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 77

Chapter 13, Problem 77

A common effect of surface tension is the ability of a liquid to rise up a narrow tube due to capillary action. Show that for a narrow tube of radius r placed in a liquid of density ρ and surface tension γ, the liquid in the tube will reach a height h = 2γ/ρgr above the level of the liquid outside the tube, where g is the gravitational acceleration. Assume that the liquid “wets” the tube and that the liquid surface is vertical at the contact with the inside of the tube.

Verified step by step guidance

Start by understanding the phenomenon of capillary action. It occurs due to the balance between cohesive forces (within the liquid) and adhesive forces (between the liquid and the tube). When the liquid 'wets' the tube, adhesive forces dominate, causing the liquid to rise in the tube.

The upward force due to surface tension can be expressed as F = 2πrγ, where γ is the surface tension, and r is the radius of the tube. This force acts along the circumference of the tube and is responsible for lifting the liquid.

The weight of the liquid column in the tube is given by W = ρVg, where ρ is the density of the liquid, V is the volume of the liquid column, and g is the gravitational acceleration. For a cylindrical column of height h and radius r, the volume is V = πr²h, so the weight becomes W = ρπr²hg.

At equilibrium, the upward force due to surface tension is balanced by the downward gravitational force of the liquid column. Set the two forces equal: 2πrγ = ρπr²hg.

Simplify the equation to solve for h: h = (2γ) / (ρgr). This shows that the height of the liquid column is inversely proportional to the radius of the tube and directly proportional to the 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 at the surface of a liquid. It arises from the cohesive forces between liquid molecules, which are stronger at the surface due to the imbalance of intermolecular forces. This phenomenon allows liquids to resist external forces and is responsible for the formation of droplets and the ability of small objects to float on a liquid's surface.

Capillary Action

Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces, primarily due to the interplay of cohesive and adhesive forces. In a narrow tube, adhesive forces between the liquid and the tube's walls can overcome the cohesive forces within the liquid, causing the liquid to rise. This effect is crucial in various natural and technological processes, such as water transport in plants and ink movement in pens.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid and is given by the equation P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In the context of capillary action, hydrostatic pressure balances the upward force due to surface tension, determining the height to which the liquid can rise in the tube.

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