Question

How much energy is stored in the surface of a soap bubble 2.1 \(\mathrm{cm}\) in radius if its surface tension is \(4.5 \times 10^{-2} \mathrm{~N} / \mathrm{m} ?\)

Short answer

The total energy stored in the soap bubble is \(U = 2.38 \times 10^{-4} \) J

Step-by-step solution

Convert the units

First, the radius of the bubble must be converted from centimeters to meters and the surface tension from N/m to J/m^2. To convert cm to m, divide the value by 100. In this case, \( r = \frac{2.1}{100} = 0.021 \, m \) and \( T = 4.5 \times 10^{-2} \, J/m^2 \). these are the units used in the international system for physics problems.

Substitute the values into the formula

Now, substitute the given values of \( r \) and \( T \) into the formula \( U = 8 \pi r^2 T \).

Calculate the energy stored

Compute the energy using the formula: \( U = 8 \pi (0.021)^2 \times 4.5 \times 10^{-2} \)

Key concepts

energy stored

The concept of energy stored in a soap bubble is tied closely to the idea of surface tension. When a soap bubble forms, a thin film of soapy water stretches across its spherical frame. This film tries to minimize its surface area due to the force of surface tension, which acts as if it's trying to "shrink" the surface. The energy stored in this surface is essentially what's needed to keep the film stretched and maintain the bubble's shape.

To determine the energy stored in a soap bubble, we use the formula:

• \( U = 8 \pi r^2 T \)

where \( U \) represents the energy stored, \( r \) is the radius of the bubble, and \( T \) is the surface tension. The factor of \( 8 \pi \) is significant because it accounts for the two distinct surfaces of a bubble (inside and outside) and incorporates the geometry of a sphere.

Understanding how this equation relates energy, radius, and surface tension helps grasp how bubbles maintain their structure and why they appear so delicately robust.

soap bubble

A soap bubble is a thin film of soapy water enclosing air that forms a spherical shape. This spherical shape occurs because it’s the form that minimizes surface area for a given volume, meaning less energy is stored due to tension in the film's surface.

The surface of a soap bubble has unique properties because it exhibits near-perfect surface tension. Surface tension is an elastic-like force existing on the surface of a liquid, influenced by the interactions of liquid molecules. In bubbles, these forces create a barrier that resists external loads, allowing bubbles to maintain their form despite being so thin and fragile.

Soap bubbles are fascinating not only for their colorful appearance but their display of physical properties that are essential in understanding fluid dynamics and surface phenomena. By examining bubbles, we gain insight into practical applications, such as minimizing materials or maximizing surface areas.

unit conversion

Unit conversion is crucial for solving problems in physics because it ensures that the quantities you're working with are in compatible units. In the context of the soap bubble problem, we start by converting the radius from centimeters to meters because the international system of units (SI) uses meters for length. This ensures consistency and allows formulas to yield results in standard units.

To convert radius:

• Radius in centimeters: 2.1 cm.
• Using the conversion, divide by 100 to get meters: \( r = \frac{2.1}{100} = 0.021 \) m.

Secondly, since the problem involves surface tension, expressed as \( \mathrm{N} / \mathrm{m} \) which is equivalent to \( \mathrm{J} / \mathrm{m}^2 \), this allows integration into the energy formula seamlessly.

Unit conversion is a foundational skill in physics that enables precise calculations and fosters a better understanding of how different systems relate.