Ch. 13 - Fluids

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 13 - Fluids

Problem 75

Chapter 13, Problem 75

Estimate the diameter of a steel needle that can just barely remain on top of water due to surface tension. (See Figs. 13–38 and 13–39a, and Table 13–1.)

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Identify the forces acting on the needle: The needle remains on top of the water due to the upward force provided by surface tension, which balances the downward gravitational force acting on the needle.

Express the gravitational force acting on the needle: The gravitational force is given by \( F_g = mg \), where \( m \) is the mass of the needle and \( g \) is the acceleration due to gravity. The mass \( m \) can be expressed in terms of the density \( \rho \), volume \( V \), and the needle's geometry.

Relate the surface tension force to the needle's diameter: The surface tension force is \( F_s = 2 \gamma L \), where \( \gamma \) is the surface tension of water (from Table 13–1), and \( L \) is the length of the needle in contact with the water. For a circular cross-section, \( L \) is approximately the circumference of the needle, \( L = \pi d \), where \( d \) is the diameter.

Set up the equilibrium condition: At equilibrium, the upward surface tension force equals the downward gravitational force, \( F_s = F_g \). Substituting the expressions, \( 2 \gamma \pi d = \rho V g \). Replace \( V \) with the volume of the needle, \( V = \frac{\pi d^2}{4} \times l \), where \( l \) is the length of the needle.

Solve for the diameter \( d \): Simplify the equation \( 2 \gamma \pi d = \rho \left( \frac{\pi d^2}{4} \times l \right) g \) to isolate \( d \). This will give you the diameter of the needle that can just barely remain on top of the water due to surface tension.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Surface Tension

Surface tension is a physical property of liquids that describes the elastic-like force existing at the surface of a liquid. It arises from the cohesive forces between liquid molecules, which are stronger at the surface due to the lack of neighboring molecules above. This phenomenon allows small objects, like a steel needle, to float on the surface of water despite being denser than water.

Weight of the Needle

The weight of the needle is the force due to gravity acting on it, calculated as the product of its mass and the acceleration due to gravity. For the needle to remain on the surface of the water, its weight must be balanced by the upward force provided by surface tension. This balance is crucial in determining the maximum diameter of the needle that can float without sinking.

Capillary Action

Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces, often observed in thin tubes or porous materials. In the context of the needle on water, capillary action can influence how the water surface curves around the needle, affecting the effective surface tension and the needle's ability to remain afloat. Understanding this concept helps in estimating the diameter of the needle that can be supported by surface tension.

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