Problem 36 - Master Resource Book in JEE Main Physics

Question

The work done in increasing the size of a rectangular soap film with dimensions \(8 \mathrm{~cm} \times 3.75 \mathrm{~cm}\) to \(10 \mathrm{~cm} \times 6 \mathrm{~cm}\) is \(2 \times 10^{-4} \mathrm{~J}\). The surface tension of the film in \(\mathrm{Nm}^{-1}\) is (a) \(1.65 \times 10^{-2}\) (b) \(3.3 \times 10^{-2}\) (c) \(6.6 \times 10^{-2}\) (d) \(8.25 \times 10^{-2}\)

Step-by-Step Solution

Answer

The surface tension is (b) \(3.3 \times 10^{-2} \text{ Nm}^{-1}\).

Step 1: Calculate Initial Area of the Soap Film

The initial dimensions of the soap film are given as \(8 \text{ cm} \times 3.75 \text{ cm}\). Calculate the area by multiplying the length and width: \[ A_1 = 8 \times 3.75 = 30 \text{ cm}^2 \]. Since the film has two surfaces (front and back), the effective initial area is \(2 \times 30 = 60 \text{ cm}^2\).

Step 2: Calculate Final Area of the Soap Film

The final dimensions of the soap film are given as \(10 \text{ cm} \times 6 \text{ cm}\). Calculate the area by multiplying the length and width: \[ A_2 = 10 \times 6 = 60 \text{ cm}^2 \]. Taking both surfaces into account, the effective final area is \(2 \times 60 = 120 \text{ cm}^2\).

Step 3: Calculate Change in Area

The change in area of the soap film can be calculated by subtracting the initial area from the final area: \[ \Delta A = A_2 - A_1 = 120 \text{ cm}^2 - 60 \text{ cm}^2 = 60 \text{ cm}^2 \].

Step 4: Convert Area Change to Square Meters

Since surface tension is typically given in \(\text{Nm}^{-1}\), convert the area from square centimeters to square meters: \[ 60 \text{ cm}^2 = 60 \times 10^{-4} \text{ m}^2 = 0.006 \text{ m}^2 \].

Step 5: Use Work-Energy Principle to Find Surface Tension

The work done is related to surface tension and change in area by the equation \( W = \gamma \times \Delta A \). Rearrange to solve for \(\gamma\): \[ \gamma = \frac{W}{\Delta A} = \frac{2 \times 10^{-4} \text{ J}}{0.006 \text{ m}^2} = 3.33 \times 10^{-2} \text{ Nm}^{-1} \].

Step 6: Select Closest Answer Choice

From the given options, the closest value to the calculated surface tension \(3.33 \times 10^{-2} \text{ Nm}^{-1}\) is option (b) \(3.3 \times 10^{-2}\).

Key Concepts

work-energy principle

The work-energy principle is a fundamental concept in physics that relates the work done on an object to the change in its energy. This principle states that the work done by external forces on an object is equal to the change in kinetic energy of that object. However, in the context of surface tension and soap films, it helps us understand how the work done is used to change the area of the film.

When you stretch a soap film, you're essentially doing work to overcome the forces of surface tension, allowing the film's area to increase. Surface tension acts like a "skin" on the surface of the film, and as you increase the film's area, you're stretching this "skin." Using the work-energy principle, the relationship between the work done (\( W \)) and surface tension (\( \gamma \)) can be expressed as:

• \( W = \gamma \times \Delta A \) - where \( \Delta A \) is the change in area.

This formula allows us to calculate the surface tension given the amount of work done and the change in the area of the soap film.

soap film

A soap film is a thin layer of liquid surrounded by air. It forms the colorful bubbles that we often see when playing with soap. Scientifically, a soap film behaves as if it is trying to minimize its surface area due to surface tension, a physical phenomenon where the surface of a liquid acts like an elastic skin.

Soap films can demonstrate various optical effects, such as iridescence, due to the interference of light reflecting from the film's multiple surfaces. As it tries to reduce its energy, a soap film naturally seeks to minimize its surface area, assuming shapes like spheres (in bubbles) when unconfined.

Importantly, when working with soap films, remember that the film has two surfaces – the one facing you and the one facing away from you. Thus, when calculating properties like area, you need to take both surfaces into account, effectively doubling the calculated single-layer area.

change in area

The concept of change in area is straightforward in the context of the problem, yet it is crucial for understanding how surface tension works. Initially, you start with a piece of soap film of certain dimensions. By applying work, the dimensions of the film change, leading to a new total area.

To calculate the change in area (\( \Delta A \)):

• First determine the initial area (\( A_1 \)) of the film based on its starting dimensions and multiply by two, accounting for both surfaces of the film.
• Next, calculate the final area (\( A_2 \)) under the new dimensions, also remembering to double it for both sides.
• The change in area is then \( \Delta A = A_2 - A_1 \).

This change in area influences how much work is needed to expand the film. In the given problem, this change is essential for calculating surface tension using the work-energy formula, highlighting the relationship between energy (or work), surface tension, and the geometric changes of the film.