Tension in the String when Soap Film is Pierced

When the massless inextensible string is placed on a horizontal film of soap solution, the tension in the string is equal to the surface tension of the soap film, which we'll denote as T.

Conversion of Soap Film into a Circular Loop

When the film is pierced inside the loop, the soap film converts into a circular loop with a diameter of d. This means that the soap film, which initially had a larger area, now forms a smaller circular loop.

Understanding Surface Tension

Surface tension is the property of a liquid that allows it to resist external forces. It is caused by the cohesive forces between the liquid molecules. In the case of a soap film, the surface tension is responsible for the film's ability to form a thin, stretched layer.

Effect of Piercing the Soap Film

When the soap film is pierced inside the loop, the cohesive forces that were holding the film together are broken. As a result, the film retracts and forms a smaller circular loop.

Change in Surface Area

The change in surface area of the soap film can be calculated by comparing the initial area of the film with the area of the circular loop formed after piercing. The initial area of the film can be represented as A, while the area of the circular loop is π(d/2)^2.

The change in surface area can be given as ΔA = A - π(d/2)^2.

Relation between Change in Surface Area and Tension

According to the Laplace's Law, the excess pressure inside a bubble or curved surface is inversely proportional to its radius of curvature. In the case of the soap film, when it forms a smaller circular loop, the radius of curvature decreases. This leads to an increase in excess pressure inside the film.

The increase in excess pressure is balanced by the tension in the string. Therefore, we can say that the tension in the string is directly proportional to the change in surface area of the soap film.

Tension in the String

Since the tension in the string is directly proportional to the change in surface area, we can write:

Tension ∝ ΔA

Using the relation for the change in surface area, we can substitute it into the equation:

Tension ∝ A - π(d/2)^2

However, we were given that the initial tension in the string is T. Therefore, we can write:

T = A - π(d/2)^2

Rearranging the equation, we get:

Tension = T + π(d/2)^2

Hence, the tension produced in the string when the soap film is pierced inside the loop is T + π(d/2)^2, which can be simplified to Td.