Motion equation in the vertical plane along a cylinder

[AntoineCompagnie]

Homework Statement

How do we find the motion equation and more specifically the motion equation of something with a mass $m$ in the vertical plane along a cylindrical path of radius $R$,

(Translation: A point of mass $m$ slides frictionless in the vertical plane along a cylindrical path of radius $R$).

The components in each direction unitaries give the following equations in the Serret Fernet referential.

\begin{cases} mr\ddot\theta = mg \sin\theta (1)\\
mg\dot\theta^2=mg\cos\theta - ||F_r|| (2)
\end{cases}

Why do we have the following motion (kinematics?) equation answer?

$$\ddot\theta -\frac{g}{R}\sin\theta = 0$$

Homework Equations

\begin{cases} mr\ddot\theta = mg \sin\theta (1)\\
mg\dot\theta^2=mg\cos\theta - ||F_r|| (2)
\end{cases}

The Attempt at a Solution

I tought we had to find $$\vec a$$, $$\vec v$$ and $$x(t)$$ but it seems to be wrong according to the answer above...

 

[haruspex]

You have not stated what the given question is.
The answer you quote is a simple reformulation of your equation (1), so is certainly true.

 

[AntoineCompagnie]

My question was at the top: how do we find the motion equation and more specifically the motion equation of something with a mass $m$ in the vertical plane along a cylindrical path of radius R.

 

[haruspex]

My question was at the top: how do we find the motion equation and more specifically the motion equation of something with a mass $m$ in the vertical plane along a cylindrical path of radius R.

Yes, but it is far from clear that is a translation of the question as asked. Are you saying the question is simply "find the equation of motion", and not perhaps "find an equation of motion" or "find the equation of motion in (a particular coordinate system)"?
The given answer is clearly a valid equation of motion in polar coordinates (though it should add r=R for completeness).