Question

When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop. We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper. If the radius of the opening of the dropper is \(r\), the vertical force due to the surface tension on the drop of radius \(R\) (assuming \(r \ll R\) ) is A) \(2 \pi r T\) B) \(2 \pi R T\) C) \(\frac{2 \pi r^{2} T}{R}\) D) \(\frac{2 \pi R^{2} T}{r}\)

Short answer

The vertical force due to surface tension is A) 2πrT.

Step-by-step solution

Identify the Vertical Force Due to Surface Tension

The vertical force due to the surface tension on the drop is the result of the surface tension acting along the circumference of the dropper's opening. Since we are dealing with the force on the drop at the moment of detachment, we focus on the part of the surface tension that has a vertical component, which acts along the circle of radius r, the radius of the dropper's opening.

Use the Simplifying Assumption

We assume that the radius of the dropper's opening, r, is much smaller than the radius of the spherical drop, R. This allows us to simplify the problem because the force due to the surface tension will primarily act on the circumference of the opening.

Calculate the Magnitude of the Vertical Force

Since the hemisphere of the drop provides the vertical component of the surface tension, and its edge acts along a circle with radius r, the vertical force due to surface tension only concerns this edge. The force is the surface tension, T, multiplied by the circumference of the opening, which is 2πr. Therefore, the vertical force due to surface tension is 2πrT.

Choose the Correct Option Based on the Calculation

According to the calculation and the physics of the problem, the vertical force due to surface tension is given by the product of the surface tension and the circumference of the dropper's opening (2πr). Therefore, the correct option is A) 2πrT.

Key concepts

Surface Tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force, due to the cohesive nature of its molecules. It arises from the fact that the molecules at the surface of the liquid have no similar neighboring atoms above, and hence exhibit stronger bonds with their neighbors on and below the surface.

This cohesive force gives rise to a 'skin' or a stretched membrane-like effect, which tries to reduce the surface area to the smallest possible size. A common example of surface tension is the ability of some insects, like water striders, to walk on the surface of water without sinking. The phenomenon can also be observed when carefully placing a needle flat on the surface of water – it floats despite being made of metal, courtesy of the surface tension.

Surface tension is typically measured in Newtons per meter (N/m) and can be affected by factors like temperature and the presence of other substances that reduce the cohesive forces between molecules.

Force Due to Surface Tension

The force due to surface tension is a vital concept when considering the behavior of liquids at boundaries and interfaces. For a liquid drop at the point of detachment, such as in the dropper example, the force exerted by surface tension acts along the line where the liquid is in contact with another surface – in this case, the edge of the dropper's opening.

Mathematically, the force due to surface tension (F) can be calculated as the product of surface tension (T) and the length (L) over which it acts, given by the formula:
\[ F = T \times L \] where in the case of the dropper, L is equivalent to the circumference of the opening (2πr). It's this tangential force that enables the spherical drop to adhere to the dropper's edge before it becomes too heavy and gravity overcomes the surface tension, causing the drop to fall.

Spherical Drops Formation

Spherical drops formation is a phenomenon that occurs due to the minimization of surface energy. When a drop of liquid forms, the molecules at the surface are subject to unequal forces, resulting in a net inward force that pulls the molecules closer together. This causes the liquid to naturally form into a shape with the smallest possible surface area relative to its volume – a sphere.

In the context of medicine being administered with a dropper, as the bulb is pressed, and the liquid emerges, the cohesive forces within the liquid tend to form a drop that is close to spherical. This is the most energy-efficient shape for the drop because it maintains the least amount of surface area. Eventually, the gravitational force exceeds the surface tension force at the point of detachment, and the drop falls from the dropper.

Understanding the precise moment that this occurs is predicated on knowledge of the weight of the drop and the vertical force due to surface tension, which can be mathematically correlated to estimate the maximum size of the drop before it detaches and falls into the eye.