Question

A drop of liquid of surface tension \(\sigma\) is in between the two smooth parallel glass plates held at a distance \(\mathrm{d}\) apart from each other in zero gravity. The liquid wets the plate so that the drop is a cylinder of diameter \(D\) with its curved surface at right angles to both the plates. Determine the force acting on each of the plates from drops under the following considerations. (1) \(\frac{\sigma \pi D}{2}\) (2) \(\frac{\sigma^{2} \pi \mathrm{D}}{2}\) (3) zero (4) None of these

Short answer

Option (1): \( \frac{\sigma \pi D}{2} \)

Step-by-step solution

Understanding the Problem

A drop of liquid with surface tension \(\sigma\) is placed between two smooth parallel glass plates held at a distance \(d\) apart. The liquid drop forms a cylinder of diameter \(D\) due to zero gravity. Our goal is to determine the force acting on each plate.

Identify the Surface Area of Contact

Since the drop is a cylinder and it wets the plates, the contact surface area with each plate is a circle with diameter \(D\).

Determine the Role of Surface Tension

Surface tension \(\sigma\) acts tangentially along the line of contact between the liquid and the plates. This forms an inward force along the surface circumference of the cylinder.

Calculate the Force Contribution from Surface Tension

The force due to surface tension around the circumference of the cylinder can be calculated by multiplying the surface tension \(\sigma\) by the circumference of the cylinder base. Therefore, the force is \(\sigma \times \pi D\).

Distribute Force to Each Plate

Since the cylinder is symmetrically placed between the two plates, the force calculated will be equally distributed to both plates. Therefore, the force acting on one plate is \(\frac{\sigma \pi D}{2}\).

Verify the Given Choices

By comparing the calculated force \( \frac{\sigma \pi D}{2} \) with the given options, it matches option (1).

Key concepts

Surface Tension

Surface tension is a property of the liquid surface that allows it to resist external forces. It is caused by the cohesive forces between liquid molecules. When a drop of liquid wets a surface, molecules inside the liquid pull each other in different directions, but at the surface, there is a net inward force. This force tries to minimize the surface area.
Surface tension can be understood through phenomena like water droplets forming spheres in zero gravity. This spherical shape minimizes surface area for a given volume due to cohesive forces acting equally in all directions.
In this problem, surface tension plays a crucial role in determining the force exerted by the liquid cylinder on the glass plates. It acts tangentially to the line of contact, which is the circumference of the cylindrical liquid drop.

Cylinder Formation in Zero Gravity

In zero gravity, different rules govern the behavior of liquids. Typically, gravity causes liquids to take the shape of their container. However, the absence of gravity means that surface tension is the dominant force.
When a liquid wets the plates in zero gravity, it spreads out into a shape that minimizes its surface energy. For our problem, this results in a cylindrical shape because the liquid drop is confined between two parallel plates. The liquid forms a cylinder with its curved surfaces perpendicular to the plates, minimizing surface area with a fixed volume.
The diameter of this cylinder remains constant, and its shape is stable due to the absence of gravitational forces that might otherwise deform it.

Force Distribution

Understanding force distribution in this context involves knowing how surface tension acts on the liquid's boundaries. The inward force caused by surface tension tries to contract the surface area of the liquid cylinder.
Given that the liquid forms a cylinder, the force due to surface tension acts along the circumference of the cylinder base, calculated as \( \sigma \times \pi D \). Since surface tension is symmetrical, this force contributes equally to each of the two plates.
Therefore, the force acting on one plate is \( \frac{\sigma \pi D}{2} \). This symmetric force distribution ensures the system is in equilibrium, balanced by the surface tension acting evenly on both sides.