An open door of inertia \(m_{\mathrm{d}}\) and width \(\ell_{\mathrm{d}}\) is at rest when it is struck by a thrown ball of clay of inertia \(m_{\mathrm{b}}\) that is moving at speed \(v_{\mathrm{b}}\) at the instant it strikes the door (Figure P11.85). The impact location is a distance \(d=\frac{2}{3} \ell_{\mathrm{d}}\) from the rotation axis through the hinges. The clay ball strikes perpendicular to the door face and sticks after it hits. What is the rotational speed \(\omega\) of the doorball system?

Data from Figure P11.85

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