Problem 992 - JEE Physics

Question

The work done in blowing a soap bubble of \(10 \mathrm{~cm}\) radius is [surface tension of soap solution is \(\\{(3 / 100) \mathrm{N} / \mathrm{m}\\}\) ] (A) \(75.36 \times 10^{-4}\) Joule (B) \(37.68 \times 10^{-4}\) Joule (C) \(150.72 \times 10^{-4}\) Joule (D) \(75.36\) Joule

Step-by-Step Solution

Answer

The work done in blowing a soap bubble of 10 cm radius is 75.36 × 10⁻⁴ Joule (option A). This is found by calculating the change in surface area and using the given surface tension of the soap solution to determine the work done, applying the formula \(W = T \Delta A\).

Step 1: Recall the formula for the work done

The formula for the work done in inflating a bubble is: \[W = T \Delta A\] where \(W\) is the work done, \(T\) is the surface tension, and \(\Delta A\) is the change in surface area of the bubble. Step 2: Calculate the change in surface area

Step 2: Calculate the change in surface area

The surface area of a sphere is given by the formula \(A = 4\pi r^2\), where \(r\) is the radius. For a soap bubble, we need to consider both the inside and outside surfaces, so the total surface area of a soap bubble is \(A_\text{total} = 2(4\pi r^2) = 8\pi r^2\). The change in surface area after inflating the soap bubble to a radius of \(10 \mathrm{~cm}\) is: \(\Delta A = A_\text{final} - A_\text{initial}\) Since the initial surface area is zero (\(A_\text{initial}=0\)), the change in surface area is equal to the total surface area of the soap bubble: \(\Delta A = 8\pi r^2\) Substitute the given radius, \(r = 10 \mathrm{~cm}\), into the formula: \(\Delta A = 8\pi (10)^2 \mathrm{cm}^2\) \(\Delta A = 800\pi \mathrm{~cm^2}\) Step 3: Calculate the work done in blowing the soap bubble

Step 3: Calculate the work done

Use the formula for the work done, and substitute the given surface tension, \(T = \frac{3}{100} \mathrm{N/m}\), and the calculated change in surface area into the formula: \(W = T \Delta A\) \[W = \left(\frac{3}{100}\mathrm{N/m}\right)(800\pi \mathrm{~cm^2})\] To convert from \(\mathrm{cm^2}\) to \(\mathrm{m^2}\), we need to multiply by \(\frac{1}{10,000}\): \[W = \left(\frac{3}{100}\mathrm{N/m}\right)(800\pi \cdot \frac{1}{10,000} \mathrm{~m^2})\] \[W = \frac{3}{100} \cdot \frac{800\pi}{10,000} \mathrm{N} \cdot \mathrm{m}\] Simplify the expression: \[W = \frac{12 \pi}{5} \mathrm{Nm}\] Step 4: Convert the work done into the desired units

Step 4: Convert the units

Convert the result from \(\mathrm{Nm}\) (Joules) to \(\times 10^{-4}\) Joules: \[W = \frac{12 \pi}{5} \mathrm{Nm} \times \frac{1}{10^{-4}}\] \[W = 75.36 \times 10^{-4} \mathrm{J}\] The work done in blowing the soap bubble is 75.36 × 10⁻⁴ Joule, which corresponds to option (A).

Key Concepts

Surface Tension

Surface tension is a fascinating concept in physics, often described as the "skin" on the surface of a liquid. It's the force that makes water striders glide on water and keeps a bubble's shape. In terms of numbers, surface tension is the force per unit length, usually expressed in Newton per meter (N/m).
For a soap bubble, this tension is doubled. Why? Because a soap bubble has both an inner and outer surface. Imagine two layers stretching as you blow the bubble. Surface tension works on both, hence multiplying the tension by two is essential to understand bubble dynamics.
To calculate the work done while inflating a bubble, you use this doubled tension along with changes in the bubble's surface area.

Sphere Surface Area

Understanding a sphere's surface area can feel daunting, but let's break it down. For any sphere, the formula to calculate its surface area is: \( A = 4\pi r^2 \)
Here, \( r \) is the radius of the sphere. But when it comes to a soap bubble, both sides of the thin soap layer contribute.

• Single sphere surface area: \( 4\pi r^2 \)
• Soap bubble (both surfaces): \( 8\pi r^2 \)

This is crucial because when you're calculating the work done in blowing a soap bubble, you consider this entire area. Imagine blowing up your bubble: the surface area grows, and so does the work needed to stretch that soapy skin. It's this increase in area that directly links to the energy or work required.

Unit Conversion

Unit conversion is the backbone of physics problem solving. Sometimes, the units you have don't match the units you need in your final answer.
In this exercise, we started with the surface area in square centimeters. However, the surface tension was in Newtons per meter, requiring us to convert to square meters for consistency.

• 1 cm = 0.01 m, thus
• 1 cm² = 0.0001 m²

To convert from cm² to m², multiply by \(\frac{1}{10,000}\) or \(0.0001\). This conversion is vital to ensure all quantities are in appropriate SI units, allowing the use of basic physics formulas without error. So, pay attention when converting units—it's often the key that unlocks the correct answer.

Physics Problem Solving

Solving physics problems often feels like fitting the pieces of a puzzle together. Start by understanding the key principles involved. In this problem:

• Grasp the idea of work done
• Use surface tension and surface area
• Perform unit conversions as needed

For our bubble, everything connects through the formula for work done: \( W = T \Delta A \)
Break it into pieces—calculate the sphere's surface area first, then double it for a bubble. Don't forget to convert units like in our example. Lastly, plug values into the formula. Step-by-step solving not only helps in getting the answer but also in understanding the problem's physics.
Revisiting these basics prepares you for almost any physics challenge, no matter how big or small.