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While studying Rotation of Rigid Bodies, I came across the following situation:

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Consider a rigid body in pure rotational motion about a fixed axis (for example the z-axis). For any particle in the object, its linear velocity is given by $$ v=r \omega=R \sin \theta \omega $$ where $\mathbf{R}$ is the position vector of the particle from the origin (see Fig. 7.9) and $\theta$ is the angle between the position vector and the $z$-axis. As shown in Fig. 7.9, the direction of $y$ is perpendicular to the plane formed by $\boldsymbol{\omega}$ and $\mathbf{R}$ where it can be verified using the right-hand rule.

Problem:

Why is the direction of $y$ perpendicular to the plane formed by $\boldsymbol{\omega}$ and $\mathbf{R}$? It is stated that this relation can be obtained by using the right hand rule. However, I am having trouble using the right hand rule in this scenario. Furthermore, I just cannot properly visualize the plane formed by $\boldsymbol{\omega}$ and $\mathbf{R}$. I would appreciate if someone could further elaborate an explanation regarding the above statements.

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    $\begingroup$ I don’t think that the direction of 𝑦 is perpendicular to the plane formed by $\omega$ and R $\endgroup$
    – Eli
    Commented Jul 5, 2021 at 12:28

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Look at this figure. The plane formed by $\boldsymbol{\omega}$ and $\mathbf{R}$ is perpendicular to y’ axis, thus $\vec e_{y’}\cdot \vec e_R=0$.

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  • $\begingroup$ There is an easy way to identify the plane containing $\omega$ and $R$. In the OP's sketch, that plane contains a double headed arrow labelled $\Theta$ and the line segment labelled $r = Rsin\Theta$. $\endgroup$
    – mmesser314
    Commented Jul 5, 2021 at 16:47
  • $\begingroup$ I still cannot see it... Is $y^{\prime}$ equal to the $X$ axis? $\endgroup$
    – Lucas
    Commented Jul 5, 2021 at 18:49
  • $\begingroup$ @Lucas no y‘ is perpendicular to x‘ and to z , you can see it an the right figure $\endgroup$
    – Eli
    Commented Jul 5, 2021 at 19:39
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The vector representing (ω) is defined as being along the axis of rotation (z). The angle (θ) is measured in the plane defined by (ω) and (R). Looking at the sketch, I would say that the (x) axis is perpendicular to that plane. In any case, such a statement requires that both the (x) and (y) axes are rotating with the rigid body.

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