Question

A drop of liquid of surface tension \(\sigma\) is in between the two smooth parallel glass plates held at a distance 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) \(\frac{\sigma^{2} \pi D}{2}\) (3) zero (4) None of these

Short answer

(1) \(\frac{\sigma \pi D}{2}\).

Step-by-step solution

- Understand the problem

A drop of liquid with surface tension \(\sigma\) forms a cylinder between two parallel glass plates in zero gravity. The aim is to find the force acting on each plate.

- Identify cylinder properties

Given the cylinder has a diameter \(\text{D}\) and is at right angles to the plates. The shape and dimensions will guide the surface area calculations.

- Calculate contact length

Calculate the total contact length of the liquid with the glass plates which equals the circumference of the cylinder's base: \[ \ell = \pi D \]

- Determine force per unit length

Surface tension \(\sigma\) acts along the contact line length, providing the force per unit length. The total force is then: \[ F = \sigma \ell = \sigma \pi D \]

- Find force acting on each plate

Since the force is distributed equally on each plate, we divide by 2: \[ F_{\text{plate}} = \frac{\sigma \pi D}{2} \]

- Verify options

Compare the derived expression with the provided options. This verifies option (1) \(\frac{\sigma \pi D}{2}\) as the correct answer.

Key concepts

force calculation

In physics, calculating force involves understanding the interaction between objects. Here, we need to determine the force exerted by the liquid drop on each glass plate. The liquid drop, due to surface tension \(\text{\(\sigma\)}\), creates a force along the boundary where it contacts the plates.
We begin by identifying the contact length between the liquid and the plates. Since the drop forms a cylindrical shape, the contact length equals the circumference of the cylinder's base, given by:
\[ \ell = \pi D \]

The surface tension acts along this contact line, producing a force per unit length. Hence, the total force is calculated by:
\[ F = \sigma \ell = \sigma \pi D \]

Finally, as the force is evenly distributed between the two plates, the force on each plate is:
\[ F_{\text{plate}} = \frac{\sigma \pi D}{2} \]
This result corresponds with option (1) from the exercise.

cylindrical shapes in physics

Cylindrical shapes are common in physics problems, especially when dealing with liquids and forces. A cylinder has a simple geometric definition: a solid with two parallel circular bases connected by a curved surface.
In our problem, the liquid forms a cylinder with diameter \(\text{\(\text{D}\)}\) between two glass plates. Key properties of cylinders include:

• Surface Area: \[ A = 2 \pi r h + 2 \pi r^{2} \] where \(r\) is the radius and \(h\) is the height.
• Volume: \[ V = \pi r^{2} h \]
• Circumference: \[ c = 2 \pi r \]

For our exercise, the circumference \(c\) is crucial. The force due to surface tension depends on the total contact length, i.e., the circumference of the cylinder's base. This interaction helps us calculate the force exerted by the liquid on the plates.

liquid behavior in zero gravity

Understanding liquid behavior in zero gravity is fascinating. In the absence of gravity, liquids can form unusual shapes due to surface tension. Surface tension is the cohesive force at the liquid's surface that pulls it into a compact shape.
In zero gravity:

• Shape Formation: Liquids tend to form spheres or other minimal surface shapes.
• Uniform Distribution: Gravity's absence means liquids don't settle but spread uniformly due to surface tension.
• Complex Interactions: With surfaces, liquids form distinct shapes, such as a cylinder between plates in our problem.

This behavior enables the liquid to maintain a cylindrical shape between the plates, facilitating our calculation of forces. Understanding these principles helps us explain why the liquid forms a cylinder and how surface tension acts, enabling us to predict the resulting force on the plates accurately.