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