\[ \int_0^{2 \pi} (P - R\, \sin \theta) (R\, d \theta) (-\cos \theta \hat{\imath} - \sin\theta \hat{\jmath}) = \alpha R^{2} \hat{\jmath} \]

Determine the value of \(\alpha\), to two decimal places.

**Clarification:** \(\hat{\imath}\) and \(\hat{\jmath}\) are horizontal and vertical unit vectors, respectively.

**Bonus:** How might this equation be interpreted in a (quasi) physical context?

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