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A liquid is poured into a spherical container of radius R up to a height h (measured from the bottom of the sphere). At this filling level the liquid surface meets the wall such that the surface is exactly horizontal at the contact edge. The contact angle of the liquid with the container wall is:

1. 0
2. cos⁻¹((R - h)/R)
3. cos⁻¹((h - R)/R)
4. sin⁻¹((R - h)/R)

Correct answer: cos⁻¹((h - R)/R)

Solution

Take the centre of the sphere O. The contact point P is at height h from the bottom, i.e. at height (h - R) relative to the centre. The radius OP makes angle phi with the upward vertical where cos(phi) = (h - R)/R. The wall's tangent at P is perpendicular to OP. The flat horizontal liquid surface and this tangent meet so that the contact angle equals phi = cos⁻¹((h - R)/R).

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