Problem 66 - Master Resource Book in JEE Main Physics

Question

The aurface tension of soap solution is \(0.03 \mathrm{Nm}^{-1}\). The work done in blowing to form a soap bubble of surface area \(40 \mathrm{~cm}^{2}\), in joule is (a) \(1.2 \times 10^{-4}\) (b) \(2.4 \times 10^{-4}\) (c) \(12 \times 10^{-4}\) (d) \(24 \times 10^{-4}\)

Step-by-Step Solution

Answer

(b) \(2.4 \times 10^{-4}\) J.

Step 1: Understand the Concept

The work done in forming a soap bubble is equal to the surface tension times the change in surface area. Since a soap bubble has two surfaces (inner and outer), the change in surface area is doubled.

Step 2: Convert Units

Convert the surface area from square centimeters to square meters. Given area: 40 cm². Conversion: \(1 \text{ cm}^2 = 1 \times 10^{-4} \text{ m}^2\). Therefore, \(40 \text{ cm}^2 = 40 \times 10^{-4} \text{ m}^2\).

Step 3: Calculate Work Done

Use the formula for work done: \(W = 2 \times T \times A\), where \(T\) is the surface tension and \(A\) is the area in square meters. Use \(T = 0.03 \text{ Nm}^{-1}\) and \(A = 40 \times 10^{-4} \text{ m}^2\).\[ W = 2 \times 0.03 \times 40 \times 10^{-4} = 2.4 \times 10^{-4} \text{ J} \]

Step 4: Compare with Given Options

The calculated work done is \(2.4 \times 10^{-4} \text{ J}\). Comparing with the given options, the correct answer is (b) \(2.4 \times 10^{-4}\).

Key Concepts

Work Done

When you perform work, you're essentially transferring energy. In the context of a soap bubble, work is needed to create and expand the bubble. The formula for calculating the work done on a soap bubble is:\[ W = 2 \times T \times A \]where:

• \( W \) stands for the work done.
• \( T \) is the surface tension, a measure of the force required to extend a liquid surface.
• \( A \) represents the change in surface area.

It's important to note that a soap bubble has both an inner and outer surface. As a result, when the bubble is formed, the surface area changes for both surfaces, leading to double the effect. By understanding this, you recognize how the energy transformation is related to the bubble's tension and area expansion.

Soap Bubble

Soap bubbles are fascinating not just for their colors and patterns, but also for their scientific properties. These bubbles consist of a thin film of soapy water enveloping air inside. The surface tension of the liquid film holds the bubble together.Surface tension is a crucial aspect here. It dictates how much force is necessary to create the bubble's surface. Specifically, surface tension influences the bubble's stability and size. In this exercise, the bubble's surface tension is given as 0.03 Nm\(^{-1}\). This characteristic helps in calculating the work needed to form the bubble.
Understanding a soap bubble also involves appreciating that the shape is typically spherical. This shape minimizes surface area for a given volume, illustrating principles of physics and geometry in a captivating way. The beauty of soap bubbles lies in these intricate plays of force and form.

Surface Area Conversion

Converting surface area units is essential to align with the standard units used in physics problems, typically square meters. This exercise begins with a surface area in square centimeters (cm\(^2\)) and requires conversion into square meters (m\(^2\)) for accurate calculation.Here's how the conversion works:

• 1 cm\(^2\) equals 1 \( \times \) 10\(^{-4}\) m\(^2\).

For a given area of 40 cm\(^2\), the conversion is calculated as follows:\[ 40 \text{ cm}^2 = 40 \times 10^{-4} \text{ m}^2 \]Understanding this conversion is vital, as it ensures the use of consistent units in equations. Physicists and engineers frequently convert units as a standard practice in calculations to maintain precision and accuracy.