An industrial robot is being used to position a small part \(P\). Calculate the magnitude of the acceleration \(\mathbf{a}\) of \(P\) for the instant when \(\beta=30^{\circ}\) if \(\dot{\beta}=10 \mathrm{deg} / \mathrm{s}\) and \(\ddot{\beta}=20 \mathrm{deg} / \mathrm{s}^{2}\) at this same instant. The base of the robot is revolving at the constant rate \(\omega=40 \mathrm{deg} / \mathrm{s}\). During the motion arms \(A O\) and \(A P\) remain perpendicular.

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300 mm A 3 200 mm P