Problem 87 - Master Resource Book in JEE Main Physics

Question

A thin liquid flm formed between a \(\mathrm{U}\) shaped wire and a light slider supports a weight of \(1.5 \times 10^{-2} \mathrm{~N}\). The length of the slider is \(30 \mathrm{~cm}\) and its weight negligible.The surface tension of the liquid film is (a) \(0.0125 \mathrm{Nm}^{-1}\) (b) \(0.1 \mathrm{Nm}^{-1}\) (c) \(0.05 \mathrm{Nm}^{-1}\) (d) \(0.025 \mathrm{Nm}^{-1}\)

Step-by-Step Solution

Answer

The surface tension is 0.025 Nm^{-1}, matching option (d).

Step 1: Understanding Surface Tension

Surface tension is defined as the force per unit length. For a liquid film stretched over a frame, it acts along the length of the frame.

Step 2: Identifying the Forces

The given weight of the slider is the force due to gravity on the system, which is supported by the surface tension of the liquid film. This force is the weight that the surface tension must balance.

Step 3: Calculate Total Length Affected

Since the film forms over both the top and bottom sides of the slider, the total length over which the surface tension acts is twice the length of the slider. Here, Length of the slider = 30 cm = 0.3 m, thus, Total length affected = 2 x 0.3 m = 0.6 m.

Step 4: Applying the Formula for Surface Tension

The formula to calculate surface tension (\(T\)) is: \( T = \frac{F}{L} \), where \(F\) is the force the surface tension supports (1.5 x 10^{-2} N) and \(L\) is the total length affected (0.6 m).

Step 5: Calculate Surface Tension

Substitute the given values into the surface tension formula: \( T = \frac{1.5 \times 10^{-2} \, \text{N}}{0.6 \, \text{m}} = 0.025 \, \text{Nm}^{-1} \).

Step 6: Choose the Correct Option

Compare the calculated surface tension with the given options. The calculated value matches option (d) 0.025 Nm^{-1}.

Key Concepts

Physics Problem Solving

In tackling physics problems, especially those involving real-world phenomena such as surface tension, it's vital to apply a structured approach. Start by clearly understanding the problem statement and what is being asked. Break down the problem into manageable steps.
Identify key elements, like **forces** and **measurements** given or required.

• **Analyze Known Values:** Recognize all given values and what you need to find. For instance, in the given problem, we're provided with the weight supported by the liquid film and the dimensions of the slider.
• **Understand Relevant Concepts:** Be familiar with underlying principles, like surface tension. It is the force per unit length acting along the surface of a liquid.
• **Use the Right Formulas:** Formulas act as tools to connect known and unknown quantities. In our problem, the surface tension equation, \( T = \frac{F}{L} \), helps in calculating the unknown tension using the known force and length.
• **Proper Unit Conversion:** Ensure all measurements are in correct units before using them in calculations, such as converting centimeters to meters.

This structured method allows for a clearer journey from problem to solution. It also maximizes problem-solving efficiency and minimizes mistakes.

Force and Motion

Force and motion are fundamental concepts connected with how objects interact and move. When considering a liquid film supporting a slider, understanding force dynamics is crucial. Here’s how they come into play:

• **Gravitational Force:** It is the force exerted by the weight of the object (the slider in this case) towards gravity. In our exercise, this force is \(1.5 \times 10^{-2} \text{ N}\).
• **Normal Forces:** These are forces that the film exerts upwards, counteracting the gravitational pull. The surface tension acts as these normal forces along the film’s surface.
• **Equilibrium:** For the slider to remain in place without moving, the forces must be balanced. This means the upward forces (due to surface tension) equal the downward gravitational force.

Comprehending these force relationships allows us to compute the surface tension required to support the slider, using principles of equilibrium and motion. Without this balance, the slider will either sink or rise depending on if the surface tension is too low or too high, respectively.

Liquid Film Dynamics

Liquid film dynamics is a fascinating area that deals with how thin layers of liquid behave. In our specific problem, the dynamics revolve around the slider and U-shaped wire.

• **Thickness of the Film:** While the problem doesn't state the film's thickness, it can influence dynamics like surface tension in more nuanced problems.
• **Surface Forces:** The liquid film exhibits surface tension, a form of force per unit length that acts parallel to the surface of the liquid. This is seen in our exercise as the film supports the slider’s weight.
• **Interactions with Boundaries:** The wire forms a closed boundary that holds the film in position. The slider's movement alters the internal distribution of forces within the film, impacting surface tension.
• **Environmental Conditions:** Factors such as temperature can affect the properties of the liquid film – though this is simplified in the problem.

Understanding these dynamics allows for a deep insight into the behavior of liquid films in practical applications, beyond just solving textbook problems. It underlines the importance of surface tension in supporting structures or separating layers in various engineering and scientific scenarios.