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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

Problem

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.

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1 Answer 1

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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

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

enter image description here

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    $\begingroup$ 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. $\endgroup$
    – Emp1
    Sep 30, 2022 at 15:13
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    $\begingroup$ I have edited my answer $\endgroup$
    – basics
    Sep 30, 2022 at 15:34
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    $\begingroup$ This makes perfect sense. Thank you! $\endgroup$
    – Emp1
    Sep 30, 2022 at 16:15

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