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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector w = ωk, where ω is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d from the axis of rotation. Let r = ⟨x, y, z⟩ be the position vector of P.

(a) By considering the angle θ in the figure, show that the velocity field of B is given by v = w × r.

(b) Show that v = −ω y i + ω x j.

(c) Show that curl v = 2w.

images

(a)

To determine

To show: The vector v=w×r .

Explanation

Given data:

w=ωk and r=x,y,z .

ω=vd (1)

From Figure, sinθ=dr .

Rearrange the equation.

d=rsinθ

Substitute rsinθ for d in equation (1),

(b)

To determine

To show: The vector v=ωyi+ωxj .

(c)

To determine

To show: The curlv=2w .

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Chapter 16 Solutions

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