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Ch 37: The Foundations of Modern Physics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 37, Problem 49c

Consider an oil droplet of mass m and charge q. We want to determine the charge on the droplet in a Millikan-type experiment. We will do this in several steps. Assume, for simplicity, that the charge is positive and that the electric field between the plates points upward. A spherical object of radius r moving slowly through the air is known to experience a retarding force Fdrag = −6πηrv where η is the viscosity of the air. Use this and your answer to part b to show that a spherical droplet of density ρ falling with a terminal velocity vterm has a radius. r=9ηvterm2ρgr = \(\sqrt{\frac{9\eta v_{term}\)}{2\(\rho\) g}}

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Start by understanding the forces acting on the spherical droplet. At terminal velocity, the net force on the droplet is zero because the downward gravitational force is balanced by the upward drag force and the buoyant force. This means: F_gravity = F_drag + F_buoyant.
Express the gravitational force acting on the droplet: F_gravity = m * g, where m is the mass of the droplet and g is the acceleration due to gravity. The mass of the droplet can be written in terms of its density and volume: m = ρ * (4/3)πr³.
Write the drag force acting on the droplet: F_drag = 6πηrvₜₑᵣₘ, where η is the viscosity of air, r is the radius of the droplet, and vₜₑᵣₘ is the terminal velocity.
Write the buoyant force acting on the droplet: F_buoyant = ρ_air * (4/3)πr³ * g, where ρ_air is the density of air. This accounts for the upward force due to the displaced air.
Combine the forces and solve for the radius r. At terminal velocity, F_gravity = F_drag + F_buoyant. Substituting the expressions for each force and simplifying, you will arrive at the formula for the radius: r = √((9ηvₜₑᵣₘ) / (2ρg)).

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Key Concepts

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

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. In the context of a droplet falling through air, it occurs when the gravitational force acting downward is balanced by the drag force acting upward, resulting in zero net force and constant velocity.
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Viscosity

Viscosity is a measure of a fluid's resistance to deformation or flow. In this scenario, the viscosity of air (η) affects the drag force experienced by the droplet as it moves through the air. A higher viscosity results in a greater retarding force, which influences the terminal velocity of the droplet.
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Drag Force

The drag force is the resistance force experienced by an object moving through a fluid, which depends on the object's shape, size, and the fluid's properties. For a spherical droplet, the drag force can be expressed as Fₔᵣₐ₉ = −6πηr v, where r is the radius of the droplet and v is its velocity. This relationship is crucial for deriving the radius of the droplet at terminal velocity.
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