Equations of motion

hi i was wondering if anyone can help me with the following part a and b of this problem, i have completed all of the other parts. in particular i'm finding it very hard to visualise part a so i dont really know where to start. and for part b is it because R is constant, and thus the result follows from there?

They want you to express the position of the object in polar coords $(r,\theta)$ then to write down the radial and tangential equations of motion, using Newton II, in polar coords.

To do so, you need to know what the expressions for radial and tangential acceleration $a_r, a_\theta$ look like. You can look these up, or better, derive them by differentiation of a general position vector $\bold{r}=r\hat{\bold{r}}$, noting that here, both $r$ and $\theta$ are in fact functions of time, since for general motion, an object can both move along a radius, and turn about the origin.

Then you have to write down equations like so:

$F_r = ma_r$
$F_\theta = ma_\theta$

Note then that this is a central force problem - so what happens in the 2nd equation?

hi i think i have done this correctly i calcuated $F_r$ correctly i got m(d^2r/dt^2-r(dtheta/dt))

is it that since r is constant? $\frac{d^2r}{dt^2}=0$

No. You have not written down the general equation of motion correctly. It should be:

$F_r = m(\frac{d^2r}{dt^2}-r(\frac{d\theta}{dt})^2)$

You were missing a square. This is a standard result and it's usually written with dot notation:

$F_r = m (\ddot{r}-r\dot{\theta}^2)$

This is the general equation for radial motion. Now you need to write down the general equation for tangential motion. Then you can start to adapt them to your specific problem. You know that $r=R$, a constant, and that the force is gravitational - that gives you the expression for $F_r$ on the LHS, and also tells you what $F_\theta$ is.