Question #184660

A particle moves so that its position vector ˜r at time t is = ˜r ˜a coswt +˜bsinwt, where w is a constant and ˜a and ˜b are constant vectors. Show that (a) ˜ ˜ r × r˙ is independent of t, ˜


1
Expert's answer
2021-04-27T01:33:41-0400

(a)


a=b=1,ab|\vec a|=|\vec b|=1, \vec a\perp\vec brr=(acosωt+bsinωt)(acosωt+bsinωt)\vec r\cdot \vec r=(\vec a\cos\omega t+\vec b\sin \omega t)\cdot(\vec a\cos\omega t+\vec b\sin \omega t)

=cos2ωt+sin2ωt=1=\cos^2\omega t+\sin^2\omega t=1


(b)


v=drdt=ωasinωt+ωbcosωt\vec v=\dfrac{d\vec r}{dt}=-\omega\vec a\sin\omega t+\omega\vec b\cos \omega t

dvdt=ω2acosωtω2bsinωt\dfrac{d\vec v}{dt}=-\omega^2\vec a\cos\omega t-\omega^2\vec b\sin \omega t




=ω2r=-\omega^2\vec r

The acceleration is everywhere towards the origin and proportional to r.\vec r.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS