Multiple Cholce Question Identify the angular momentum about \( G \) of a particle of the rigid body shown in the figure. \[ v_{i}^{\prime \Delta m_{i}} \] O \( r_{i}^{\prime 2} \Delta m_{i} \) \( r_{i}^{\prime} \times v_{i}^{\prime} \Delta m_{i} \) \( \frac{r_{i}^{\prime 2}}{\Delta m_{i}} \)

To find the angular momentum \( \mathbf{L} \) about point \( G \) of a particle in a rigid body, we consider the formula \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \), where \( \mathbf{p} = \Delta m_{i} \mathbf{v}_{i}^{\prime} \) is the linear momentum of the particle. Hence, the angular momentum can be expressed as \( \mathbf{L} = \mathbf{r}_{i}^{\prime} \times (\Delta m_{i} \mathbf{v}_{i}^{\prime}) \), leading us to the conclusion that the correct option is \( r_{i}^{\prime} \times v_{i}^{\prime} \Delta m_{i} \). It's important to note that the angular momentum depends not only on the position vector and velocity but also on the mass of the particle. Incorrect calculations often arise from neglecting either the mass or the correct application of the cross product, which can lead to significant discrepancies in results.