Describing the motion of a particle using polar coordinates

[NATURE.M]

1. Problem
Consider a particle that feels an angular force only of the form:
F_θ = 3mr'θ'. Show that r' = ± (Ar^4 + B)^(1/2), where A and B are constants of integration, determined by the initial conditions. Also, show that if the particle starts with θ' ≠ 0 and r' > 0, it reaches r = ∞ in a finite time.

The Attempt at a Solution

So I understand the first part of the question and can easily show r' = … using θ' = L/mr^2, where L is the angular momentum. Now my issue is with the second part. I know I probably should set up an integral to evaluate from r_0 to r = ∞. I tried starting at F_θ = 3mr'θ' = m(dθ'/dt). I think this equation is correct (although may not be what I should be using). From here I would separate variables, as 3r'dt = (1/θ')dθ'. But this doesn't seem to be right. I'm really stuck here. Any ideas ?

 

[Orodruin]

What do the equations of motion look like in polar coordinates?

 

[NATURE.M]

Isn't it just 3mr'θ' = m(dθ'/dt), since the radial force is 0 ?

Edit: Or are you referring to the general form:
F_θ = m(rθ' + 2r'θ')

 

[Orodruin]

No, you have a problem in curvilinear coordinates and have to take into account that the basis vectors change. I suggest writing down the equations of motion in Cartesian coordinates and transform them to polar. You will have both radial and tangential acceleration in general (if the radial acceleration was zero, how would you get to infinity in finite time?).

Also, your can see that your equation is wrong by dimensional analysis, the left hand side has units mass x length / time^2 and the right hand side mass / time^2.

 

[NATURE.M]

Okay but I assumed since the problem specified that the particle feels only an angular force that would mean the radial force is 0, an thus the radial acceleration was 0.

 

[Orodruin]

In curvilinear coordinates, there is a difference between force and (coordinate) acceleration being equal to zero.

 

[NATURE.M]

Okay I'm a bit confused. Since the force is given in polar coordinates why would I need to go back to Cartesian ?

 

[Orodruin]

You don't need to, but you must figure out what the equations of motion are in one way or another.

 

[NATURE.M]

okay so the radial component of the force in polar coordinates would generally be given as
F_r = m ( r'' − r θ' ^2 ). But since F_r = 0, we have mr'' = r θ' ^2 and m(rθ' + 2r'θ') = 3mr'θ'. Am I on the right track ?
From here i would still need to integrate from 0 to infinity I would think. This seems to be a real challenge with the above equations however.
I guess my real issue is I'm not entirely sure how I would show that r goes to infinity in a finite time period.

 

[Orodruin]

Your force equations look ok now. Remember, you are not tasked with computing the finite time - only with showing that it is finite. Your integral is also not from r=0 as it is difficult to have a tangential velocit in that particular point.

And just to check: Did you use that angular momentum is conserved in your solution to the first part? Since you have a tangential force, this will not be true in general.

 

[NATURE.M]

I used L = θ'mr^2, where L is angular momentum. I arrived at the right answer, although it was only a direct substitution of θ' into the F_θ. So I didn't completely rely on the conservation property.