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