Problem 39 - College Physics

Question

Find the gauge pressure in pascals inside a soap bubble 7.00 \(\mathrm{cm}\) in diameter. The surface tension of this soap is 25.0 dynes/cm.

Step-by-Step Solution

Answer

The gauge pressure inside the soap bubble is approximately 2.857 Pa.

Step 1: Understand the Problem

We are given the diameter of the soap bubble, which is 7.00 cm, and the surface tension, which is 25.0 dynes/cm. We need to find the gauge pressure inside the bubble.

Step 2: Convert Units

First, let's convert the given values into the SI units. The surface tension is given in dynes/cm, where 1 dyne = 10^{-5} N, so we convert it to N/m: \[ 25.0 \text{ dynes/cm} = 25.0 \times 10^{-5} \text{ N/cm} = 25.0 \times 10^{-3} \text{ N/m} = 0.025 \text{ N/m} \]Next, convert the diameter into meters: \[ 7.00 \text{ cm} = 0.07 \text{ m} \]

Step 3: Calculate the Radius

Calculate the radius of the bubble by dividing the diameter by 2: \[ r = \frac{0.07 \text{ m}}{2} = 0.035 \text{ m} \]

Step 4: Use the Pressure Formula for a Bubble

For a soap bubble, the gauge pressure \( P \) is given by the formula: \[ P = \frac{4\gamma}{r} \]where \( \gamma = 0.025 \text{ N/m} \) is the surface tension and \( r = 0.035 \text{ m} \) is the radius. Substitute the values into the formula to calculate the pressure.

Step 5: Perform the Calculation

Substitute the known values of \( \gamma \) and \( r \) into the formula:\[ P = \frac{4 \times 0.025}{0.035} \]Calculate:\[ P = \frac{0.1}{0.035} \approx 2.857 \text{ N/m}^2 \]Thus, the gauge pressure inside the bubble is approximately 2.857 Pa.

Key Concepts

Soap Bubble Physics

Soap bubbles are fascinating structures, and understanding the physics behind them can be captivating. At the basic level, soap bubbles are formed by trapping air inside a thin film of soapy water. This film is very thin, and its surface tension is what allows the bubble to maintain its shape.
Bubbles occur naturally in spherical shapes due to surface tension. A sphere is ideal because it has the smallest surface area for a given volume, creating a balance that uses the least amount of energy.
In the context of physics, a soap bubble is interesting because it involves concepts like pressure, volume, and surface tension, making it a great subject for study.

Surface Tension

Surface tension is a vital concept in understanding soap bubbles. It's the property that makes the surface of a liquid act like a stretched elastic membrane. This happens because the molecules at the surface of a liquid bond together more tightly than those inside.

• Surface tension is measured in units such as dynes/cm or N/m.
• It affects the shape and stability of the bubble. Stronger surface tension will hold a bubble together more robustly.

For a soap bubble, the formula for gauge pressure involves surface tension twice because the bubble has two surfaces: the inner and outer layer of soap. This emphasizes how critical surface tension is in maintaining a bubble's integrity.

Unit Conversion

Unit conversion is often a crucial step in problems involving physical calculations. In this exercise, we converted the surface tension from dynes/cm to N/m and the diameter from cm to meters.
Here's a quick guide to understanding and performing such conversions:

• 1 dyne = 10^{-5} N. Multiplying dynes by 10^{-5} converts them to newtons.
• To convert cm to m, remember that 1 m = 100 cm, so you divide the number of centimeters by 100.

Unit conversion is essential because it standardizes different measurements, making them compatible in calculations that require specific units like meters and newtons, commonly used in the International System of Units (SI).

Physics Problem Solving

Solving physics problems, such as finding the gauge pressure in a soap bubble, involves a systematic approach.
Here is a simple pathway to tackle such problems:

• Understand the given data and what needs to be calculated.
• Convert all measurements to compatible units, using SI units as a standard where possible.
• Apply relevant formulas, in this case, the formula for gauge pressure in a soap bubble, \( P = \frac{4\gamma}{r} \).
• Substitute the values and solve the equation to find the unknown value, like pressure.

This approach highlights critical thinking and demonstrates how complex problems can be broken into manageable steps, ensuring a streamlined and logical progression toward an accurate solution.