# Understanding coordinate system for looping pendulum equations of motion

I'm interested in the phenomenon of the looping pendulum, which is a system consisting of one large and one small mass connected by a string passing over a rod. In particular, I've been looking at this paper: https://arxiv.org/abs/2103.14752

What I've been struggling to understand is how the position vector of the mass $$M_1$$ was expressed as

$$\vec{r}_1=(R \cos \theta-\ell \sin \theta) \hat{x}+(R \sin \theta+\ell \cos \theta) \hat{y}$$

on page 3 of the paper. I would greatly appreciate if someone could go through the process by which the position was expressed like this.

You only need to project the dashed radius and the plain line of length $$\ell$$ on horizontal (for $$x$$-component) and vertical directions (for $$y$$-component),

Let's call $$\alpha = \theta - \dfrac{\pi}{2}$$ the angle between the dashed radius and the vertical axis, that is the same angle as the angle between the wire of length $$\ell$$ and the horizontal axis.

Thus $$x$$- and $$y$$- coordinates of the mass $$M_1$$ reads

$$x_1 = - \ell \cos \alpha - R \sin \alpha = -\ell \sin \theta + R \cos \theta$$
$$y_1 = -\ell \sin \alpha + R \cos \alpha = \ell \cos \theta + R \sin \theta$$

• I don't think I understand. What do you mean by projecting the dashed radius and the line of length l? I understood the bit with alpha and it being the angle the line of length l makes intersecting the horizontal axis. My trigonometry is awful though so I still don't quite understand where the sines and cosines are coming from. Could you possibly (only if convenient) help me with a diagram? Thank you so much.
– Emp1
Sep 30, 2022 at 15:13
• I have edited my answer Sep 30, 2022 at 15:34
• This makes perfect sense. Thank you!
– Emp1
Sep 30, 2022 at 16:15