1. **Problem statement:** Find the coordinate of the tip of the suspended rod when the rod is displaced by an angle $\theta$ from the vertical and the disk rotates by an angle $\phi$. 2. **Setup and assumptions:** - Let the disk center be at the origin $(0,0)$. - The rod is attached at the disk center and has length $L$. - The rod initially points vertically downward (along negative $y$-axis). - The rod rotates by angle $\theta$ from vertical. - The disk rotates by angle $\phi$ around its center. 3. **Coordinate system and rotation:** - The disk rotation $\phi$ rotates the entire system about the origin. - The rod angle $\theta$ is relative to the vertical line fixed to the disk. 4. **Finding the tip coordinates:** - The rod tip relative to the disk center before disk rotation is: $$x_r = L \sin(\theta), \quad y_r = -L \cos(\theta)$$ - After disk rotation by $\phi$, apply rotation matrix: $$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \cos(\phi) & -\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{pmatrix} \begin{pmatrix} x_r \\ y_r \end{pmatrix}$$ 5. **Calculate final coordinates:** $$x = L \sin(\theta) \cos(\phi) - L \cos(\theta) \sin(\phi) = L (\sin(\theta) \cos(\phi) - \cos(\theta) \sin(\phi))$$ $$y = L \sin(\theta) \sin(\phi) + L \cos(\theta) \cos(\phi) = L (\sin(\theta) \sin(\phi) + \cos(\theta) \cos(\phi))$$ 6. **Simplify using angle addition formulas:** - Recall $\sin(a - b) = \sin a \cos b - \cos a \sin b$ - Recall $\cos(a - b) = \cos a \cos b + \sin a \sin b$ Therefore, $$x = L \sin(\theta - \phi)$$ $$y = L \cos(\theta - \phi)$$ 7. **Interpretation:** - The tip coordinate of the rod is at $$\boxed{(x,y) = \bigl(L \sin(\theta - \phi), L \cos(\theta - \phi)\bigr)}$$ This gives the position of the rod tip in the fixed coordinate system after both rotations.