Thread: 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 r²θ ̇ = 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 −8b²h²r⁻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

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

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^-2cos
r..=-2bhr^-2sinθ-8b²h²r^-5cos²θ

then for the radial component
(-2bhr^-2sinθ - 8b²h²r^-5cos²θ - 2bh²r^-4sinθ)

which isnt what I want lol