A glass capillary tube has the shape of a truncated cone (apex angle alpha), so its two ends have circular cross-sections of different radii. When it is dipped vertically into water, the water rises to a height h, at which level the radius of the tube's cross-section is b. If the surface t

Question

A glass capillary tube has the shape of a truncated cone (apex angle alpha), so its two ends have circular cross-sections of different radii. When it is dipped vertically into water, the water rises to a height h, at which level the radius of the tube's cross-section is b. If the surface tension of water is S, its density is rho, the contact angle with glass is theta, and g is the acceleration due to gravity, then the value of h is:

Accepted Answer

(2S/(b*rho*g)) cos(theta - alpha/2). For a vertical tube, equilibrium gives h = 2S cos(theta)/(rho g r). In a conical tube the wall makes a half-angle alpha/2 with the vertical, which tilts the surface-tension vector. The vertical component of the surface tension at the meniscus then carries cos(theta - alpha/2) instead of cos(theta). With the local radius b at the meniscus, h = 2S cos(theta - alpha/2)/(b rho g).

Solution

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