How Does a Particle Reach Infinity in Finite Time with Angular Force?

[Ertosthnes]

Homework Statement

Consider a particle that feels an angular force only, of the form F$$_{\theta}$$ = 3m$$\dot{r}$$$$\dot{\theta}$$. Show $$\dot{r}$$=$$\pm$$$$\sqrt{Ar^{4}+B}$$, where A and B are constants of integration, determined by the initial conditions. Also, show that if the particle starts with $$\dot{\theta}$$$$\neq$$0 and $$\dot{r}$$>0, it reaches r=$$\infty$$ in a finite time.

Homework Equations

F$$_{r}$$=m($$\ddot{r}$$-r$$\dot^{\theta}$$^2)=0
F$$_{\theta}$$=m(r$$\ddot{\theta}$$+2$$\dot{r}$$$$\dot{\theta}$$)

The Attempt at a Solution

I've already shown that $$\dot{r}$$=$$\pm$$$$\sqrt{Ar^{4}+B}$$. What I need to do now is show that it reaches r=$$\infty$$ in a finite time. I'm not sure what I need to do here... any thoughts?

 

[tiny-tim]

I've already shown that $$\dot{r}$$=$$\pm$$$$\sqrt{Ar^{4}+B}$$. What I need to do now is show that it reaches r=$$\infty$$ in a finite time. I'm not sure what I need to do here... any thoughts?

Hi Ertosthnes!

(have a theta: θ and a square-root: √ and an infinity: ∞ )

(ooh … and use dashes rather than dots on this forum … they're easier to read!)

You need to solve dr/√(Ar⁴ + B) = dt.

(or you could "sandwich" it between two integrals that are easier)

 

[Ertosthnes]

Thanks Tim! Okay, obviously the integral as is would be pretty tough to solve. Could I say that dt = dr/√(Ar^4 + B) $$\leq$$ dr/(Ar^2), and then integrate to show that t<infinity?

 

[tiny-tim]

Thanks Tim! Okay, obviously the integral as is would be pretty tough to solve. Could I say that dt = dr/√(Ar^4 + B) ≤ dr/(Ar^2), and then integrate to show that t<infinity?

Hi Ertosthnes!

(you could have used the ≤ a also )

… and it's always positive, so … yes, that's fine!