Circular motion (plane polars)

[Klaynos]

Doing some revision and I've got a question I struggling with, it's not assessed.

 

I'm having real issues with it:

 

A particle of unit mass moves under a central force in the logarithmic spiral orbit,

[math]r=ke^{\alpha \theta}[/math], where k and [math]\alpha[/math] are constants. Given the expression for accleration in plane polar coordinates

 

[math]\bold a = (\ddot r - r \dot \theta^2)e_r + r(r \ddot \theta + 2 \dot r \dot \theta)e_\theta[/math]

 

deterimine [math]\theta (t)[/math and r(t).

 

Derive an expression for the force as a function of radias.

 

My first idea was to differntiat [math]r=ke^{\alpha \theta}[/math] twice, reform it into [math]\theta =[/math] and differntiate that twice and shuve it back into the orignal acceleartion expression. Is this the right method or am I missing something?

 

Cheers

[Klaynos]

Oh, and in:

 

[math]

r=ke^{\alpha \theta}

[/math]

 

is [math]\theta[/math] a function of t, as I would assume?