Question

A small spherical droplet of density $d$ is floating exactly half immersed in a liquid of density $\rho$ and surface tension $T$. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet):

• A. $r=\sqrt{\frac{2T}{3\left(d+\rho\right)g}}$
• B. $r=\sqrt{\frac{T}{\left(d+\rho\right)g}}$
• C. $r=\sqrt{\frac{T}{\left(d-\rho\right)g}}$
• D. $r=\sqrt{\frac{3T}{\left(2d-\rho\right)g}}$

Answer 1

The Correct Option is D

To solve this problem, we must analyze the forces acting on the spherical droplet when it is in equilibrium, floating exactly half-immersed in the liquid.

The key forces to consider are:

1. Buoyant Force (\(F_b\)): This is the upward force exerted by the liquid on the immersed part of the droplet. The volume of the immersed portion is half the volume of the sphere because the droplet is half-immersed.
2. Weight of the Droplet (\(F_w\)): This is the downward gravitational force acting on the entire droplet.
3. Force due to Surface Tension (\(F_t\)): This force acts upwards along the perimeter of contact. The expression for this force is \(F_t = 2\pi r T\), where \(r\) is the radius of the droplet.

Step-by-step Calculation:

1. The weight of the droplet is given by: \(F_w = \frac{4}{3}\pi r^3 d g\) where \(d\) is the density of the droplet and \(g\) is the acceleration due to gravity.
2. The buoyant force (since the droplet is half submerged) is given by: \(F_b = \frac{1}{2} \times \frac{4}{3}\pi r^3 \rho g = \frac{2}{3}\pi r^3 \rho g\) where \(\rho\) is the density of the liquid.
3. At equilibrium, the following condition holds: \(F_w = F_b + F_t\) which gives us: \(\frac{4}{3}\pi r^3 d g = \frac{2}{3}\pi r^3 \rho g + 2\pi r T\)
4. Rearrange the terms: \(\frac{4}{3} d g = \frac{2}{3} \rho g + \frac{2 T}{r\pi r^2 g}\)
5. Simplify and solve for \(r\): \(2d - \rho = \frac{3T}{r^2 g}\) Therefore, \(r = \sqrt{\frac{3T}{(2d - \rho)g}}\)

This matches the provided correct answer option: \(r = \sqrt{\frac{3T}{(2d - \rho)g}}\).

Therefore, the correct answer is the expression for the radius given by:

\(r=\sqrt{\frac{3T}{\left(2d-\rho\right)g}}\)