Calculate the Curl of a Velocity vector field

  • #1

Homework Statement



The velocity of a solid object rotating about an axis is a field [tex]\bar{v} (x,y,z)[/tex]
Show that [tex]\bar{\bigtriangledown }\times \bar{v} = 2\,\bar{\omega }[/tex], where [tex]\bar{\omega }[/tex] is the angular velocity.

Homework Equations



3. The Attempt at a Solution [/B]

I tried to use the Stoke's theorem using an infinitesimal element with trapezoidal shape, but i was stuck with calculations. Which is the best way to resolve the equation ? It would be fantastic if you could explain me the geometric intuition behind the problem
 
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Answers and Replies

  • #2
Delta2
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Just use the relationship ##\vec{v}(x,y,z)=\vec{\omega}\times(x\vec{i}+y\vec{j}+z\vec{k})## and some identities of vector calculus about the curl operator.

The main identity you ll use is the first one found here : https://en.wikipedia.org/wiki/Curl_(mathematics)#Identities. Notice that you ll treat ##\vec{\omega}## as a constant vector in this identity so it will be ##\nabla\cdot\vec{\omega}=0## , ##\vec{F}\cdot\nabla \vec{\omega}=0## e.t.c
 
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  • #3
Thanks so much =)
 
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