Question

If the excess pressure inside a soap bubble is balanced by oil column of height \(2 \mathrm{~mm}\) then the surface tension of soap solution will be. \((\mathrm{r}=1 \mathrm{~cm}\) and density \(\mathrm{d}=0.8 \mathrm{gm} / \mathrm{cc})\) (A) \(3.9 \mathrm{~N} / \mathrm{m}\) (B) \(3.9 \times 10^{-2} \mathrm{~N} / \mathrm{m}\) (C) \(3.9 \times 10^{-3} \mathrm{~N} / \mathrm{m}\) (D) \(3.9 \times 10^{-1} \mathrm{~N} / \mathrm{m}\)

Short answer

The surface tension of the soap solution is approximately \(3.9 \times 10^{-2} \,\text{N/m}\) (Option B).

Step-by-step solution

Write down the given values

We are given: - Radius of the soap bubble (r) = 1 cm - Height of the oil column (h) = 2 mm = 0.2 cm (converted to centimeters for consistency) - Density of the oil (d) = 0.8 gm/cc

Excess pressure formula and balanced pressure

We know the excess pressure inside a soap bubble can be calculated using the formula: \(P_{excess} = 4 \times \frac{T}{r}\) where \(P_{excess}\) is the excess pressure, T is the surface tension we need to find, and r is the radius of the soap bubble. The pressure inside the soap bubble is balanced by the pressure due to the oil column, which can be calculated using the formula: \(P_{oil} = h \times d \times g\) where \(P_{oil}\) is the pressure due to the oil column, h is the height of the column, d is the oil's density, and g is the acceleration due to gravity (approximately 9.81 m/s² or 981 cm/s²). As the excess pressure inside the bubble is balanced by the oil column, we can set these pressures equal to each other: \(4 \times \frac{T}{r} = h \times d \times g\)

Solve for surface tension (T)

Now we will solve the equation for T (Surface tension): \(T = \frac{r \times h \times d \times g}{4}\) Using the given values: \(T = \frac{1\,\text{cm} \times 0.2\,\text{cm} \times 0.8\,\text{g/cc} \times 981\,\text{cm/s²}}{4}\)

Calculate the surface tension

Calculate T using the values in the equation: \(T = \frac{1 \times 0.2 \times 0.8 \times 981}{4} = 0.0392\,\text{N/m}\) In scientific notation with correct units, it is: \(3.92 \times 10^{-2}\,\text{N/m}\) Among the answer choices, this matches closest to option (B): \(3.9 \times 10^{-2} \,\text{N/m}\) So, the surface tension of the soap solution is approximately \(3.9 \times 10^{-2} \,\text{N/m}\).

Key concepts

Soap Bubble

Soap bubbles are fascinating thin films of liquid surrounding pockets of air. They are usually made from a mix of soap and water, forming a double layer with air trapped on the inside. Soap bubbles appear due to the surface tension of the liquid. Surface tension is the tendency of liquid surfaces to shrink into the minimum surface area possible. In a soap bubble, this causes a spherical shape, which is the most efficient shape in terms of surface area for a given volume of air trapped inside.

Bubbles can be a great way to explore concepts of physics! They exhibit behaviors like thin-film interference, causing their unique vibrant colors. Also, soap bubbles demonstrate how different pressures inside and outside the bubble balance. When there's excess pressure inside the bubble, it can push outwards against atmospheric pressure.

Pressure Calculation

Pressure inside a bubble and the pressure acting against it determines its stability. Calculating these pressures involves understanding forces acting per unit area.

• Excess Pressure: Caused by surface tension, calculated using the formula \(P_{excess} = 4 \times \frac{T}{r}\).
• Oil Column Pressure: Balancing the bubble's pressure, calculated using \(P_{oil} = h \times d \times g\).

The two pressures balance to form the equation: \(4 \times \frac{T}{r} = h \times d \times g\). This allows us to solve for unknown variables like the surface tension \(T\). Calculating these pressures is crucial for predicting the behavior of bubbles and understanding the concept of balanced forces.

Density of Fluids

Density is a property that expresses how much mass is contained in a given volume. It's crucial in understanding how different fluids interact and balance each other's pressures.

• Definition: Density \(d\) is defined as mass per unit volume, often expressed as gm/cc or kg/m³.
• Role in Pressure: In problems involving pressure balance, like the soap bubble exercise, density is part of the equation \(P = h \times d \times g\) for calculating the pressure a fluid exerts.
• Importance: Knowing the density of a fluid, such as oil in this case, helps us determine how much pressure it can apply to balance another pressure source.

Density shows why some substances float while others sink, and it affects the buoyancy of objects and fluids.

Physics Problem-Solving

Physics problem-solving involves identifying known values, understanding the principles at play, and using equations wisely to find unknowns. Here's how to effectively tackle physics problems like the soap bubble exercise:

• Understand the Problem: Read it carefully to grasp what is being asked and what information is given.
• Identify Formulas: Recognize which physics principles and formulas apply. For example, excess pressure and density-related equations.
• Plug in Known Values: Substitute the given numbers into your identified formulas.
• Calculate and Analyze: Carry out mathematical computations carefully. Check if your answer makes sense and matches physical reality.
• Review: Double-check your math and ensure all units are correct. This ensures an accurate and reliable solution.

Good physics problem-solving involves a meticulous approach and thoughtful application of basic principles to get to the correct outcome.