Position of particle in inertial reference frame

[jasonchiang97]

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.

 

[Andrew Mason]

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

 

[jasonchiang97]

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 = R²where 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

 

[jasonchiang97]

Never mind I figured it out.

Thanks for the hint!