Chapter 16.5, Problem 37E

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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** = *ω***k**, 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** = 2**w**.

(a)

To determine

**To show:** The vector

Explanation

**Given data:**

From Figure,

Rearrange the equation.

Substitute

**(b)**

To determine

**To show:** The vector

**(c)**

To determine

**To show:** The

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