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Position of particle in inertial reference frame

  • #1

Homework Statement


The position of a participle in a fixed inertial frame of reference is given by the vector

r = i(x0 + Rcos(Ωt)) +j(Rsin(Ωt))


where x0, R and Ω are constants.

a) Show that the particle moves in a circle with constant speed

Homework Equations



F = mv2/r

The Attempt at a Solution



r = r'

where r' is the non-inertial reference frame

dr/dt = i(-RΩsin(Ωt)) + j(RΩcos(Ωt))

I can transform it to a non-inertial reference frame v' using

v = v' + (ω × r')

but since r = r' then

v = v' + (ω × r')

But I'm not sure where that leads me

I also had another thought where if the curl of the velocity in the inertial frame is non-zero does that prove the object is moving in a circular motion? Since the curl is a circulation density.
 

Answers and Replies

  • #2
Andrew Mason
Science Advisor
Homework Helper
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Write the equation for a circle at the origin. Then, if you rewrite the given equation as:
##\vec {r} = x_0\vec {i} + R\cos (Ωt)\vec {i} + R\sin (Ωt)\vec {j}##
does that help? Hint: this is not a physics question. Just math.

AM
 
  • #3
Write the equation for a circle at the origin. Then, if you rewrite the given equation as:
##\vec {r} = x_0\vec {i} + R\cos (Ωt)\vec {i} + R\sin (Ωt)\vec {j}##
does that help? Hint: this is not a physics question. Just math.

AM
sorry, do you mean a circle centered at the origin? if that's what you mean then

x2 + y2 = R2


where x(t) = Rcos(Ωt) and y(t) = Rsin(Ωt) in polar coordinates

then

##\vec {r} = x_0\vec {i} + R\cos (Ωt)\vec {i} + R\sin (Ωt)\vec {j} = x_0\vec{i} + x(t)\vec{i} + y(t)\vec{j}##

or is the last step not needed
 
  • #4
Never mind I figured it out.

Thanks for the hint!
 

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