polar coordinate components and acceleration

I'm stuck on the second part of this question:

Suppose that a particle moves in a plane with trajectory given by thepolar equation r = 2b sinθ for some constant b > 0.

(i) Show that this can be written in Cartesian coordinates as,x2 + (y − b)2 = b2,

This is the equation for a circle of centre (0, b) and radius b.

[Hint:recall that r2=x2+y2 and y=rsinθ]

(ii) Suppose that the transverse component of the acceleration is zero.

(a) Prove that r^{2}θ ̇ = h is constant.

(b) Assuming that r =/ 0, show that r ̇ = 2bhr^{−2 }cos θ and hence find r ̈.

(c) Use your answers to (b) to show that the radial component of the acceleration is −8b^{2}h^{2}r^{−5}.

So far ive got:

r=2bsinθ

r ̇=2bcosθθ ̇

so the transverse co-ordinate is (4bcosθθ ̇^2+2bsinθθ ̈) the dots indicating 1st and second derivative, usually above the r's and theta's

Re: polar coordinate components and acceleration

Hope that makes sense, I had a little trouble with the formatting

Re: polar coordinate components and acceleration

Ok iv worked through to part b) but ive made a mistake somewhere n cant spot it, could someone help me out please:

r.=2bhr^{-2}cos

r..=-2bhr^{-2}sinθ-8b^{2}h^{2}r^{-5}cos^{2}θ

then for the radial component

(-2bhr^{-2}sinθ - 8b^{2}h^{2}r^{-5}cos^{2}θ - 2bh^{2}r^{-4}sinθ)

which isnt what I want lol