planet of mass

m moves in a circular orbit of radius R around a Sun (of mass M),

under the influence of the Sun’s gravitational force, given by

$\displaystyle F=\frac{-GMm \hat{\rho}}{R^2}$

where G is Newton’s gravitational constant and $\displaystyle \rho$ is radial basis vector.

(i) Write down Newton's second law.

My answer: $\displaystyle m\left(R \ddot{\theta} \hat{\theta} -R \dot{\theta}^2 \hat{\rho}\right) = \frac{-GMm \hat{\rho}}{R^2}$

ii) Show $\displaystyle \hat{\theta}$ component of second law leads to $\displaystyle \ddot{\theta} = 0$ and the $\displaystyle \hat{\rho}$ leads to $\displaystyle \dot{\theta}^2 = \frac{GM}{R^3}$

There's more I'll add later