) =0.0053m=5.3mm

Answer 2

Given that,

Radius = 1 mm

Tension of mercury = 0.465 N/m

Contact angle = 135°

(a). We need to calculate the depression of mercury column in the capillary

Using formula of height

$h=\dfrac{2T\cos\theta}{r\rho g}$

Where, r = radius

h = height

T = surface tension

g = acceleration due to gravity

Put the value into the formula

$h=\dfrac{2\times0.465\cos135}{1\times10^{-3}\times13.6\times10^{3}\times9.8}$

$h=-0.00493\ m$

$h=-4.93\ mm$

(b). If the length dipped inside is half

We need to calculate the depression of mercury column in the capillary

Using formula of height

$h'=\dfrac{-h}{2}$

Put the value into the formula

$h'=\dfrac{-4.93}{2}$

$h'=-2.465\ mm$

(c). We need to calculate the angle made by the mercury surface at the end of the capillary with the vertical

Using formula of angle

$\cos\theta=\dfrac{h'\rho g r}{2T}$

Put the value into the formula

$\cos\theta=-\dfrac{2.465\times10^{-3}\times13.6\times10^{3}\times9.8\times1\times10^{-3}}{2\times0.465}$

$\theta=\cos^{-1}(-0.35326)$

$\theta=110^{\circ}$

Hence, (a). The depression of mercury column in the capillary is -4.93 mm

(b).  If the length dipped inside is half then the depression of mercury column in the capillary is -2.465 mm

(c). The angle made by the mercury surface at the end of the capillary with the vertical is 110°